•
THE DECENNIAL PUBLICATIONS OF
THE UNIVERSITY OF CHICAGO
THE DECENNIAL PUBLICATIONS
ISSUED IN COMMEMORATION OP THE COMPLETION OP THE PIRST TEN
YEARS OP THE UNIVERSITY'S EXISTENCE
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EDITED BY A COMMITTEE APPOINTED BY THE SENATE
EDWAKD CAPPS
STAER WILLAED CUTTING EOLUN D. SAiilSBCRY
JAMES ROWLAND ANGELL WILLIAM I. THOMAS SHAILER MATHEWS
CARL DARLING BUCK FREDERIC IVES CARPENTER OSKAR BOLZA
JULIUS STIEGLITZ JACyUES LOEB
THESE VOLUMES ARE DEDICATED
TO THE MEN AND WOMEN
OP OUR TIME AND COUNTRY WHO BY WISE AND GENEROUS GIVING
HAVE ENCOURAGED THE SEARCH AFTER TRUTH
IN ALL DEPARTMENTS OF KNOWLEDGE
LECTURES ON THE CALCULUS OF
VARIATIONS
•A
OA/c^' ^'^^"--^
LECTURES ON THE CALCULUS
OF A'ARIATIONS
OSKAK BOLZA
OF THE DEPAETMEXT OF MATHEMATICS
THE 'DEC -EX MA L PUBLICATIONS «
SECOND SERIES VOLUME XIV
CHICAGO
THE UNIVERSITY OF CHICAGO PRESS
1904
v\
' Copy rig lit. V.io-J
BY THE UNIVERSITY OF CHICAGO
September, 1901
PREFACE
The principal steps in the progress of the Calculus of
Variations during the last thirty years may be characterized
as follows:
1. A critical revision of the foundations and demonstra-
tions of the older theory of the first and second variation
according to the modern requirements of rigor, by Weier-
STEASS, Erdmann, Du Bois-Eeymond, Scheeffer, Schwarz,
and others. The result of this revision was: a sharper for-
mulation of the problems, rigorous proofs for the first three
necessary conditions, and a rigorous proof of the sufficiency
of these conditions for what is now called a "weak" extre-
nium.
2. Weieestrass's extension of the theory of the first and
second variation to the case where the curves under consid-
eration are given in parameter-representation. This was an
advance of great importance for all geometrical applications
of the Calculus of Variations; for the older method implied
— for geometrical problems — a rather artificial restriction.
3. Weiersteass's discovery of the fourth necessary con-
dition and his sufficiency proof for a so-called "strong"
extremum, which gave for the first time a complete solution,
at least for the simplest type of problems, by means of an
entirely new method based upon what is now known as
" Weierstrass's construction."
These discoveries mark a turning-point in the history of
the Calculus of Variations. Unfortunately they were given
by Weierstrass only in his lectures, and thus became
known only very slowly to the general mathematical public.
Chiefly under the influence of Weierstrass's theory a
vigorous activity in the Calculus of Variations has set in
ix
Pkeface
during the last few years, which has led — apart from exten-
sions and simplifications of Weierstrass's theory — to the
following two essentially new developments:
4. Kneser's theory, which is based upon an extension of
certain theorems on geodesies to extremals in general. This
new method furnishes likewise a complete system of suffi-
cient conditions and goes beyond Weierstrass's theory,
inasmuch as it covers also the case of variable end -points.
5. Hilbert's (I priori existence proof for an extremum
of a definite integral — a discovery of far-reaching impor-
tance, not only for the Calculus of ^^ariations, but also for the
theory of differential equations and the theory of functions.
To give a detailed account of this development was the
object of a series of lectures which I delivered at the Collo-
quium held in connection with the summer meeting of the
American Mathematical Society at Ithaca, N. Y., in August,
1901. And the present volume is, in substance, a reproduc-
tion of these lectures, with such additions and modifications
as seemed to me desirable in order that the book could serve
as a treatise on that part of the Calculus of Variations to
which the discussion is here confined, viz., the case in which
the function under the integral sign depends upon a plane
curve and involves no his/her derivatives than the first.
With this view I have throughout supplied the detail argu-
mentation and introduced examples in illustration of the gen-
eral principles. The emphasis lies entirely on the theoretical
side: I have endeavored to give clear definitions of the fun-
damental concepts, sharp formulations of the problems, and
rigorous demonstrations. Difficult points, such as the proof
of the existence of a "field," the details in Hilbert's exist-
ence proof, etc., have received special attention.
For a rioforous treatment of the Calculus of Variations
the principal theorems of the modern theory of functions of
a real variable are indispensable; these I had therefore to
Preface xi
presuppose, the more so as I deviate from Weiersteass and
Kneser in not assuming the function under the integral sign to
be analytic. In order, however, to make the book accessible
to a larger circle of readers, I have systematically given ref-
erences to the following standard works: Encyclopdedie dev
mathematischen Wissenschaftcn (abbreviated £".), especially
the articles on ''Allgemeine Functionslehre" (Prixgsheim)
and ''Differential- und Integralrechnung'' (Voss); Jordan,
Coins <r Analyse, second edition (abbreviated J.) ; Genocchi-
Peano, Differcnticdt'echnung mid Grundziigc dcr Iidcgird-
recJmung, translated by Bohlmann and Schepp (abbreviated
P.); occasionally also to Dini, Theoric der Fnnctioncu eiiicr
verdnderlicJtca reelleii Grossc, translated by Luroth and
Schepp; Stolz, Grundzugc der Differential- luid Integral-
rechnung. The references are given for each theorem where
it occurs for the first time ; they may also be found by means
of the index at the end of the book.
Certain developments have been given in smaller print in
order to indicate, not that they are of minor importance, but
that they may be passed over at a first reading and taken up
only when referred to later on.
A few remarks are necessary concerning my attitude
toward Weierstrass's lectures. Weierstrass's results and
methods may at present be considered as generally known,
partly through dissertations and other publications of his
pupils, partly through Kneser\s Lelirbi(ch der Variations-
rechniiiig (Braunschweig, 1900), partly through sets of notes
("Ausarbeitungen") of which a great number are in circula-
tion and copies of which are accessible to everyone in the
library of the Mathematische Verein at Berlin, and in the
Mathematische Lesezimmer at Gottingen.
Under these circumstances I have not hesitated to make
use of Weierstrass's lectures just as if they had been pub-
lished in print.
xii Preface
My principal source of information concerning "Weiek-
STRASs's theory has been the course of lectures on the Cal-
culus of Variations of the Summer Semester, 1879, which
I had the good fortune to attend as a student in the Uni-
versity of Berlin. Besides, I have had at my disposal sets
of notes of the courses of 1877 (by Mr. G. Schulz) and of
1882 (a copy of the set of notes in the '-Lesezimmer" at
Gottingen for which I am indebted to Professor Tanner), a
copy of a few pages of the course of 1872 (from notes taken
by Mr. Ott), and finally a set of notes (for which I am
indebted to Dr. J. C. Fields) of a course of lectures on the
Calculus of Variations by Professor H. A. Schwarz
(1898-99).
I regret very much that I have not been able to make
use of the articles on the Calculus of Variations in the
EncDclopaedie dcr mcdhe.matisclien Wissenschaftcn by
Kneser, Zermelo, and Hahn. When these articles ap-
peared, the printing of this volume was practically com-
pleted. For the same reason no reference could be made to
Hancock's Lectures on the CalcuJiis of Variations.
In concluding, I wish to express my thanks to Professor
G. A. Bliss for valuable suggestions and criticisms, and to
Dr. H. E. Jordan for his assistance in the revision of the
proof-sheets.
OSKAR BoLZA.
The University op Chicago.
August 28, 1904.
TABLE OF CONTENTS
CHAPTER I
XX. PAGE
i^(a', y, y')dx
§ 1. Introduction -------- 1
§ 2. Agreements concerning Notation and Terminology - 5
if 3. General Formulation of the Problem - - - - 9
§ 4. Vanishing of the First Variation ----- 13
§ 5. The Fundamental Lemma and Euler's Differential
Equation -------- 20
§ 6. Du Bois-Reymond's and Hubert's Proofs of Euler's Dif-
ferential Equation ------- 22
§ 7. Miscellaneous Remarks concerning the Integration of
Euler's Differential Equation - - - - 26
§ 8. Weierstrass's Lemma and the E-function - - - ;i3
§ 9. Discontinuous Solutions - - . - - 36
§10. Boundary Conditions - ------ 41
CHAPTER II
The Second Vaeiation of the Integral I F(x,y,y')dx
§11. Legendre's Condition ----.. 4.4
§ 12. Jacobi's Transformation of the Second Variation - 51
§ 13. Jacobi's Theorem - ----- 54
§14. Jacobi's Criterion - - - . - - - 57
§15. Geometrical Interpretation of the Conjugate Points - 60
§16. Necessity of Jacobi's Condition - - - - - 65
CHAPTER III
r
Sufficient Conditions for an Extremum of the Integral
F(x, y, y')dx
§17. Sufficient Conditions for a Weak Minimum - - 68
§18. Insufficiency of the Preceding Three Conditions for a
Strong Minimum, and Fourth Necessary Condition 73
xiii
xiv Table of Contents
§19. Existence of a Field of Extremals - - . - 78
§20. Weierstrass's Theorem ------ 84
§21. Hubert's Proof of Weierstrass's Theorem - - - 91
§22. Sufficient Conditions for a Stronfj Minimum - - 94
§23, The Case of Variable End-Points - - - - 102
CHAPTER TV
Weierstrass's Theory of the Problem in Parameter-Rep-
resentation
§ 24. Formulation of the Problem . - . . - 115
§25. The First Variation ------- 122
§26. Examples --------- 126
§27. The Second Variation ----- 1.30
§28. The Fourth Necessary Condition and Sufficient Condi-
tions ------- 138
§29. Boundary Conditions - ------ 148
§30. The Case of Variable End-Points - - - - 153
§.31. Weierstrass's Extension of the Meaning of the Definite
Integral I Fi.r,n,.r',y')dt- - - - - 156
CHAPTER V
Kneser's Theory
§.32. Gauss's Theorems on Geodesies ----- 164
§3.3. Kneser's Theorem on Transversals and the Theorem on
the Envelope of a Set of Extremals - - - 166
§34. Construction of a Field ------ 175
§35, Kneser's Curvilinear Co-ordinates - - - 181
§36. Sufficient Conditions for a Minimum in the Case of
One Movable End-Point ----- 187
§ 37. Various Proofs of Weierstrass's Theorem—The Assump-
tion F(i, a) +0 ------- 193
§38. The Focal Point ------- 199
CHAPTER VI
Weierstrass's Theory of the Isoperimetric Problems
§39. Euler'sRule - ------- 206
§40. The Second Necessary Condition - . - - 213
Table of Contents xv
§•41. The Third Necessary Condition and the Conjugate
Point - - 218
§i± Sufficient Conditions - - 232
CHAPTER VII
Hilbert's Existence Theorem
§ 43. Introductory Remarks .--... 245
§44. Theorems concerning the Generalized Integral J* - 247
§45. Hilbert's Construction ..--._ 253
§46. Properties of Hilbert's Curve - - - - - 259
ADDENDA 265
INDEX - - - 269
CHAPTER I
THE FIRST VARIATION
§1, INTRODUCTION
The Calculus of Variations deals with problems of maxima
and minima. But while in the ordinary theory of maxima
and minima the problem is to determine those values of the
independent variables for which a given function of these
variables takes a maximum or minimum value, in the Cal-
culus of Variations definite integrals^ involving one or more
unknown functions are considered, and it is required so to
determine these unknown functions that the definite inte-
grals shall take maximum or minimum values.
The following example will serve to illustrate the char-
acter of the problems with which we are here concerned, and
its discussion will at the same time bring out certain points
which are important for an exact formulation of the general
problem :
Example I : In a plane tJiere are (jicen tivo points A, B
and a straight line S. It is required to determine, among all
curves which can he drawn in this plane beticeen A and B,
the one luhich, if revolved around the line 2, generates the
surface of minimum area.
We choose the line S for the ic-axis of a rectangular
system of co-ordinates, and denote the co-ordinates of the
points A and B hj Xq, iJq and x^, y^ respectively. Then for
a curve
y=f{x)
1 The problem of the Calculus of Variations has, however, been extended beyond
the domain of definite integrals (viz., to functions defined by differential equations)
by A. Mayer, Leipziger Bei-ichte, 1878 and 1895. Compare Knesek, Lehrbuch, chap. vii.
1
2 Calculus of Variations |Chap. I
joining the two points A and B, the area in question is given
by the definite integral ^
J=27r f \jVl-^ij'-dx ,
where ij' stands for the derivative f'{jr). For different
curves the integral will take, in general, different values ;
and our problem is then analytically : among all functions
/ [jr) which take for x= Xq and x = Xi the prescribed values
ijq and iji respectively, to determine the one which furnishes
the smallest value for the integral J.
This formulation of the problem implies, however, a
number of tacit assumptions, which it is important to state
explicitly :
a) In the first place, we must add some restrictions con-
cerning tlie nature of the f mictions f {x) which we admit to
consideration. For, since the definite integral contains the
derivative y ' , it is tacitly supposed that / (.r) has a deriva-
tive ; the function / (j") and its derivative must, moreover,
be such that the definite integral has a determinate finite
value. Indeed, the problem becomes definite only if we
confine ourselves to curves of ci certain class, characterized
by a well-defined system of conditions concerning continuity,
existence of derivative, etc.
For instance, we might admit to consideration only func-
tions /' (x) with a continuous first derivative ; or functions
with continuous first and second derivatives ; or analytic
functions, etc.
b) Secondly, by assuming the curves representable in tlie
form y ^=f{x), where /(a;) is a single-valued function of x,
we have tacitly introduced an important restriction, viz., that
we consider only those curves which are met by every ordi-
nate between Xq and Xi at but one point.
la being a real positive quantity, y a will always be understood to represent
the positive value of the square root.
§1] First Variation 3
We can free ourselves from this restriction by assuming
the curve in parameter-representation : '
x = <i>{t) , U = ^{t) .
The integral which we have to minimize becomes then
J=27r C ' yVx'-~\- y'-'dt ,
where .r'= <^'(/), })' ^^ "^ \t), and where /q and i^ are the
values of / which correspond to the two end-points.
c) It is further to be observed that our definite integral
represents the area in question only when ij ^ 0 throughout
the interval of integration. The problem implies, there-
fore, the condition that ilic ciirrcs shall lie in a ccvtaui
region' of the -x, //-plane (viz., the upper half-plane).
d) Our formulation of the problem tacitly assumes that
there exists a curve which furnishes a minimum for the area.
But the existence of such a curve is by no means self-
evident. We can only be sure that there exists a lower
limit ^ for the values of the area; and the decision whether
this lower limit is actually reached or not forms part of the
solution of the problem.
The problem may be modified in various ways. For
instance, instead of assuming both end-points fixed, we may
assume one or both of them movable on given curves.
An essentially different class of problems is represented
by the following example :
1 Compare chap. iv. Even in this generalized form the analytic problem is not
quite so general as the original geometrical problem. For the area in question may
exist and be finite, and yet not be representable by the above definite integral. This
suggests an extension of the problem of the Calculus of Variations, first considered
by Weieesteass. Compare §§ 31 and 44.
- A restriction of the same nature, but from other reasons, occurs in the problems
of the brachistochrone and of the geodesic; compare §26.
3 Compare E. I A, p. 72, and II A, p. 9; J. I, No. 25; and P., No. 20.
4 ■ Calculus of Variations [Chap. I
Example II : Aviong all closed plane curves of given
perimeter to determine the one lohicJi contains the maximvin
area.
If we use parameter-representation, the problem is to
determine among all curves /or which the definite integral
i
V^x'- + y'-dt
lias a given value, the one which maximizes the integral
'J=\ I (-i"//'- x' y)dt .
JtQ
Here the curves out of which the maximizing curve is to be
selected are subject — apart from restrictions of the kind
which we have mentioned before — to the new condition of
furnishing a given value for a certain definite integral.
Problems of this kind are called "isoperimetric problems;"
they will be treated in chaj). vi.
The preceding examples are representatives of the simplest
— and, at the same time, most important — type of problems
of the Calculus of Variations, in which are considered defi-
nite integrals depending upon a plane curve and containing
no higher derivatives than the first. To this type we shall
almost exclusively confine ourselves.
The problem may be generalized in various directions :
1. Higher derivatives may occur under the integral.
2. The integral may depend upon a system of unknown
functions, either independent or connected by finite or
differential relations.
3. Extension to multiple integrals.
For these generalizations we refer the reader to C. Jordan,
Cours d' Analyse, 2eed., Vol. Ill, chap, iv ; Pascal-(Schepp),
Die Variationsrechnung (Leipzig, 1899) ; and Kneser, Lelir-
buch der Variationsrechnung (Braunschweig, 1900), Ab-
schnitt VI, VII, VIII.
§2] First Variation 5
§2. agreements* concerning notation and terminology
a) We consider exclasively real variables. The ''inter-
val (a 6)" of a variable x — where the notation always
implies rt<6 — is the totality of values x satisfying the
inequality a^x^b. The ^^vicinitu (5) of a point Xi=ai.
X2,^=(i-2, ' ' ' , Xn^o,n^ is the totality of points a"i, .ro, • • • . ./■„
satisfying the inequalities:
I -^'i — "i 1 < S J I a:'2 — a2 [ < S , • • • , | a^,,— a„ | < 8 .
The word "(/o»^rt^^" will be used in the same sense as
the German Bereich, L e., synonymous with "set of points"
(compare E. II A, p. 44). The word ''region " will be used :
(a) for a "continuum," i. e., a set of points which is "con-
nected" and made up exclusively of "inner" points; in this
case the boundary does not belong to the region ("open"
region) ; (6) for a continuum together with its boundary
("closed" region) ; (c) for a continuum together with part
of its boundary. The region may be finite or infinite ; it
may also comprise the whole /i-dimensional space.
When we say : a curve lies "/u" a region, we mean : each
one of its points is a point of the region, not necessarily an
inner point.
For the definition of "inner" point, "boundary point"
{front iere), and "connected" {cVun seal tenant) we refer to
E. II A, p. 44 ; J. I, Nos. 22, 31 ; and Hurwitz, Verhand-
luncjen des ersten internationalen Mathematilxercongresses
in Zurich, p. 94.
h) By a "function''' is always meant a real single-valued
function.
The substitution of a particular value x=Xq in a function
^{x) will be denoted by
(t>(x)\ = <f> (xo) ;
iThe reader is advised to proceed directly to §3 and to use §2 only for reference.
6 Calculus of Variations [Chap. I
similarly
also
L Jj-o
Instead we shall also use the simpler notation
<^(.r)f, <f>{x,y)\^', {<f>U)l
where it can be done without ambiguity, compare e).
We shall say: a function has a certain property in' a
domain ^ of the independent variables, if it has the property
in (juestion at all points of the domain #. no matter whether
they are interior or boundary points.
A function of a'l, x-,,- ■ ■, x^ has a certain property in flic
vicinity of a point Xx=ax, 0C2 = 02, • ■ • , ic„ = a„, if there exists
a positive quantity S such that the function has the property
in question in the vicinity (S) of the point a^, Oo,---, «„.
If L<^{Jt)^0, we shall say: <f>{Ii) is an ^^ iufinitesimar''
(for Lh = 0); such an infinitesimal will in a general way
be denoted by (//). Also an independent variable // which
in the course of the investigation is made to approach zero,
will be called an '•infinitesimal."
c) Derivatives of functions of one variable will be denoted
by accents, in the usual manner :
df{x) dy{x)
f («^) = 1 ? / (^) = , o , etc.
• ^ ^ dx ^ ^ dx^
For brevity we shall use the following terminology" for
various classes of functions which will frequently occur in
the sequel. We shall say that a function f{x) which is
defined in an interval {xqXi) is
1 Or, with more emphasis, " throughout."
2 The letters C, D are to suggest "continuous," "discontinuous; " the accents
the order of the derivative involved.
§2] First Variation
of class C7 if/(a.^) is continuous 1
of class C' it fix) and/'(x) are continuous i . / . ^\
of class C<"' if f{x) , f {x),- • ■ and /""(.<■) are continuous J
with the understanding concerning the extremities of the
interval that the definition of f{x) can be so extended
beyond (-^o-*'!) ^^'^^^ ^^^ above properties still hold at
Xq and x^.
If f{x) itself is continuous, and if the interval (j'V'i) ^'^n
be divided into a finite number of subintervals
\X^)Cl) , (C1C2) J • • • , \Cn-\-^i) >
such that in each subinterval f{x) is of class C (C"), whereas
f'{x) {f"{x)) is discontinuous at c^, Co,- • ■, c„_i. we shall say
that/(.r) is of class D'(D"). We consider class C'(C") as
contained in D'(D"), viz., for ii^l.
From these definitions it follows that, for a function of
-t-
class -D'. the progressive^ and regressive derivatives /'(c^,),
/'(Cy) exist, are finite and equal to the limiting values'
/' (c. + 0), /' (c, - 0) respectively.
d) Partial derivatives of functions of several variables
will be denoted by literal subscripts (Kneser):
F,j{x, I/, p) = q'^' ' ,
FyA^^y^p) = ^\ ^ j' ^tc. ;
also
J. ( , ^F{x,y,p)
y=vi)
dy
Also of a function of several variables we shall say that
it is of class C^'"-^ in a domain ^ if all its partial derivatives
IE. II A, p. 61; DiNi, Grutidlaf/en, etc., §68: and Stolz, GrundzUge, etc.. Vol.
I, p. 31.
•■^E. II A, p. 13.
8
Calculus of Variations
[Chap. I
up to the-91*^ order inclusive exist and are continuous in'
the domain ^.
e) The letters x, y will always be used for rectangular
co-ordinates with the usual orientation of the positive axes,
i. e., the positive y-a.xis to the left of the positive a'-axis. It
will frequently be convenient to designate points by num-
bers : 0, 1, 2, • • • ; the co-ordinates of these points will then
always be denoted by .Tq? 2/o 5 ^i? 2/i 5 ^2? 2/2 j ' ' " respectively;
their parameters, if they lie on a curve given in parameter-
representation, by /q? ^ij hi' ' '•
A curve^ (arc of curve)
will be said to be of class C, C, etc., if the function / (.i;)
is of class C, C", etc., in [xoor^). In particular, a curve of
class D' is continuous and made up of a finite number of
arcs with continuously turning tangents, not parallel to the
y-axis. The points of the curve whose abscissoe are the points
of discontinuity C\, C2, ■ • • ,
C^-i of/'(;r), • • • will be called
its corners. At a corner the
/ i\ curve has a progressive and a
regressive tangent, and.
_^
+
tana
X
FIG. 1
:/'(c) , tana=/'(c) .
(See Fig. 1.)
/) The integral
J=\ F{^, y, y')dx
taken along the curve
6 : y=f{x) , Xo
iWhen ^ contains boundary points, an agreement similar to tliat in the case
of one variable is necessary with respect to these points.
2 The corresponding definitions for curves in parameter-representation will be
given in §24.
§3] First Variation 9
from the point A{jrQ, jJq) to the point B {.r^, j/i), i. e., the
integral
f ' f(x, f{x), f\x))dx
will be denoted by J^ {^B) (more briefly J^^ or J {AB)^ ; or
by J^y, if the end-points are designated by numbers: ^l, v.
(j) The disfancc between the two points P and Q will be
denoted by | PQ |, the circle with center O and radius r by
(O, r) (Harkness and Morley). The angle which a vector
makes with the positive a^-axis will be called its amplitude.
§3. GENERAL FORMULATION OF THE PROBLEM*
a) After these preliminary explanations, the simplest
problem of the Calculus of Variations may be formulated in
the most general way, as follows :
There is given :
1. A well-defined infinitude M of curves, representable
in the form
y =f(x) , Xo^x'^Xi ;
the end-points and their abscisses Xq , x^ may vary from curve to
curve. We shall refer to these curves as "admissible curves."
2. A function F{x, y, 2^) of three independent variables
such that for every admissible curve 6, the definite integral
F{x,y,y')dx (1)
-0
has a determinate finite value.
1 Until rather recently a certain vagueness has prevailed with respect to the
fundamental concepts of the Calculus of Variations. The most important contribu-
tions toward clear definitions and sharp formulations of the problems are due to
Du Bois-Reymond, '"Erlauterungen zu den Aufangsgriinden der Variationsrech-
nung," Mathemutische Annalen, Vol. XV (1879), p. 283; Scheeffee, " Ueber die
Bedeutung der Begriffe 'Maximum und Minimum' in der Variationsrechnung,"
ibid.. Vol. XXVI (1886), p. 197; Weieksteass, Lectures on the Calculus of Variation,
especially those since 1879. Compare also Zermelo, Untersuchungen zur Varia-
tionsrechnung, Dissertation (Berlin, 1894 1, p. 24; Kn'eser, Lehrbuch.%Vi, and Osgood,
"Sufficient Conditions in the Calculus of Variations," Annals of Mathematics (2),
Vol.11 (1901), p. 105.
10 Calculus of Variations [Chap, i
The set' of values J",, thus defined has always a lower
limit, K, and an upper limit, G (finite or infinite"). If the
lower (upper) limit is finite, and if there exists an admissible
curve 6 such that
J^ = K , (t/,( = 6r) ,
the curve 6 is said to furnish fhe absolute minimum (ma.vi-
mitm) for the integral J (with respect to M). For everv
other admissible curve ^ we have then
Jz^^J, , (J^^J,) . (2)
The word ''extremum" ^ will be used for maximum and mini-
mum alike, when it is not necessary to distinguish between
them.
Hence the pvohlem arises : to determine all admissi])le
curves which, in this sense, minimize or maximize the inte-
gral ./.
6) As in the theory of ordinary maxima and minima, the
problem of the absolute extremum, which is the ultimate
aim of the Calculus of Variations, is reducible to another
problem which can be more easily attacked, viz. , the problem
of the relative extremum:
An admissible curve 6 is said to furnish a rcJaflrc mini-
mum* [maximum) if there exists a '' neighborliood II of fJie
curve 6," however small, such that the curve 6 furnishes an
absolute minimum with respect to the totality Mi of those
curves of M which lie in this neighborhood ; and by a
neighborhood II of the curve 6 we understand any region'
which contains 6 in its interior.
1 By "set" we translate the German Punktmenge, the French ensemble, J. I,
No. 20.
2 The upper limit is +oo, if for every preassigned positive quantity .-1 there
exist curves g for which J(^ > A; see E. II A, p. 9.
3 Du Bois-Reymond, Mathematische Amialen, Vol. XV, p. 564.
*In the use of t^e words "absolute" and " relative" I follow Voss in E. II A,
p. 80. Many authors call the isoperimetric problems "problems of relative maxima
and minima."
■'For the definition of the term " region," see p. 5.
§31 First Variation 11
According to Stolz, the relative minimum (maximum)
will be called proper, if there exists a neighborhood M such
that in (2) the sign > (<) holds for all curves 6 different
from (5: improper if, however the neighborhood II may be
chosen, there exists some curve (S different from 6 for which
the equality sign has to be taken.
A curve which furnishes an absolute extremum evidently
furnishes a fortiori also a relative extremum. Hence the
oriiJ-inal problem is reducible' to the problem: fo defermine
all flios^e cni-res witicli fnriiislt o, relative minim mii ; and in
this form we shall consider the problem in the sequel.
We shall henceforth always use the words "minimum,"'
"maximum"' in the sense of relative minimum, maximum;
antl we shall confine ourselves to the case of a minimum,
since every curve which minimizes J, at the same time maxi-
mizes — ./, and vice I'ersa.
c) In the abstract formulation given above, the problem
would hardly be accessible to the methods of analysis; to
make it so, it is necessary to specify some concrete assump-
tions concerning the admissible curves and the function F.
For the present, we shall make the following assumptions:
A. The infinitude M of admissible curves shall be the
totality of all curves satisfying the following conditions:
1. They pass through two given points A (xq, ijq) and
B{xi,?h)'
2. Thev are representable in the form
y =/(.r) , x^^x^Xi ,
f{x) being a single-valued function of x.
3. They are coiitiirHoio^ and consist of a finite number of
1 After the relative problem has been solved, it merely remains to pick out among
its solutions those which furnish the smallest or largest value for J. Only if the
relative problem should have an infinitude of solutions, new difficulties would arise.
For a direct treatment of the problem of the absolute extremum compare Hilbert's
existence proof (chap, vii) ; Daeboux, TMorie des surfaces, Vol. Ill, p. 89; and Zer-
MELO, Jahresbericht der Deutschen Mathematiker-Verehugung, Vol. XI (.1902), p. 184.
12 Calculus of Vaeiations [Chap. 1
arcs with continuously turning tangents, not parallel to the
y-axis; i. e., in the terminology of §2, c),f[x) is of class D' .
4. They lie in a given region* ?S of the x, ?/-plane.
B. The function F{x,y,p) shall be continuous" and
admit continuous partial derivatives of the first, second, and
third orders in a domain^ QI which consists of all points*
(x, y, p) for which (x, y) is a point of U, and^; has a finite value.
Under these assumptions the definite integral t/g taken
along any admissible curve 6 is always finite and determi-
nate,' provided we define, in the case of a curve with corners,
the integral as the sum of integrals taken between two suc-
cessive corners. Since we suppose the end-points A and B
fixed and the curves representable in the form y=f(x), the
curves 6 all lie between the two lines x = Xq and x=Xi,
with the exception of the end-points, which lie on these
lines.
Hence it follows that we may, in the present case, give
the following simpler definition of a minimum : An admis-
sible curve 6 : // ^/ [x) minimizes the integral J, if '^ there
1 Compare §2, a).
21 follow here the example of Pascal, loc. clt., p. 21, and Osgood, loc. cit., p. 105.
W^EIERSTRASS, JORDAN, and Kneser suppose the function F {x, y,p) to be analytic.
3 If we interpret p as a third co-ordinate perpendicular to the x, 2/-plano, Qt is the
cylinder, infinite in both directions, whose base is the region R.
1" Point" in the sense of the theory of "point-sets." Compare E. II A, p. 44,
and J. I, No. 20.
5 If the curve has no corners, this follows at once from elementary theorems on
continuous functions (J. I, Nos. 60, 66). If the curve has corners, the integral Jg has
no immediate meaning. But the two integrals
F(x,f(x),f{.c))dx and I F(x,f{x),f'ix))dx
are finite and determinate and equal to each other, and at the same time equal to the
sum of integrals mentioned in the text. Compare Dini, loc. cit., §62; §187, 2; §190,9;
and §190, 2.
6 In admitting the equality sign in the inequality (2), I deviate from the conven-
tions generally adopted in the Calculus of Variations and follow Stolz {Grundzuge
der Differenzialrechnung, Vol. I, p. 199), whose definition is more consistent with
the usual definition of absolute minimum. If the equality sign were omitted, it
could not be said that every curve which furnishes an absolute minimum furnishes
a fortiori also a relative minimum.
§41
FiEST Variation
18
exists a positive quantity p such that J^.^J^, for every
admissible curve 6: //=/(,/•) which satisfies the inequality
\y — y\<P ior .r^ ^x^Xi . (3)
This means geometrically that the curve (S lies in the interior'
of the strip of the ./", ^-plaue between the two curves
y=fi'^) + p , y=f{jc)- 9
on the one liand. and the two
lines ./■ = £("o, x = j\ on the
other hand. This strip we
shall call "the neighborhood'
(p) of the curve ^," the points
A and B being included, the
rest of the boundary excluded.
FIG. 2
§A
VANISHING OF THE FIRST VARIATION
We now suppose we have found a curve (5 : y^fipc) which
minimizes the integral
J = ( F{x, y, y')dx
in the sense explained in the last section. We further sup-
pose, for the present,^ that f {x) is continuous in (a"o-<"i) and
that 6 lies entirely in the interior of the region S.
From the last assumption it follows that we can construct*
a neighborhood (p) of 6 which lies entirely in the interior
of IJ.
1 Except, of course, the points A and B.
2 Compare Osgood, loc. cit., p. 107.
3 These restrictions will be dropped in g§9 and 10.
■* About any point P of t^ we can construct a circle (P, ?•) which lies entirely in
S, since P is an inner point of S. Let pp be the upper limit of the values of r for
which this takes place. Then pp varies continuously as P describes the curve G
(Weiekstrass, Werke, Vol. II, p. 204) and reaches therefore a positive minimum
value pj, (compare E. II A, p. 19 and J. I, No. 6-4, Cor.), If we choose p < p^) the neigh-
borhood (p) of (i will lie in the interior of SI.
14 Calculus of Variations [Chap. I
We then replace' the curve 6 by another admissible curve
6: Z/=7(.r), I
lying entirely in the neighborhood (p). ■
The increment
\, ^iJ=~U — !J=f ( •^■) - / (•'«•) ,
which we shall denote by co, is called the total van'atiuu of ij.
Since S and 6 pass through A and B, we have
(o(^j = 0 , a,(.r,) = 0 , (4)
and since 6 lies in (/j),
\<»{x)\ < p in (a-oa:"i) . (4a)
The corresponding increment of the integral,
is called the total variation of the integral J ; it may be written :
AJ= C '\F{x,y + .^,u'+u>')-F{:x,y,y')\dx .
Since 6 is supposed to minimize ./, we shall have
AJ^O ,
provided that p has been chosen sufficiently small.
For the next step in the discussion of this inecpiality two
different methods have been proposed:
a) Application of Taylor's formula: If we a})ply Tay-
lor's^ formula to the integrand of A ,/, we obtain, in the nota-
tion of §2, (I),
1 The process of replacing 15 by (5 is called " a variation of the curve G ; " the same
term is frequently applied to the curve S itself, which is sometimes also called "the
varied curve," or " a neighboring curve."'
2The conditions for the applicability of Taylor's formula are fulfilled, com-
pare E. II A, p. 77, and J. I, No. 253. F^., Fyy-, etc., are synonymous with F^^, F^^.titc,
The method here used was first given by Lageange. See Oeuvrcs, Vol. IX, p. 297.
Compare also Du Bois-Reymond, Mathematische Annalen, Vol. XV (1879), p. 292, aud
Pascal-Schepp, Die Variationxrechnung, p. 22.
Instead of Taylor's formula with the remainder-term, Weieesteass (Lee.
tures), Knesee (Lehrbuch der Variationsrechnung, §8), and C. Joedan (Cours
crAnalyse,YoLIJI, No. .350), who suppose Fix, y , p) to be analytic, use Taylor's
expansion into an infinite series. Here, however, the question of integration by
terms should be considered. .
§4]
First Variation
15
+ \f' {^\u ^' + 2 F„, coo, ' + F,.,. <.'')dx ,
where the arguments of Fy and Fy^ are x, y, jj', those of
Fyy, Fyy, Fyy : X, y + eco, ij' + eco', e being a quantity
between 0 and 1.
We now consider, with Lagrange,' speciaP variations of
the form
W = €7} , (5)
where i] is a function of x of class D' which vanishes for
£c = .ro and x^^x-^, and e a constant whose absolute value is
taken so small that (4a) is satisfied.
Then A,/ takes the form'
J^e] r\F,r}^F,.r}')dx + {c)
(6)
where (e) denotes an intinitesimal for Z.€ = 0.
Hence we infer that we must have
f \F,^ri + F„r,')dx = 0
(7)
for all functions i] of class D' which vanish at ^o and x^;
1 Oeuvres, Vol. IX, p. 298.
2For the purpose of deriving necessary conditions, we may specialize the
variations as much as convenient. It will be different when we come to sufficient
conditions (comijare §17).
i Proof ^ We suppose first that i\' (a-) is continuous in (vt\)a'i) and denote by m and
/li' the maxima of 1 1 (a-) | and | i\'{x) \ in {.Xf^{), and by g a quantity greater than the
maximum of \f'{x) \ in (.rnj-,). Having once chosen the function r\ (.r), we can then
determine a positive (juantity h such that the point (.c, y ) lies in the neighborhood (p)
of (; and that ~q<'li <.1 for every x in (.(Vi-,), provided that | e | < 6. On the other
hand, the three functions | F
-van
finite fixed quantity G. Hence, by the mean
I (f 1? ^'IF . low' + ^ ■<»'■) f/j
remain, in this domain, below a
ue theorem,
^ e^ G (/ + 2/01 M+f*^) (.<-i -Xj
If ri'(x) is not continuous in (xqX^), apply the same reasoning to the integrals
taken between two successive corners of 6.
X
If) Calculus of Variations [Chap. I
for otherwise we could make A ,/ negative as well as positive
by giving e once negative and once positive sufficiently small
values.
6) Differeniixdion icith respect io e: The same result (7)
as well as formula ((3) can be obtained by the remark, due
to Lagrange,' that by the substitution of er] for co, the inte-
gral J becomes a function of e. say J (e), which must have a
minimum for e= 0. Hence we must have''' ./'(O) 0. If
r](.r) is of class C in (.roO^i), it follows from our assumptions
concerning the function F and the curve 6 that
dF{x,y (x) H- €77 (•^)> y ' i^) + ^^ ' (■^))
is a continuous function of x and e in the domain,
Xq'^x'^X]^, |e|^eQ, eg being a sufficiently small positive
quantity, and therefore the ordinary rule' for the differen-
tiation of a definite integral with respect to a parameter may
be applied. Hence we obtain
dJ(e)
ch
This proves (7) and at the same time ((V), since by the defi-
nition of the derivative,
A J = J (e) - J (0) = e ( J ' (0) + {€)) .
If r){x) is of class D' , decompose the integral J in the
manner described in §3, c), and then proceed as above.
c) The STjmbol B: We now make use of the following
permanent notation introduced by Lagrange* (1760).
Let (f>{x, y, y', y","-) he a function oi x,y and some of
the derivatives of y, whose partial derivatives with respect
1 Oeuvres, Vol. X, p. 400. This method has been adopted by LindelOf-Moigno,
DiEXGEE, and Osgood.
2 Moreover J"(0) must be g 0. This condition will be discussed in chap. ii.
3 Compare E. II A, p. 102; J. I, No. 83.
i Oeuvres, Vol. I, p. 336. Compare also J. Ill, No. 348.
§4] First Variation 17
to //. //'. //"•••• up to the ]i^^ order exist and are continuous
in a certain domain. Then if we replace // by // = //-]-ej/,
and accordinii:ly //' l)y //'=//' :£»;'. etc.. we can expand the
function
<f) = <t> ' "*■ •// + «>?•//' + e ^ '.•■• )
accordiuii; to powers of e and obtain an expansion of the form
2 »
^ = cf>-\-\cf>,+ ^d>,^ \-^<f>„ + ^"i^) ,.
J. -J > /v.
where (e) denotes as usual an infinitesimal, and
The quantities ecf)^, e-ify-i, • • • are called fJie frsf. second,
• . • ran'afio)i of cf) and are denoted by 8(f), 8-(f). ■ ■ ■ respect-
ively.
It is easily seen that
Again, if (f> does not contain e. S'^'cf) may be detined by
= 0
c^-
Similarly. B'^'J is defined as the term of order /.-. multi])lied
by k!, in the expansion of
J = I i^ (a\ /y + e t; , ?/ ' + £7? ' ) fir
according to powers of e, the possibility of this expansion up to
terms of order A- being, of course, presupposed. Accoitlingly
8y "'-''"
ch"
t'-
18 Calculus of Vaeiations [Chap. I
It follows immediately' that
In particular
8j=e ('^\F„r, + F,_,.rj')<h- . (8)
We may therefore formulate the result reached above as
follows : For an cxtremuni if is necessary iliaf Ihc Jirsi
variation of the integral J sliaU vanisli for all (((hiiissihie
variations of the function //.
d) More general type of variations : For many investigations
it is necessary to extend the important formula (6) to variations of
the following more general type :-'
' (.r , e) , (5a)
CO = W I
where w(a', e) is a function of x and e which vanishes identically
for e==0. We suppose that '»'(.r, e) together with the partial deriva-
tives w.c, Wt, W(.« are continuous in the domain
e„ being a sufficiently small positive quantity.
Moreover, in the case when Ijoth end-points are hxed
to (ajo , e) =0 and to {.x\ , e) = 0
for every | e [ ^ e„ . If we denote ^^(aj, 0) by i?(a7), formula (6) holds
also for variations of type (5a). This can be most easily proved by
the method explained under 6).
For the function
Fix, y{x) + i>y{x, e), y'{x) + ia^{x, c)) dx
must have a minimum for e = 0, and therefore J'(0) = 0. From the
above assumptions concerning w (x, e) it follows that differentiation
ander the sign is allowed and that Uex exists and is equal ^ to w.,e .
1 Provided always that the limits are fixed and that the ordinary rules for the
differentiation of a definite integral with respect to a parameter are applicable.
2 Such variations were already considered by Lagkange, Oeuvres, Vol. X, p. 400.
3 Compare E. II A, p. 73.
§4] First Vaeiation 19
Hence we ol^tain' also in the present case
J'(0)= ( ' {F,^-n + F„rt')dx ,
which leads immediately to (6).
For variations of type (5a) the definition of the s3'mbol 5 nnist
be modified. In order to cover also the case of variable end-points,
we suppose that av and I'l are functions of e which reduce to .r„ and
Xi respectively, for e = 0. Putting then as before
we define - " '^ ^'
^'"^ = 717^) F{x,y,Tj')dx
and similarly if 4> is a function oi x, y, y' , •■• and I'u, Xi,
e = 0
^ 9^<^ (a- , y , y' , ■•• , x^,, a-,)
a,
.A-
e^-
The definition of the symbol 5 given under b) is a special case
of this general definition.
The method of differentiation with respect to e, especially when
combined with the consideration of variations of type (5a), seems
to reduce the problem of the Calculus of Variations to a prol)lem of
the theory of ordinary maxima and minima ; only ajsparently, how-
ever ; for, as will be seen later, the method furnishes only necessary
1 For variations of the special type (5) equation (6) may also be written
(6a)
This formula remains true for variations of the more general type (oa'). For from
the properties of w (.c, e) it follows that the quotients
(<o(x, €)-u,(a-,0))/e and (<-j;(-r, «) - "j;(a-, 0))/«
approach for Le = Q their respective limits a)^(a;,0) and <^j.^(-f,0) uniformly fov all
values of x in the interval (.ryj-i) (compare E. II A, pp. 18, 49, 52, 65; .7. I, Nos. 62, 78
and P., Nos. 45, 100). Hence it follows that
{FyO> + Fy.<o')dx=e I ^Fyr, + F,i.r,')dx + ^(^) ,
which proves the above statement.
2 Always under the assumption that all the derivatives occurring in the process
exist and are continuous.
20 Calculus of Variations [Chap. I
conditions, but is inadequate for the discussion of sufficient condi-
tions, whereas the method based iqDon Taylor's formula, though less
elegant, furnishes not only necessary but also sufficient conditions,
at least for a so-called weak minimum (compare §17, b).
r) Ti'diisfoniiafioii of flic Jirxt rorlafloii hij iiif('(/i-(iii<>ii
hij jxuis:
For the further discussion of equation (7) it is customary
to integrate the second term of Bj by parts:
8./ = .|[,F,J + X>('-.-,^^.)"'-|
(9)
Since t) vanishes at .-t'o and .r^, this leads to the result that
for an extremum it is necessary that
for all functions rj of class D' which vanish at x'q and .rj.
The integration by parts presupposes, however, that not
only ij' but also ij" exists and is coiifiinions in (^o^i)? fi"^^
for the present we shall make this further restricting assum[)-
tion' concerning the minimizing curve.
§5. THE FUNDAMENTAL LEMMA AND EULER's EQUATION
To derive further conclusions from the last equation we
need the following theorem, which is known as the Funda-
mental Lemma of the Calculus of Variations :
If M is a function of x which is coutinuons in {xxyr^},
and if
riMdx = 0 (11)
^0
iThe necessity of this assumption was first emphasized by Du Bois-Reymond in
the paper referred to on p. 9). If y" does not exist, the existence of — F^, becomes
doubtful. The restriction will be dropped in §6. Discontinuities of rj' of the kind
here admitted do not interfere with the above results (9) and (10), since ij itself is
continuous. For the principles involved in the integration by parts, compare E. II A,
p. 99, and J. I, Nos. 81, 84.
§5] First Variation 21
for all functions rj icJiich vanish at .Vq and .t\ and icJiicJi
admit a continuous derivative in {or(fii\), flicii
... M = 0 (12)
For suppose Ji" (d? ' ) =1= 0 , say > 0 , at a point .r ' of the
interval (j"of'i) 5 then we can, on account' of the continuity
of M, assign a subinterval (|oli) o^ (^o^i) containing .r' and
such that Jf>0 throughout (^oli)- Now choose 7; = 0 out-
side of (foil) and 7/ = (a- — |o)^(^ — fi)" "^ (lofi) ; this function
admits a continuous derivative in {X(fc-^, vanishes at .Tq and .Vy.
and nevertheless makes
£^\
rjMdxyO ,
contrary to the hypothesis (11); therefore Jf (a:"')=t=0 is
impossible.^
The conditions of this lemma are fulfilled for equation
(10); for, since we suppose y" to exist and to be continuous
in (xryr-,), the function d
is continuous^ in {X(fiCy). ^^
1 Compare P., Xo. 17.
2This proof is due to Du Bois-Retmond {Mathematische Annalen, Vol. XV
(1879), pp. 297, 300). In the same paper he proves that the conclusion ilf = 0 remains
valid even if the equation (11) is known to hold only :
1. For all functions r/ having continuous derivatives up to the nti> order, inclusive :
proceed as above and choose, for (f(j^i),
2. For all functions having ctU their derivatives continuous.
H. A. ScHWARZ goes still farther and proves the conclusion valid if the rj's are
supposed regular iti (Xn-r,), i. e., developable into ordinary power series J {x - .r ) in
the vicinity of every point x' of the interval (.ruJ-j) Lectures on the Calrulus of Varia-
tions, Berlin, 1898-99, unpublished.)
On the other hand, the proof given in most text-books, in which
rt-i.X- Xo) {Xy -x)M
is used, assumes that (11) holds for all continuous functions ») vanishing at .>„ , j-, ,
or else, if the assumptions of the lemma concerning rj are not changed, that M' exists
and is continuous. This last assumption would, in our case, imply that y" exists
and is continuous.
Also Heine's proof (Mathemntische Annalen, Vol. II (1870), p. 189) could be
applied to our case only after further restricting assumptions concerning y,
3 Compare J. I, Xo. 60, and P., Xo. 99.
22 Calculus of Variations [Chap. I
Hence we obtain the frsf ncccssari/ coiidi/ion for an
extremum :
Fundamental Theorem I:' Ever ij fund ion // n-hich min-
imizes or maximizes f/ie integr'ol
J= f ' Fi,r,y,y').
must saiisfi/ the (Ji(ferenfi(il ('(jiuttion
F,-I^F, = i,. (I)
This differential equation was first discovered by Euler"
/ in 1744, and will be referred to as Euler''s {(lijfereniiol)
equation.^
*» §0. DU bois-reymond's and hilbert's proofs of euler's
^ EQUATION
The preceding method, which was ]:>ased upon the integration
by parts of §4, furnishes only those solutions of our problem which
admit a continuous second derivative. The question arises: Do
there exist any other solutions and if so, how can we
find them?
In order to answer this question, we return to the equation
SJ-0 in the original form (7) and, with Du Bois-Reymond and
HiLBEET, integrate tJie first, instead of the second, term by parts.
Since -n vanishes at both end-points, we get :
v'i^y- 1 Fydx)dx = 0 . (13)
1 We have prored this theorem only for functions y having a continuous second
derivative. The extension to functions having only a continuous first derivative
follows in g6, to functions of class Z)' in §9.
2EuLER, Methodus inveniendi lineas curvas maximi minimive proprietafe
gaiide7i1es, chap, ii, art. 21 ; in Stackel's translation in Ostwald's Klasiiker der
exakten Wissenschaften, No. 46, p. 54,
sKneser, HiLBERT, and others call it "Lagrange's Equation." Lagrange him-
self attributes it to Euler. See Oeuvres de Lagrange, Vol. X, p. 397 : " cette 6quati<)ii
est celle qu'EuLER a troutee le premier."
gBj First Variation 23
This iutegratiou by parts is leg-itimate, even ii y" should not exist,
since it presupposes only the continuity' of Fy and v' .
We are thus led to the problem :
If N{x) be continuous i)i (.ru^'i), and if
I X
C r)'Ndx = 0 (14)
for all fnucfions v of class C which vanish at .r„ andoTi, what
follows w'ith respect to iV ?
The answer is that N ynust be constant in (x,j.r'i).
a) Du Bois-Reymond- reaches this result ))y the following
device :
Let f be any function which is continuous in (.ru-j^i) and satisfies
the condition
\dx = 0 ; (15)
then the function
dx
is of class C in (a'uJ^i) and vanishes for x — x^ and x = .»■,, and
therefore, according to our hypothesis, satisfies (14), that is.
£
CNdx = 0 . (16)
Thus it follows from our hypothesis that every continuous func-
tion which satisfies (15) necessarily satisfies (16) also.
Now let fi be any continuous function of x ; and c the following
constant :
c
then the function
Xi Xq •^.Tq
^ = C.
is continuous and satisfies (15), hence it must satisfy also (16),
therefore
1 The continuity of F follows from the continuity (compare the beginuingof §4)
of y' and from our assumption {B) concerning F; and v' may be supposed continuous,
since (9) must hold for all functions rj of class D' which vanish at Xq and .r, . and
therefore a fortiori for all functions r) of class C which vanish at Xq and .r, .
2Loc.cit., p. 313.
24 Calculus of Variations [Chap. I
f \Ndx=: C \i{N~X)dx = 0 , (17)
if we denote bv X the constant
X =: I Ndx/(Xi — X^))
*' r,.
But from (17) it follows by the Fundamental Lemma that'
N = \ ,
i. c, constant, Q. E. D.
b) Another, more direct, proof has been given by Hilbert" in
his lectures (summer 1899). He selects arbitrarily foiu* values,
a. /3. a . /3 satisfying the inequalities
£ro<a</3<a'<y8'<X, ,
and then builds up a function' v of class C which is equal to zero
in (a*oa); increases from 0 to a posi-
tive value k as X increases from a
'• • ^ • ^ '. to /3; remains constant, = A; in (/3a');
decreases from A; to 0 as a* increases
from a to /i . and finally is equal to zero in {^'Xi):
Substituting this function in (1-4), we obtain
r)'Ndx-\- I r]'Ndx = 0 ;
v' being positive in the first, and negative in the second, integral
we can apply to both the first mean-value theorem* which fiu-nishes
k^N{a^e{(i-a))-N(a'-\-d'{(3'-a'))l =0 ,
where O<0<1 and 0<e'<l.
Finall}', let /3 and ^' approach a and a' respectively; then it
follows, since A' is continuous, that
1 This result is a special case of the isoperimetric modification of the Funda-
mental Lemma, see below chap. vi.
2 See Whittemoee, Annals of Mathematics (2), Vol. II (1901), p. 1.32.
•* Nothing more than the existence of such a function — which is a priori clear — is
needed for the proof: Hilbert gives a simple example, see Whittemoee's presenta-
tion.
* Compare E. II A, p. 97; J. I, No. 49; and P., No. 191, IV.
§6] First Variation 25
N{a) = N(a'),
i. €., N is constant in (a^o^^i).'
c) Applying this lemma to (13) we get
a constant ; or
The right-hand side of this equation is differentiable and its
derivative is I<\j ; hence the same must be true of the left-hand
side, i. e., the function
is differentiable in (.ivri) and
±F. = F
dx " " '
Tlius we find the important corollary to Theorem I that every
sol lit ion of our problem u'ith contimtous fir.'^t derivative — not
only those admitting a second derivative — »i»s^ satisfy Euler's
equation.
From the fact that F,y is differential:)le folloics the existenee'
of the second derivative y" for all values of x for ivhich
F,y(.v,y{,x^,y'(x))^0 . (19)
For, if we put
y{x + h)- y{x) = k , y'{x + h)-y'(x) = l ,
then, since the theorem on total differentials' is applicable under
our assumptions, and since y ' is continuous, we have
1 Hilbert's proof can easily be extended to the case where iV, while finite in
(.(,-f,Xj), has a finite number of discontinuities. For, if a and a' are points of con-
tinuity, we can always choose P and fi' so near to a and a' respectively that N is
continuous in (aP) and (a/3') ; it follows then as above that iV^(a) =N{a.'), i. e., under
the present (tssumiHions N has the same constant value in all points of continuity.
Hence it follows further that in a point of discontinuity, c:
N{c-0)-N{c + 0) .
2 First pointed out and emphasized by Hilbert in his lectures; see Whitte-
MOHE. loc. cit.
3 Compare E. II A. pp. 71. 7.3; J. I, Nos. 86, 127; and P., No. 10.").
20 Calculus of Variations [Chap. I
where a, |3, 7 apj^roacli zero as // approaches zero. Hence it follows
that if (19) is satisfied,
exists, and that
F — F — 11' F ■
y"= " 1-: " "" ; (20)
moreover, (20) shows that y" is coniinuous iu (.r„j-]).
^7. MISCELLANEOUS EEMARKS CONCERNING THE INTEGRATION
OF euler's equation
a) Euler's differential equation (Ij is of the second
order ^ as can be seen from the developed form
F,-F,,,-y'F,,^-,j"F,.,. = ^ ; (21)
its general solution contains, therefore, two arbitrary con-
stants,
ij=fU-,a,li) . (22)
The constants a. I3 have to be determined^ by the condition
that the curve is to pass through the two points A and B :
y,=f{x,, a, (3)
yi=f(Xi, a- ^) •
Every solution of Euler's equation (curve as well as
1 Unless i^.. (.r, y, J/) should be identically zero. In this case Euler's differ-
ential equation degenerates either into a finite equation or into the identity : 0 = 0
but never into a differential equation of the first order. For if F ■ . = 0, F must be of
the form : L(x , y^-r ^lyx ■, y) y' and (21) reduces to :i — J/_j, = 0. See also below,
under d).
If Euler's differential equation degenerates into a finite equation, it is in
general impossible to satisfy the initial conditions when the end-points are fixed.
Also in the general case when F contains higher derivatives, Euler's differ-
ential equation can never degenerate into a differential equation of odd order;
compare Frobexius, JoMr?ia[/iir J/«^/ieniafifc, Vol. LXXXV (1878), p. 206, and Hirsch,
Malhematische Annalen, Vol. XLIX (.1897), p. 50,
2 This determination may be impossible ; in this case there exists no solution of
the problem which is of class C and lies in the interior of S.
§7] First Variation 27
ftinction) is called, according to Kneser, an extremal; there
is then a double infinitude of extremals in the plane.
In the S2)ecial case ichoi F does not contain x explicifJij,
a first integral of (I) can be found immediately.' For, if F
does not contain x explicitly, we have
and therefore every solution of (I) also satisfies
F - y'F,,. = cons\. (24)
, Vice versa, every solution of (2-1:), except ?/ = const., also
satisfies (I).
b) Example I (see p. 1):
F=yVl-i-y" .
Hence
and E u 1 e r ' s equation becomes :
d yij'
or. after performing the differentiation,
By putting -r^ =J>, the integration of this differential equation
is reduced to two successive quadratm-es, and the general integral
is easilv found to be
11 ^ a cosh — - — - .
^ a
The extremal!^ are therefore catenaries n-ith the X-axis for rl irectrix.
Since F does not contain x, a first integral could have been
obtained directly by the corollary (24);
F-y' F,,.^ , ^^ = a .
' Vl + y''
1 Noticed already by Euler, loc. cit., p. 56, in St.\ckel's translation.
28 Calculus of Variations [Chap. I
If a =j= 0, this leads to the same resiih as above; for a = 0 we obtain
y = 0, which, however, though a sohition of (24), is not a sohitiou
of Euler's equation.
The general solution of (I) being found, the next step would be
so to determine the two constants of integration that the catenary
passes through the two given points.'
c) Tlir()ii(/Ji a (jiven point a, b in the interior of the
region' iS one and hut oiw e.rtrcnifiJ of class C can hedrairn
in a (jircn direction of anipIitiKle^ <« ( =t= — ^), provided tliat
F,„{a,h,h')^0 , (2.-,)
ivliere It' — tan (o .
For, if we solve (I) with respect to^", we obtain for /y" a
function of or, [/, ij' which, according to our assumptions (B|.
is continuous and has continuous partial derivatives with
respect to y, y' at all points of the domain' 01 which satisfy
(25). Hence the statement follows from Cauchy's general
existence theorem* for differential equations. :
1 For this interesting problem we refer to: Lixdelof-Moigno, Joe. rit.. No. 103;
DiENGER, loc. cit., pp. 15-19; Todhuxter, Researches in the Culrulus of Variatio>ui.
pp. 55-58 ; Caeul, A Treatise on the Calculus of Variations, Nos. 60, 61. For Schwarz's
solution see Hancock, '"On the Number of Catenaries through Two Fixed Point>;."'
Annals of Mathematics (1), Vol. X (1896), pp. 159-174.
■■!See§3, c). •iSee§2,sr).
*" Suppose the functions/,. U' , !/i, i/o- ' ' ' • ^n^ ^"'^ their first partial tlerivatives
with respect to y^, y2,- • • • l/„to be continuous in the domain
1 X - a I ^ P . //, - 6i ; ^ r , • • • , I !/„ - ^„ , 5 '• ;
let M be the maximum of the absolute values of the functions f- in this domain, and
let I denote the smaller of the two quantities p and r M.
Then there exists one, and but one, system of functions y, (x), i/jCa-).- - • , //„ i.')
which in the interval \ x — a \ < / are continuous and differentiable, satisfy the differ-
ential equations
-^=/,U-,!/i,i/,>- ••••'/„) > (' = 1,2. •■•.«)
and the inequalities \ y^M —b^\ ^ r , and take for x = a the values
Compare E. II A, pp. 193 and 199, and .J. Ill, Nos. 77-80; also Picakd, Tr<u,e
d' Analyse, Vol. II, chap. xi.
In order to apply the theorem in the present case, replace (21 ) by the equivalent
system.
s
7] First Variation 'j!9
If, therefore,
for every finite value of 2>, one extremal can be drawn from
[a, b) in every direction, except the direction of the ^-axis.
A problem for which
at every point [x, ij) of the region jR for every finite value of
p, is called, according to Hilbert, a regular prohlet)!.
d) We consider next the exceptional case in whicJi Eiilevs
differe)itial equation degenerates into an identiti/.
Suppose the left-hand side of (21) vanishes for every system of
values X, y, y , u . Then, since y ' does not occur in the three first
terms, it follows that the coefficient oi y" must vanish identically,
so that we must have separately
i/ , = 0 F — F — ii'F = 0
for every x, y, y . From the first identity it follows that F must
he an integral linear function oiy', say
F{x,y,y')=M{x,y)+N{,r,y)y' .
Substituting this value in the second identity, we get
the well-known in tegrability condition for the differential expression
Mdx -\- Ndy .
Hence we infer : If M and N and their first partial derivatives are
single-valued and continuous in a simply-connected region ^ of
the X, 7/-plane, then there exists* a function V{x, y), single-valued
and of class C in ^ and such that
y, = M , V, = N ,
and therefore
F{x, y, y') = F,+ V,y' = ^ V{x, y) .
Hence if S : y=f(oo) be any curve of class C drawn in S> between
the points A{Xo, y^) and B{xi, y/i) our integral Jy^ has the value
1 See PiCARD, TraiU cf Analyse, 2d ed., Vol. I, p. 93.
30 Calculus of Variations [Chap. I
F{x, y, y')dx= F(.r,, y^} — V{xo, //o) ,
and is therefore iudepeudent of the i^ath of integration (S and
depends only upon the position of the two end-points.
On account of the continuity of V{x,y), the result remains
true for ciuves 6 with a finite numljer of corners, as is at once seen
by decomposing the integral J in the usual manner.'
Vice versa : If the value of the integral Jq is independent of
the path of integration 6 as long as 6 remains in the interior of a
region g* contained in S, then the function F must be of the form
31 (x, y)^X{.v, y)y' , where M„==Xj., for every point {x, y) in the
interior of ^ for which a^o ^ •*' < -^"i •
For let (^2. Vi) be any inner point of §> whose abscissa Xi lies
between a'o and a^i and yi, yi' two arbitrarily prescribed values;
then we can always draw in g> a curve Q.y = f{jc), of class C which
passes through (.^o, ^o), (.ri, v/i), (jc-i, 7/2). and for which f'{jc.^ = ij2,
f"'{jc2) = y2 •
According to our hypothesis, A J must vanish for every admis-
sible variation of 6, whence we infer by the method of §§ 4, 5 that
y=zf(^x) must satisfy Euler's differential equation. The left-
hand side of the latter must therefore vanish for the arbitrary
.system of values x = X2, y= 1/2, y' - yl , y" ^y-i , which proves the
above statement.
We thus reach the result : -
In order that the value of the integral
F{x, y, y')dx
may he independent of the x>ath of integration it is necessary and
sufficient^ that Euler's differential equation degenerate into an
identity.
It is clear that in this case there exists no proper* extremum of
the integral J.
e) We conclude these remarks by considering briefly the inverse
problem : Given a doubly infinite system of curves {functions)
y=f{^, «> /3) ,
1 Compare p. 12.
2 Compare J. Ill, Nos. 362, 363, aud Kxesek, Lehrbuch, §51.
3 Sufficient only if the region ^ is simply-connected.
♦ Compare §3, b).
§7] First Variation 81
to determine a function F(x, y,y') ^^^ ^/'«^ '^e given system of
vnrres shall he the e.vtrenials for the integral
J= C 'F{jc,y,y')~
This problem has always an infinitude of solutions which can
be obtained by quadratures.^
For if
y"=G{x,y,y') (26)
is the differential equation of the second order" whose general
solution is the given function y=f{x, a, ^,) (with a, /3 as constants
of integration), then we must so determine the function F{x, y, y')
that (26) becomes identical with Euler's differential equation for
F, i. €., according to (21)
F,~F,,,-F,,,yy'=GF,^,, . (27)
If we differentiate (27) with respect io y' , we get for M=Fyy-
a linear partial differential equation of the first order, viz.,
If
a = (}>{.r,y,y') , ^ = i}j (,r, y, y')
is the solution of the two equations
y=f{x,a,f3), y' =f:c{x,a., IB)
with respect to a and ^, and if further
and
x(-^' y^ y') = ^(«^> </>('^. y, v')^ ^{-^^ y> y')) »
1 Daeboux, TMorie des surfaces. Vol. Til, Nos. 604, 605. For the analogoiis problem
in the more general case when F contains hij^her derivatives, compare Hirsch, Mathe-
matische Annalen, Vol. XLIX (1897), p. 49.
2 Obtained by eliminating a, p between the three equations
compare, for inst., J. I, N'o. 1G6.
32 Calculus of Variations [Chap. I
the general integral of (28) is found to be, according to the general
theory' of linear partial differential equations of the first order,
-1/X = ^{cf>U-, y, y'), xl>{.v, y, y' )) ,
where * is an arbitrary function of <f> and --f.
After the function M has been found, F is obtained ])y two
successive quadratures from the differential equation
.^--, = M(x. y, y ) .
Finally the two constants of integration X, m (which are functions
of X and y), introduced by the latter process, must be so determined
that F satisfies the original partial differential equation (27) from
which (28) was derived by differentiation.
Example:- To determine all functions F for which the ex-
tremals are straight lines
y = ft.r -\- /? .
The differential equation (26) becomes, in this case,
y" = 0 .
Accordingly, we obtain
ct>^ y' , i{, = y- .ry' , x = ^^onst.
Hence
M = ^{y' , y - .ry') ,
and therefore
F= j {y' - t) ^ {t , y - xt) (It -j- y X {x , y) + h- {x, y) .
The condition for X and m becomes in this case
9X 9/A
dx dy
The most general expression for X and m- is therefore
dv 9v
^^Vy^ ^" = 91''
where v is an arbitrary function of x and y.
1 Compare, for inst., J. Ill, Xo. 242.
2 Compare DAEBorx, loc. cit.. No. 606.
§8J First \'ariation 83
§8. WEIERSTRASS'S LEMMA AND THE E-FUNCTION
Before proceeding to the consideration of so-called
discontinuous solutions, we must derive a lemma, due to
Weierstrass,^ which is of fundamental importance for many
investigfations in the Calculus of Variations.
Suppose there are given, in the region H, an extremal ©
of class '^C": >J=f{^'), and a curve 6 of class C : // /(r).
meeting G at a point ^ 2 : (.ro, >/■>). Besides there is given a
point 0: (.ro, 2/o) on @, before 2, that is, Xq<j-2. Let 3 be
that point of 6 whose abscissa is a-o + Z', h being a positive
infinitesimal, and select arbitrarily a function v of class C"
satisfying the conditions
Then we can so deter-
mine e that the curve
-- - , FIG. 4
which necessarily passes through the point 0, also passes
through the point 3. For this purpose we have to solve the
equation
f{X2 + h) + er) ( .r, + h) = f{x, + //)
with respect to e. Since /(j-o) =/(j"2)? we have
f(x, + h) -f{.c, + h) = (jj: - U-l ) h-t h (// ) ,
where ijo =/'(^2), Th^I" (•^'2) and (//) is an infinitesimal for
Lh=0. Hence we obtain
= /-[^+w]
It is proposed to compute the difference
-^ '-/ = Jffi — ( " 02 I "23) J
iThe lemma here given is a modification of the correspond ing lemma given by
"Weieesteass in his lectures U879) for the case of parameter-representation ; see §2.S.
2 This assumption must be made on account of the integration by parts which
occurs below ; compare §4.
3 For the notation compare §2, e).
84 Calculus of Variations [Chap. I
the integrals J, J, J being taken along the curves (5", (5, ^
respectively, from the point represented by the first index to
the point represented by the second.
A./ may be written
^ ' (F - F) dx+ i ' {F - F) dx ,
where F,F,F or F[x],F[x],F[x] stand for F(x,ij{x),y'{x)),
F{x, y{x), y'{.r)\ F[x, Tj(.r), y'{x)) respectively.
The first integral, treated by the method of §4, becomes,
since G is an extremal,
X, ' ^^' ~ ^'^ ''''■ = '"'^ ^^' t^-^^] + ' * '-'
= h[{y.:-y;)F,,[.v,-\ + {h)] .
To the second integral we apply the first mean-value
theorem and obtain, on account of the continuity of i^^[.i"]
and -F[.r],
J[ ' (F-F) dx = h [f [x,'] - F [.r,] + (/«)] .
Collecting the terms, we reach the result
Jo:. - (Jo2 + J23) = h \ {]/: -yi)F^. [.r,] + F M - F [.n] + {h)\.
Similarly let 4 be that point of 6 whose abscissa is
X2 — 1i , and determine e ' so that the curve
6 : y = y + ^'v
passes through 4. Then we obtain by the same process
J,u + 'J.2 - J02 = - /' ) (5/2' - 2/2 ) Fy. [x,-] + F [x,-] - F [x,-] + (h) I .
If we put for brevity
F{x, y,p)- F{x, y,p) - (p - p) Fy.{x, y , p>)
= E{x,y; p,p) , (29)
X, y,2^jP being considered as four independent variables,
thp preceding results may be written:
f!8] First Variation 35
Jo, - {Jo-2 + ^2.) = - /< } E {x, , !J,; v/; , 7/; ) + (/') { ' I ^^j.
J.,+ {'l:-J.2) = ^h\E(j-,,!j,; y2,U2) + {l>)\-\
We shall refer to these two formulae as Weiersfrass's
Lemnid. The function E(.r. ij; p,p) defined by (29) will
play a most im})ortant part in the sequel ; it is called Weicr-
strass's 'E-fntiction}
The same results (80) -hold if the curves 03 and 04 are of
the more general type (5a):
y=f{:r)-^ia{x, e) ,
where the function <w(.t, e) vanishes identically for e = 0, has
the continuity properties enumerated on p. 18, and satisfies
besides the conditions:
a)(j"u, e) = 0 for every e, and oy^[,x'2, 0) =p 0 .
For the determination of e we have, in this case, theequation :
/ {x, + //) + (u {x, + h , e) - fix, + A) = 0 .
The resulting value of e is of the same form as above.
This follows from the theorem^ on implicit functions; for if
1 Compare Zermelo, Dissert ait ion, p. 66.
-"If f(x, y) is of class C in the vicinity of {.r„, j/^) and
then a positive quantity A; being chosen arbitrarily but sufficiently small, another
positive quantity h^. can be determined such that for every x in the interval (xu— h/^,
x„-\-hfJ) the equation /(.1-, 2/) =0 has one and but one solution y between y^— fc and
The single-valued function 2/-=i//(a') thus implicitly defined by the equation:
fix . //I = 0, is of class C in the interval (.Cq— hj., ^q+Zi^.) and
dy _ fx
Hence
<lx f,j
!/-y,)= (^--eo)
where i o=:0.''
x=x„
(Compare E. II A, p. 72; J. I, No. 91; P., No. 110).
If f{x, y) is regular in the vicinity of (a; , j/ ), also the function 2/- i/* [x) is regu-
lar in the vicinity of x^,. (Compare E. IT B, \>. 103, and Harkness andMorley,
Introduction to the Theory of Analytic Functions, No. 156.) For the extension of the
theorem to a system of m equations between m-]-n unknown quantities, see the ref-
erences just given.
30 Calculus of Variations [Chap. 1
we denote the left-hand side of the preceding equation by
F{li, e), this function is of class C" in the vicinity of A = 0.
e = 0; further: F(0, 0) = 0 and finally jPJO, 0)=^0.
Incidentally we notice here the formula
{F-F)dx+ Fdx
= /' \iK - //; ) F,^. [a-,] + F [.r,] + (7^)] ,
which holds for negative as well as for positive values of It.
Hence it follows that if the arc 02 of the extremal Q mini-
mizes the integral ./, the end-point 0 being fixed while the
end-point 2 is movable on the curve ^, then the co-ordinates
of the point 2 must satisfy the condition
F + ry'-u')F/=0 .
{''■Condition of transvcrsalittj," compare the detailed treat-
ment of the problem with variable end-points in ^28.)
§1>. DISCONTINUOUS SOLUTIONS
We must now free ourselves from the restriction' imposed
upon the minimizing curve at the beginning of ^4, viz., tV t
ij' should be continuous in (^Vi)? ^^^ we propose to deter-
mine in this section all those solutions of our problem which
present corners — so-called "(lisco)itinuous solutions."
(i) In the first place, the theorem holds that (dso (lisrf)ii-
tinuous solutions must satisfi/ Uuler's differential equation.
Suppose for simplicity" that the minimizing curve 6 has
only one corner C(x2, }Jz) between A and B. According to
§3, c) the integral J",, is then defined by
J,= P F{x, ij. y')d.r+ P.Fic:, ij , y')dx . (31)
iThe assumption that the curve shall lie entirely iu the interior of the re^cn
S will still be retained in this section.
^The results can be extended at once to the case of several corners.
§9]
First \ariatiox
37
the notation indicating that Ij'U'^) is defined in the tirst
integral by y' {.r^ — 0). in the second by u'{x2,-\-Q).
The theorem in question is most easily proved by the
wliich is very nseful in
FIG. -)
method of partial variation
many investigations of the Cal-
culus of Variations:
We consider first such spe-
cial' variations ADC of type
(•")) as leave the arc CB un-
changed and vary only A C.
To such variations all the con-
clusions of ^'i^4:-() can be applied, and it follows as before
that for the interval (.r^, .ro — 0) Euler's equation must
hold. The sam*^' result follows for (.r2 — ^*. •■•"i) from the
consideration of variations which leave A C unchanged;
hence it is true for the whole interval (-ro^i)."
h) A discontinuous solution with one corner is therefore
composed of two extremals involving in general different
constants of integration:
y =f{.r, a,, /3i) iu (.r„, a-,— 0) ,
y =f(.r, a,, 13,) in (.ro + 0,a",) .
For the determination of x-> and of the constants of inteo^ra-
tioii we have in the first place the initial conditions
fnrther the condition that y is continuous at x-i'.
f{,i\, a,, /?,) =/(.«•.,, a.,, p.j) ;
and finally two further conditions which are furnished by the
following: theorem due to Weierstrass and Erdmanx:^
J Compare the remark on p. 15, footnote 2).
2Withthe same understanding as iu (31) concerning the meaning of y' at the
corner.
:i Weierstrass, Lec/wres at least as early as 1877; Erdmaxx, Journal fur Mathe-
iixitik, Vol. LXXXII (1X77), p. 21. Another demonstration has been deduced by
38 Calculus of Variations [Chap. I
Theorem: At every corner of a minimizing curve the
two limiting values of Fy- are equal. -^
F ■
— F ,
a;2+0
)
(32)
X2+0
y'Fy .
(33)
and likewise f"^ "
F-ij'F„\ = F
To prove (32) consider a variation ^4 GB of type (5) for
which the function 77 is of class C in {x^fic^ and 1] (a^o)=t=(>.
The integral At/ breaks up into two integrals taken between
the limits {jcq, Xo — 0) and (a^oH 0, x^) respectively. Apply-
ing to each of these the methods of §4 we find that also in
this case 8J=0, and further we obtain" from (9), since (I)
is satisfied:
8j^cy, (x,) (Fy. Ix, - 0] - F^. [x, + 0]) ,
where fy[x] stands again for Fy(x, f(x), f (x)). Since
Sj=0, (32) is proved.
The proof of (33) follows from Weierstrass's Lemma
(30) if we identify the arcs A C and CB of Fig. 5 with the
arcs 02 and 24 of Fig. 4, respectively, and consider suc-
cessively the variations 034 and 04234 of the arc 021. The
corresponding values of the total variations A J" are given by
the two equations (30), the values of ij-l, T/^' being in the
present case
yl = y ' ('^2 — 0) = yi ; V2 = y' U2 + O) = //,' .
Hence it follows that for an extremum it is necessary that
Whittemoee, loc. cit., from Hilbert's proof of Euler's equation: By means of
the extension of the lemma of §6 to discontinuous functions (see p. 25, footnote 1), it
can be shown that equation (18) holds with the same value of the constant A for both
segments {x^^x.^-d) and (j-2 + 0,a:,). Hence follows Euler's equation as well as
equation (32). This method can be applied to discontinuities of a much more com-
plex character and even to the case of an infinitude of points of discontinuity; see
Whittemoee, loc. cit.
iFor the notation compare §2, 6).
2 The integration by parts is legitimate since by the method of §6 the existence of
-r- F . is established for each of the two segments (^q, a-g -0) and (x^ + 0, r,) .
§9] First Variation 30
and on account of (32) this is equivalent to (33).
c) Example' III: To minimize the integral
Here
Hence a first integral of Euler's differential equation is
4^" + %"+ 2//' = const. ;
therefore
rj = a.r + /3 .
i.e., the extremals are straight lines, and the line AB joining the
two given jDoints is a possible continuous solution.
In order to obtain all discontinuous solutions with one
corner, we have to find all solutions pi, m of the two equations
ky\ + ^lA + 2i>, = ^pi + h^\ + 2p, .
-^p\-4.p\-p\=-'^p\- Apl-iA ,
where
Pi = ^/ ' (c — 0) and p2=ij' (c -\-0) and p^ 4= p^ .
Dividing out hy pi— p^ and putting
i>i + P2 = » . in + Piih + pi = ^t-'
we get
2h- + 3(f + 1 = 0
-3u' + 6aH- + 4H-+ *( =0 .
These equations have one real solution, // = — 1 , ?c = + 1 , from
which we obtain
Pi = 0 , p2 = - 1 ,
or
jpi = — 1 , pi = 0 .
1 A special case of the example given by Erdmann, loc. cit., p. 24.
40 Calculus of Variations [Chiip. I
Every discontinuous soluUou must therefore be composed of
straiglit lines making the angles 0 or 2>-n- / 4c with the positive x-a,vis.
If the slope m, = ( ^i — !/„)/{-)Ci — x„) of the line A B lies between 0 and
- 1, there are indeed two such solutions, A d JBand A CiB with one
corner and an infinity with n ^ 2
corners.
Since F = y'^ {y' -\-Yf, these
discontinuous solutions furnish
B for J the value zero and there-
fore the absolute minimum}
FIG. 6
d) In many cases the impos-
sibility of discontinuous solu-
tions can be inferred from the following
CoroUarij :~ If {xo, 2/2) '^' " corHer of a minimizing curve,
then the function
J'^nri'*'^, !h,p)
must canisli for some finite value of p.
For the function
is a continuous function of j) admitting a finite derivative
for all finite values oi j); further, if we put
y ' {X2 - 0) = 2h , y ' (^2 + 0) = p-2 ,
we have px^ p-y, find, according to (32),
<^ (Pl) = "^ {P-2) •
Hence by Rolle's Theorem the derivative
<^'{p) ^ F,j.,y{x.2, y2,p)
must vanish for some value of 2^ between 2^1 and P2 •
If therefore the problem is a "regular problem," /. e., if
for every point in the interior of 2J and for all finite values
'The minimum is, however, " improper " (compare §3, b)), because in every
neighborhood of AC^B (or A C^ B) broken lines can be drawn, joining A and 6, whose
segments have alternately the slopes 0 and - 1. For such a curve A J = 0 .
2 Compare also Whittemoee, loc. cit., p. 136.
§10] First Variation 41
of j>. we infer that no discontinuous solutions are possible in
the interior of U.
Example I (see p. 1) : F =ii V \-\-y"\''B>. is the upper half-plane
{y^O).' Here
F..= y
II u
is =1=0 in the interior of S. and consequently no discontinuous
solutions are possible in the interior- of S.
§10. BOrXDARY CONDITIONS
In all the preceding developments it was assumed" that the
minimizing curve should lie entirely in the interior of the region
S. But there may also exist solutions of the problem as formulated
in §3 which have points in common with the boundary of S. To
determine these solutions is the oliject of the present section.
For this investigation it is convenient to make use of the idea of
a point by point variation, of a curve which played an important
part in the eai'lier history of the Calculus of Variations.
Between the points of the two curves
and 6: y^y^Ay
we may establish a one-to-one correspondence by letting two points
correspond which have the same abscissa x. And we may think
of the second curve as being derived from the first by a continuous
deformation in which each individual point moves along its ordinate
according to some law, for instance, if in
we let a increase from 0 to 1.
A point of 6 whose abscissa is x', is called a point of free
variation if ^.y{x') may take any sufficiently small value; other-
wise, a point of unfree variation.
For a curve 6 vihich lies entireh' in the interior of S al]
points except the end-points are points of free variation.* and this
freedom was essential in the conclusions of §§4 and 5.
1 Compare §1, c). 3 See the beginning of §4.
2 Compare the next section. *Iu our formulation of the problem, §3.
42
Calculus of Variations
[Chap. I
This is not true for a curve which has points in common with
the boundary. For simplicity let us suppose that the ]x)undary of
S contains an arc 6 representable in the form
d=fU) ,
f(x) being of class C". In order to fix the ideas suppose that S
lies above 6. Then if 6 has a point P in common with (5. the
variation of P is unfree and restricted l^y the condition
A^/^O. (34)
Suppose the minimizing curve 0231 has the segment 23 in com-
mon with the boundary.
Then the method of partial varia-
.tion applied to 02 and to 31 shows that
these two arcs must be extremals.
Consider next a variation of type (5 )
which leaves 02 and 31 unchanged and
varies only 23. Since A// — e?; must be
^0, V cannot change sign and if we
choose v^O then e must be taken posi-
fire ; hence we can no longer infer from
(6) that 5j- = 0, but only that
SJ^O . (35)
After the integration by parts of § 4 we obtain therefore
d
£'{f^-t^^
0
for all functions v of class D ' which vanish at x^ and x-, and satisfy
besides the condition
7?^0 .
The lemma of § 5, slightly modified, leads in the present case
to the
1 Moreover at the end-points 2 and 3 the following condition must be satisfied :
E(J-.2,2/2; 2/2''»2') = 0; E(>3,2/3; yi,y^') = Q.
The proof follows easily from Weierstrass's Lemma (see Fig. 7). Compare also
the treatment of the problem in parameter-representation, §29. The question of
sufficient conditions for one-sided variations has recently been considered by Bliss in
a paper read before the Chicago section of the American Mathematical Society. He
finds that for a so-called regular problem (§7, c) the arc 23 of the curve T furnishes a
§10] FiKST Variation 43
Theorem:^ If the minimizing curve has a segment^ 23 in
common xcith the boundary of S, then along this segment tlie
folloti'i)ig condition must be satisfied .•
:Fy-~F,,.^0 , if a lies above 23 , (3Ga)
(XdC
F,j-^F„.^0 , if a lies below 23 . (36b)
smaller value for the integral J than any other curve of class D' ji .i.:ii«' the two
points 2 and 3, lying in a certain neighborhood of the arc 23 and saf"ifijinij the comli-
tion A 2/50, provided that the condition
u dx y
is fulfilled along the arc 23.
The proof is based uiDon the construction of a "field " (see §§19, 20, 21) of extrem-
als each one of which is tangent to the curve Q and lies entirely on one side of 'e.
1 Of the properties specified above.
CHAPTER II
THE SECOND VARIATION
■^11. legendre's condition
The integration of Euler's differential equation and the
subsequent determination of the constants of integration'
yield in general a certain niimber" of curves 6 as the only
possible solutions of our problem; that is, if there exist at
all curves which minimize the integral J, they mu^t be con-
tained among these curves.
We have now to examine each one of these curves sepa-
rately and to decide whether it actually furnishes a minimum
or not.
We confine ourselves in this investigation to curves which
lie entirely in the interior of the region U and have no
corners.
a) Goieyulities concern in (j the second variation.
We suppose then we have found an extremal
©0 : y=M'^), Xo^jc^x^ (1)
of class C which passes through the two points A and B,
and which lies entirely in the interior of the region U.
Then we replace, as in § 4, the curve @o t>y ^ neighboring curve
y = y + ^
and apply to the increment A J" Taylor's formula,^ stopping,
iBy the initial conditions (23), the corner conditions (32) and (33), and the
boundary conditions.
2 The number may be infinite (see Example III, p. 40) ; but it may also be impos-
sible so to determine the constants as to satisfy the conditions imposed upon them ;
this happens, for instance, in Example I for certain positions of the two given points ;
see the references given on p. 28.
3If jP is an analytic function, regular in the domain ST, expansion into an infinite
series may be used instead.
44
§11J Second Variation 45
however, at the terms of the third order. If we put for
brevity
Fyy {x,f,{x),f;{x))=P'
Fyy\x,f,{x),n{x)) = Q I- (2)
Fy;j\x , /o {x) , fo {x))=R
and remember that Si/:=0, since G^ is an extremal, we obtain
A J = 1 ) {Pio' + 2^0)0)' + J?a)'2) da- + ( (oy, io'), dx , (3)
(o), a)')3 being a homogeneous function of dimension three
of CO, 0)' .
Considering again special variations of the type (o = €r] and
reasoning as in §4, we obtain
A J = r [i f ' {Prj' + 2Qr,r,' + Br}") dx + (ej] , (4)
where (e) is again an infinitesimal.
Hence we infer the theorem:
For a miiii)iium {inaximu)n) if is necessary that the
second variation he positive (tiegatire) or zero:
SV^O (^0) (5)
for (lU functions v of class D' irlticJi vanisJi at Xq and x^.
For according to the definition given in ^4, c),
gV = £2 r ' ^p^2 _^ 2(^)r,r,' + Pri'-) dx . (oa)
The same result can also be obtained by the method of differ-
entiation with respect to e, explained in ■§4, h); see p. ll),
footnote 2.
From our assumptions concerning the functions -F(.j" , /j , j))
and /o(.r) it follows' that the three functions P, Q, R are
continuous in the interval (a^oTi). We suppose in the sequel
that they are not all three identically zero in {X(yX\).
1 Compare J. I, Xo. 60, and P., No. 99.
46 Calculus of Variations (Chap. I]
h) Legendre' s condifioit.
For the discussion of the sign of the second variation,
Legendre' uses the following artifice: He adds to the second
variation the integral
I [2rjr]'ir + r]'tc')dx ,
where ir is an arbitrary function of jt of class C in (jTcyri).
This integral is equal to zero ;" for it is equal to
• '■^11 </.<■ L J-i-ii
and r] vanishes at ,/o and .r^.
He thus obtains S-J in the form
8M = e' i '' \(P + 'r') rj' + 2 ((? + W) -qrj' + i?r,"^1 d.V .
And now he determines the arbitrary function u) by the con-
dition that the discriminant of the quadratic form in ?;. t)'
under the integral shall vanish, /. c.
This reduces S'-,7 to the form
from which he infers that R must not change sign in {x(fc^)
and that S-,/ has then always the same sign as R.
These conclusions are, however, open to objections. For,
as Lagrange-^ had already remarked, Legendre' s trans-
formation tacitly presupposes that the differential equation
1 Legendre: "M6moire sur la manifere de distinguer les maxima des minima
dans le calcul des variations," Mimoires de V Acadimle des Sciences, 1786; in
Stackel's translation in Ostwald's Klassiker der exacten Wissenschaften. No. 47,
p. 59.
^This holds true also when >j has discontinuities of the kind which we have
admitted (§3, c)); compare p. 12, footnote 5), and remember that rj and w are con-
tinuous in (.ry.rji.
3 In 1797; see Oeiwres, yol. IX, p. 303.
§11] Second Variation 47
(()) has an integral which is finite and continuous in the
interval {JCffiCi), and that B does not vanish in {oC(fCi).
Nevertheless, by a slight modification' of the reasoning,
the first part of Legendre's conclusion can be rigorously
proved, /. c, the
Fundamental Theorem II: For a minhiniiii [mcurimum)
it is necessavn fix if
R{x) = F,y^.ix,Mx),f:{x))^0{^0) in {x,x,) . (II)
For, suppose jR{c) < 0 for some value c in (.>Vi) ; then we
can assign a subinterval (lo^i) of (xf^i) for which the follow-
ing two conditions are simultaneously fulfilled:
1. R(,r) < 0 throughout (fnli ) :
2. There exists a particular integral w of ((i) which is of
class C in (loli)-
For, since B{x) is coxitiniious in (.ro-Ti) and i?(c)<0, we
can determine a vicinity {c — ^, c -r ^) of c in which i?(a^) < 0.
Hence it follows that if we write the differential equation (G)
in the form
^=_P + (£±i£)!, (P,a)
dx R
the right-hand side, considered as a function of .r and //', is
continuous and has a continuous partial derivative with
respect to w in the vicinity of the point x-=c, ir = WQ, iuq
beincr an arbitrary initial value for ir.
Hence there exists, according to Cauchy's existence
theorem/ an integral of (6) which takes for .r = c the value
. u- = Wq, and which is of class C in a certain vicinity (c — S',
c-rS') of r. The interval (fo^i) in question is the smaller
of the two intervals (c — S, c r 8) and (c — S', c-^S').
This point being established, we choose for -?; a function
which is identically zero outside of (fo^i), ^^^^^ eqnnl to
iThe proi>f in the text follows Weieestbass's exposition, Lectures, 1879.
2 Compare p. 28, footnote i.
48 Calculus of Variations [Chap. II
(.r — ^o) (<^ — li) i^^ (loll)- The function -q thus defined fur-
nishes an admissible variation of the curve @o> since it is of
class D' in {-r^^x), and vanishes
at .r,) and .rj.
For this particular function
?;. ^'J becomes
To this integral Legendre's transformation is aj^plicable.
FIG. 8
Accordinglv
^--''XX^'+^f^-
The function -q ' H ^ — 77 is certainly not identically zero
throughout (loli)* f<^i' it is different from zero for .r=^^o and
Hence if i? (c) were negative, a variation of @o could l3e
found for which S2J"< 0, which is impossible if @o minimizes
the integral J. Therefore i?(j")^0 in {.r^^, Q. E. D.
Leaving aside the exceptional case^ in which R{x) has
zeros in the interval {x^x\, we assume in the sequel that for
the extremal ©0 the condition
i?> 0 in {x^^ (II')
is fulfilled.
A consequence of this assumption is that not only f^{.r)
but also/Q'(ip) is continuous in {x^fic^)^ as follows immediately
from equation (20) at the end of ^6. Hence we infer that
not only the functions P, Q, R themselves but also their
first derivatives are continuous in (a:v'i)-
Example^ I (see p. 27): F—yVl + y-; hence
1 An example of this exceptional case is considered by Eedmann, Zeitschriff filr
Mathematikund Physik\ Vol. XXIII (1878), p. 369. viz.,
F = y cos X and Jy < ,3 < .I'j .
2 All the square roots are to be taken positive, see p. 2, footnote 1.
§11] Second Variation 49
F —0
7.^ - ^'
F - ^
nil '
FurtluM-
"'■' v\ + !r'
'■"■' (. 1 + /0'
e,:
y = «„ eosli —
hence
"u
P = 0 .
^ = tanh — ,
i?-a„/cosh^^''~^" .
a„
Since we suppose .y>0, it follows that a„ >0 and therefore i?>0
for every .r.
c) Jdcohi's form of Lcgendrcs dijfercitfial ('<{nnfioii.
We have mow to examine the second part of Legendre's
conclusion, viz., that, if E > 0 throughout (/Vi), then 8-J^()
for all admissible functions ?;.
The conclusion is correct, as follows immediately from
the preceding developments, whenever there exists an in-
tegral of the differential equation ((>) which is finite and
continuous' throughout (iro^rj); it is wrong, as will be seen
in §10, if no such integral exists.
It is therefore necessary to enter into a discussion of the
differential equation (C)). For this })ur})ose Jacobi" reduces
the differential equation (()) to a homogeneous linear differ-
ential equation of the second order by the substitution^
w=-Q-R-^ , (8)
which transforms (Cy) into
iP-Q')n-~iRH') = 0 . (9)
We shall refer to this differential equation as J(i,cohi"'s
(lifferential cqtiatioii and shall denote its left-hand side
by ^(u):
1 Since i?4=0, the continuity of in implies tlie continuity of ir\ coin pare (Oa).
2"ZurTiioorie der Variations-RechnunRund der DifFerentialKli'ichungen,",/oMr-
7iiil fiir Matheinatik,\o\. XVII (.1837), p. 68; also OsfirdhVn Kldssiker, etc., Nt). 47, p. 87.
s Notice that also the derivatives of Q, R exist and are continuous, as shown
above.
50 Calculus of Variations [Chap. II
If we write (9) in the form
d:?+Rd^- + -I^"=''' (^">
the coefficients are continuous in (./Vj). Hence it follows,
according to the general existence theorem' on linear dif-
ferential equations, that every integral of (10) is con-
tinuous and admits continuous first and second derivatives
in (./V'l)-
Hence we can infer that if the condition ; i? > 0 in {X(^i)
is s((tisfic(l (111(1 if tlic dijfcreniial equation (9) has an
infegi'al n n-Jtich is (liffcrcnt from zero ttn'OHf/hont (a^o^i),
tJicii 8-./>"0 for evcri/ (idmissibte function i] not ideniicaUy
zero.
For if n is such an integral, then (8) furnishes an inte-
gral u- of (t)) of class C in {X(yi\), and therefore h-J^i). In
order to show that the equality sign must be excluded, we
introduce n instead of ir in (7), and obtain
This shows that 3-,/ can be equal to zero only when
1]' II — T)ii' = 0 throughout (.ro-ri), /. e., when 77 = Const. 11,
which is impossible since r] vanishes at Xq and j\, and n
does not.
If, on the contrary, every integral of (9) vanishes at least
at one point of (.ro-rj), Legendre's tranformation is not
applicable to the whole interval. We shall see (in §!<))
that in this case h-J can. in general, be made negative.
1 Compare E. II A, p. 194, aud Picard, Traiti d' Analyse, Vol. Ill, pp. 91, 92. If
F and consequently also P, Q, R are analytic functions, the existence theorems
for analytic diEEerential equations may be used instead. For linear differential
equations in particular, sec ScuhESlSGER, Haitdbuch der Theorie der linearen Differ-
enUalgleichungen, Vol. I, p. 21.
^12] Second Variation 51
i^l"2. .TACOBl's TRANSFORMATION OF THE SECOND VARIATION
The proof of the statement made at the end of the pre-
ceding section is ))ased upon a second transformation of
S-J due to Jacobi.'
(() Let (|,i^i) be either the interval (j^V'i ) itself or a sub-
interval of (•'o'l). and let ?/ be identically zenj outside of
(lo^i), and in (^y^i) equal to some function of class C" which
vanishes at fo and Ij.
Then if we denote by 211 the quadratic form of i], i]' :
and ap})ly Euler"s theorem on homogeneous functions, we
may write 8'-J in the form
The second term can be integrated by parts since rj" Ls con-
tinuous, and we obtain
,,(r dny^ rh /an d an\
1 Journal fitr Mathematik, Vol. XVII (1837). p. 6x. Jac obi derives (8) as well as
the iiitejrratiou of (10) from the remark that S'J = S(SJ), hence
\ r ■'■• f'"'
Jo )
where
M= r - -^ F . .
." d.r y
But
&M=*[&!/) = £*(>) "I .
Jacobi's paper, which is not confined to the simple case which we are here
considering, but which also treats the case in which the function F contains higher
derivatives of y of any order, marks a turning point in the history of the Calculus of
Variations. It gives, however, only very short indications concerning the proofs:
the details of the proofs have been supplied in a series of articles by Delauxay.
Spitzer, Hesse and others (see the list given by Pascal, loc. cit., p. 6.3). Among
these commentaries on Jacobi's paper, the most complete is that by Hesse
{Journal fiir Mathematik, Vol. LIV (18.57), p. 2.55), whose presentation we follow in
this section.
Jacobi's results have been extended to the most general problem involving
simple definite integrals by Clebsch and X. Mayer (lee the references given in
Pascal, loc. cit., pp. 64, 6.5, and C. Jordan', Cour.s cfAyialyse^Xol. Ill, Nos. 373-91).
52 Calculus of \'ariations [Chap. Ii
But 'q vanishes at ^^) and fj, and
drj dx Or) dx ^ '
Hence we obtain Jacobi's expression for the second
variation :
S-'J = e2 \\^P{r,)cl.r , (12)
which leads at once to the following result:
// there exists an integral n of the differential eqH(d{()ii
(.9) icliich vanishes at tiro })(>ints |o «'"' li <'f (^'o**!), we can
niake^ B'^J^^O, viz., by choosing
_ ( n in (Li^) ,
''"(() outside of iUi) .
/*) In th(» sequel we shall need an extension of form n la
(/V) to the ease wJien i] is of class D" . Let Cj, rv, • • •. r„ be
the points of discontinuity of t]' or ?;", Then the integral
for ^-J must be broken up into a sum of integrals from ^q ^o
c,, from Cj to c-j, etc., before the integration by parts is
applied. Hence we' obtain in this case
ao
or, if we substitute for tt-^ its value and remember that ??,
oiq
Q, R are continuous at c^, co, • • •, c,^ :
+ I ■q<if{r))dx[ . (12a)
»' so 1
c) From (12) a second proof' of (11) can be derived ; this
proof is based upon the following property of the differen-
1 It will be seen later on that it follows from this result that, in general, there can
be no extremuni in this case, see s§l+ and 16.
2 Due to .Jacobi. see the references on p. 51, footnote 1, in particular to Hesse.
§121 Second Variation 53
tial operator ^: If u and /• are any two functions of class
C", then
u<lf{v)-r^{n)=-~R{nr- ~n'c) . (13)
Heuce if it satisfies the differential equation
we get
*(r) r= -Aji{uv'-u'v) ,
dx
...^
"t
and if we imt , \^
J) being any function of class C" , and multiply by p, we
obtain
(2,u)^{pu)= -pj^(Rp'ir)
= ^£(Rpp'ur)+R(p'Hr-. (14)
But since
Pv' + 2^ri-'+ Re'- = c * (c) + ^'v ((^r + i^f')
we obtain from (14):
= i?(p'»)^ + -'|.(yr^,((^H + i?H')) . (15)
Now suppose moreover t1t(d ii is different from zero
fhrouf/houf (luli). Then we may substitute in (15) for the
arbitrary function j) the quotient
^' = «'
and since 77 vanishes at f,, and ^,. also j) will vanish at
54 Calculus of Variations [Chap, ti
fo and ^1. Hence, on integrating (15) between the limits
^0 and ^i, and substituting for ^) its value, we obtain'
c I 7, — - — -dx . (ii'i)
8- J = e-
§13. JACOBl's THEOREM
By the developments of the last two sections, the decision
reofardinij the sig^n of the second variation is reduced to
the discussion of Jacobi's differential equation (U). It is
therefore a theorem of fundamental importance, discovered
by Jacobi'^ in 1837. that the general solution of the differ-
ential equation "^(^fj^Q can Vje obtained l^y mere processes
of differentiation, as soon as the general solution of Euler's
differential equation is known.
a) Assum2:)tious^co)iC(')-itiii[/ fJu'i/cncrnl solution f{.r. a . /3)
of EuJei's differi'iiiial ('(jiictfioii :
We suppose for this investigation that the extremal Qq is
derived from the general solution by giving the constants
a. /3 the special values a^). /3^^. so that
Further, we suppose that the function f(x, a, ^), its first
1 Notice that iu the present proof we have to suppose -q to be of class C" ia <.^„li) .
It can, however, be easily proved that the result is true also for functions r) of class
C and even D . iu accordance with the results of §11, c). This follows from the fact
that ;j ■ does not occur in the identity (15) and that p'^u {Qu-^Ru) is continuous even
at the points of discontinuity of r;' or jj".
2 See the reference on p. 51, footnote.
3If the interval {x^^^) is sufficiently small, these assumptions are a conse-
quence of our previous assumptions concerning the function F (p. 12), the
extremal ('„ (p. 44) and the function R (p. 48). This follows from the theorems con-
cerning the dependence of the general solution of a system of differential equations
upon the constants of integration; compare Paisleve in E. II A, pp. 195 and "200,
and the references there given to Picard, Bexdixsox, Peaxo, Xicoletti, and
V. Escheeich; also Xicoletti. Atti della R. Ace. dei Lincei Rendiconti, 1895, p. 81ii.
For the case when F is an analytic function, compare E. II A, p. 202, and
Kn"Eser, -Leftrdw:?!., §27.
For certain special investigations concerning the "conjugate points." the addi-
tional assumption is necessary that also/<ia,/a^./^(3 exist and are continuous in A;
compare p. 59, footnote 1, and p. 62, footnote 4.
§13] Second Variation 55
partial derivatives and the cross-derivatives fj-a^fx^ ai't^ <-"<>ii-
tiiinous. and that/,.j. exists in a certain domain
A : A'n ^ u- ^ A', , I a — a„ j ^ d , I /3 — /3„ ^ r/ ,
where Xq<^Xq, ^Y^ >./'i and (/ is a positive quantity.
From these assumptions, together with our previous
assumptions concerning the function F, the assumption that
©Q lies in the interior of the region U and the assuni[)tiou
that i?(.r)>0 in (./Vi) it follows:
1. That' also the partial derivatives /„,.,/j3,. exist, are con-
tinuous and equal iof^,^,/,.^ respectively, throughout A;
2. That if we replace in the first and second partial deriva-
tives of F the arguments ij , //' l)y /(•'■, a. ^),f\.{.i\ a. /3),
these pai'tial derivatives are changed into functions of j-, a. /9
which are continuous and have continuous first partial deriva-
tives with respect to a and /S;
3. That-
F,.„.(.r,/(.r, a, fi),fjj-, a, ^))>0 , (16)
the last two statements being true throughout the domain
A provided that the quantity d and the differ-
ences Xq — Xq, J^i — j'l be taken sufficiently small;
4. The quantities d, jTq — Xq, Xi — .ri being so selected,
it follows further from equation (20) in ^6 that also the
partial derivatives /j.^,, f[,._,.a, fxx? exist and are continuous
in A.
h) The general intcyrdl of Jacohi's (lijjfereiifidl eqiia-
Hon (9) can now be obtained according to Jacobi {ioc. cif.)
as follows :
If we substitute in Euler's differential equation for ij
the general integral f{x, a. ^) we obtain
1 Compare E. II A, p. 73, and Stolz, GrundzUge der Different Uxl- und InteqraU
rechnung. Vol. I, p. 150.
2Since R{x) has a positive minimum value iu (-Vi) and F^y^y (.r,/(.r, a,^),
f^ix^a.^^) is uniformly continuous in A.
5(3 Calculus of Variations [Chap. II
-£^F^[.r,f{.V, a, ft).fj.r, a, ft)) = 0 ,
an identity which is satisfied for all values of j' , a , /3 in the
domain A and which may therefore be differentiated with
respect to a or yS. On account of the preceding assumptions,
the order of differentiation with respect to j- and a (or /3)
mav be reversed' and we obtain
where the accents denote again differentiation with respect
to j:
If we o-ive in (17) to a. /3 the particular values a = a^y
^ = ^0 and remember the definition of F. Q. B in ^11
equation (2), we obtain
Jacobi"s Theorem : If
y=f(.V,a,ft)
Is the (jeneral solution of Eider's differeniidl rqudtiou, iheii
the (liffevential equation
^{h) = (P-Q')u--^(Ru') = 0
admits the two part icidar int('(/)-(ds
^'2 = fp U' > a,, . A. I •
Corotlarijr Tlie tiro imrticular integrals r^ and r-. are.
in general, linearhj independent.
For, in order that r^ and r^ may be linearly independent,
iFrom the existence and continuity of ^ (-Fy^/a^) and li^u'W ^o^^""^'* ^^^
existence and continuity of f„^^ on account of (16).
2 See Pascal, loc. cit., p. 75.
%li] Second Variation 57
it is necessary and sufficient that their ''Wronskian deter
D{.r)
minaiit"'
I
y-i (./•) -r, (.r)
'■/ (^) '2' (^) I
be nut identically zero.
On the other hand, since /(./■, a, /3) is sup[)osed to be the
general solution of Euler's differential equation, it must
be possible so to determine a and yS that y and //' take
arbitrarily prescribed values /j-> and 2/2 ft)r a given non-
singular value of a', say .I'j.
The two functions /(d'-o, a, /3) and/,.(.r2, a, /3) of a. /3 must
therefore ))e independent, and consequently' their Jacobiau
9 (/./.,.
Jxa /.rfl
9(a, /«)
cannot be identically zero for all values of a, ^. But for
a- ttQ, /3--^^Q, this Jacobiau is identical with the determi-
nant D(.t), since fax'^La, /^^•=fx^, and therefore r^ and r.2
are linearly independent, except, possibly, for singular sys-
tems of values a^, /S^, /. r., for singular positions of the two
given })oints A and B.
We exclude in the sequel such exceptional cases and
assume that )\ and Vo are linearly independent. Then fJie
(ji'iicnil iiitcgral of JacobTs (liff'crenfial equation is
tt = CV: + C,r, , _ (19)
C\, C-2. being two arbitrary constants.
^14. JACOBI'S CRITERION
By J ac obi's theorem the further discussion of the sign
of S'-'J is reduced to the question: Under what conditions is
it possible so to determine the two constants C\, C-> that the
function u = Ci?'i + CoVo shall not vanish in {xqXi) ?
1 Compare E. II A, p. 2fil, and J. Ill, No. 122.
2Compare P., No. Vl'l, IV and J. I, No. 94.
58 Calculus of Variations [Chap. II
In order to answer this question, we construct the expres-
sion'
A (,r , ,r„) = )\ (.r) n (.r,) — r.^ (.r) /■, (.r„) ; (20)
it is a particular integral of (9) and vanishes for x = ji-q; if
it vanishes at all for values of ./■ > j-q, let Xq be the zero next*
greater than .Tq, so that
A {xq , Xo) = 0 ,
A (x , .ro) ^ () for x, < x < x^ , (21)
A(^;, .r„) = 0 .
Then it follows from a well-known theorem on homosene-
ous linear differential equations of the second order dues to
Sturm'" that every integral of (9) independent of A (./•. Xq)
vanishes at one and but one point between ./-(j and .ro .
We have now to distinguish two cases :
Case I : Xq ^ Xi .
Then every integral of (9) vanishes at some point of (./Vi)
and we obtain according to §12, a) the
Theorem: I/xq^Xi, it is j^ossible to })i((ke 8'-J:=0 hi/ a
proper choice of the function v-
1 Compare Hesse, loc. cit., i). 258, and A. Mayer, Journal fiir Matheinaiik, Vol.
LXIX (1868), p. 250.
2 "If iij , M.2 are two linearly iudependent integrals of
d u , du ,
where p and q are functions of j-, then between two consecutive zeros of u^ there is
contained one and but one zero of u.^, provided that these zeros are comprised in an
interval in which p and q are continuous." See Sturm, " M6moire sur les Equations
diH:'6rentielles du second ordre" {Journal de Liouville, Vol. I (1H.36), p. 131); also
Sturm, Coui-s d' Analyse, 12th ed.. Vol. II, No. 609. The theorem follows easily from
the well-known formula
du
I ^ -fpdx (2''*)
' dx '■ dx
where Cis a constant =1=0. From the same formula it follows that if, and Mj cannot
dMj
vanish at the same point, and that Mj and —r- cannot vanish at the same point.
Compare also Darboux, TMorie des Surfaces, Vol. Ill, No. 628, and Bochee,
Transactions of the American Mathematical Society, Vol. II (1901), pp. 150, 428.
It seems that W'eiersteass was the first who used Sturm's theorem in this
connection. Hesse (loc. cit., p. 2.57) reaches the same results in a less elegant way
by making use of the relation (22).
■^ Compare Addenda at end of book.
§1-1] Second Variation 59
For instance, by taking 77= A (,r, Xq) in f./Vo' ) and identi-
cally zero in (xQ.ri).
Hence Jacobi inferred that an extremum is impossible if
•^"o'^A ; foi'j ^'^ ^iid ^"'^ being zero, the sign of A J depends
npon the sign of 8^J which can be made negative as well as
positive by choosing the sign of e properly. This conclusion
is, however, legitimate only after it has been ascertained'
that the particular variation which causes S-J" to vanish does
not at the same time make 8^J=0.
Case II: Xq >d^i or else Xq non-existent.
In this case the particular integral
A (x, X,) = )\ (x) i\ (j-i) - /•, (x) r, (.r,)
of (9) is linearly independent of A(.r,.ro) since A(j'o, j^o) — 0,
whereas
A (Xo, x,)= — A (a-i , X^) :^ 0 .
Hence it follows from Sturm's theorem that A(^-, Xjj4=0
for XQ^x<ixi, and therefore also (on account of the con-
tinuity of A(j-, a-i)) for Xq — S^j-<ri, h being a sufficiently
small positive quantity. Now choose x^ between Xq — S and
Xq and so near to ^'0 that' Jro<j"^< j-g. Then we can apply
Sturm's theorem to the two particular integrals A (.r, a^j)
and A {x , x^) = r-^ [x) Vo (x^)
— Voix) ri(x^) and obtain
the result that
A (x, J7°) =t= 0 iu {x„Xi) .
iThe value of sl/ for this particular function t; has been computed by Erdmaxx
{Zeitschrift fur Mathematik und Physik, Vol. XXII (1877), p. 327). He finds, in the
notation of § 1.")
6'' J = - ^^R {■>■„' ) '<t>y (a-„', 7o) <t>yy{^(i\ y„) ; ( 23 )
R(x^')and<t>yU\,'.,yQ) are always different from zero; and i<>yy (j;(,', y,,) is also different
from zero except when the envelope of the set (28) has a cusp at A' or degenerates
into a i>oint. With the exception of these two cases then, Jacobi's result is correct.
Compare also §1(5.
2See §13, a). On account of (16), R{x)>0 and, therefore. r^[.r) and r,i.r) are
continuous not only in (.r„.rj ) but also in the larger interval uY|,A',).
00 Calculus of Variations [Chap. II
We obtain, therefore, according to §11 c), the
Theorem: 7/i?>0 throKgJioiii (d"o*"i), and either Xi<.Xq
or j'o non-existent, then S'J is 2)osifire for all admissible
functions tj.
Hence Jacobi inferred that in this case a minimum
actually exists, and this was generally believed until Weier-
STRASS showed the fallacy of the conclusion (1879) (see §17).
The above two theorems constitute " Jacobi's Criterion."
The value Xq is called tJie conjugate of the vcdue x^^; and the
point A' of the extremal ©o whose abscissa is Xq, the con-
jiigote of the point A whose abscissa is x^^.
§1.-). GEOMETRICAL INTERPRETATION OF THE CONJUGATE
POINTS
Jacobi' has given a very elegant geometrical interpreta-
tion of the conjugate points, which is based upon the con-
sideratioii of the set of extremals through the 'point A.
(I) This set is detined by the two equations
y =f{.v, a. ^) ,
y„=fU\n «•, A) •
The second equation is satisfied by a
at least one of the two partial derivatives
fa i-r, , a„ , p,) = 7-1 (x„) aud /p {x^ , <
is 4=0 since r^ix) and r.^ix) are two independent integrals of
(0) and E{xq)^0 (see p. 58, footnote 2). According to the
theorem- on implicit functions we can therefore solve (25)
either with respect to a or with respect to yS. But we
obtain a more symmetrical result if we express a and /3 in
terms of a third parameter 7.
If we choose, for instance,
^Loc. cif., and VorlesungenUber Dynamik, p. 46; also Hesse, loc. ciL, p. 258.
2 Compare p. 33, footnote 2.
(24)
(25)
ao, ^'
-/So;
and
A,) =
V2 U',)
§15J Second Variation 01
y=A(.ro,a,/8) (20)
and denote by 7o the value
we can solve' the two equations (25) and (20)) witli respect
to a and /3, and obtain a unique solution
a = a(y) . /3 = fS{y) ,
which is continuous in the vicinity of the point 7 = 7o and
satisfies the condition
«o = « (y<) , -^11 = /3 (ju) ■
Moreover the functions a (7), /3(7) admit, in the vicinity of
of 7o, continuous first derivatives.
Hence it follows that if we put
f(x,a(y). (i(y)) = <^ (.r , y) ,
the function (^(r. 7), its first partial derivatives and the
derivatives" ^,.,., (f),.y will be continuous in the domain
X^^x^ X^ , I y — y„ I ^ di ,
di being a sufficiently small positive quantity. Further-'
more, the equation
//„ = <^ (;r„ , y) (27)
is satisfied for all sufficiently small values of [7 — 70 | .
The equation
// = <^(.r.y) (28)
represents, therefore, the set of extremals through A in a
certain vicinity f)f the extremal (?o- ^b^ latter itself being
represented by
(v„: !/ = <t>{.r. y„) . (29)
By differentiation with respect to 7 we get
'All the conditions of the theorem on implicit functions are fulfilled at the
point a = aj|, j3=^^|, y = 7,|. In particular, the .Jacobian of the two functions
/'(x„, a, ^)-y,|andf_p(J•(,,a,^)-7with^espect toaand/3 is +Of<)ra = an, p =p„,y^y,,,
its value beinj? D (j-,,) =rj (j-q) j-j' (.Cq) — rgC-i'o) rj' (.r|j), which is different from zero,
since r, , r.2 are linearly independent and x^ is a non-singular point of the differential
equation (9).
2Also<|) will be continuous if /aa- fafi^f^a ■"■'' continuous in A.
62 Calculus of Variations [Chap. II
and thei-efore, on putting 7^7o,
<P7 l-^ ' y- ~ ^,^ (^^^ ^^/ ^^^^ _ j.^ ^^^^ ^-' (^^^^
The functions </)y('', 7o) f^nd ^Us ■'''o) differ, therefore, only
by a constant factor:'
^y {x , y„) = C A (.r , a-,,) , C =^ 0 (30)
and consequently the conJiKjufc raliic .r,, 7*/r^// (dso he
defined' as the root next grecdcr ihaii .i\^of the cqudfioii
«^y(^-,yn) = 0 . (80a)
From (30) and the properties' of A(r, .ro) it follows further
that
'^yx (. -^'o , To) ^ 0 4>^, { .r,: . y„ ) 4= 0 (31)
//) According to the" preceding results, the co-ordinates
irj, <7o' of the conjugate point A' satisfy the two equations
^ (•<■,', z/u, y..) = <l> (•'■»', y,,) — i/„' = 0 , •
% [x', , Z/n' . y,) = <^v (a-,,' , y,j) = 0 ,
and the determinant
is different from zero for x = Xq, y = Uo, 7=^7o5 its value
being ^^^•(•^'05 7o)- Hence we obtain, according to the theory
of envelopes,* the following geometriccd interpretation:
1 The same results concerning <t> (.c, 7) hold if, instead of the particular parame-
ter y chosen above, we introduce another parameter 7' connected with 7 by a relation
of till' form
7 = X(7') 1
where x (7) and its first derivative are continuous in the vicinity of 7,,, and x'(y^,) +0.
-Compare Eedmann, Zeitschrift filr Mathematik und Physik, Vol. XXII (1877),
p. 32.J.
'•' Compare p. ")8, footnote 2.
^Compare E. Ill D, p. 47. The proof presupposes the continuity orf
* , * . "t^. *^,. . *v,;' *vv i" t^^ vicinity of the point x = j-\,, y = y\^, 7 = 7o- These
§!•■>]
Second Variation
r,3
Consider the extremal
e,.: y = <!>(■''. y,>)
and a neighboring extremal of the set (28):
G: y = <f^U,y„-\-k) .
Then if 'A'! ^^ chosen sufficiently small, the curve G will
meet Gq at one and bnt one point P in the vicinity^ of ^'.
And as k approaches zero, the
point P approaches A' as lim-
iting position. Hence we have
the
TlicorcDi : The conjugate A'
of fhejioint A is ihe j)oini ii'liere
tJie exfremal Gq meets for the
firxt time tJte envelope of the set of extreinats tItroiKjh A.
d) ExAJiPLE IV : F — g(y').-d function of y' alone.
The extremals are straight lines ; the set of extremals (28) is the
pencil of straight lines through A : hence there exists no conjugate
point.
The same result follows analytically: The general .solution of
Euler's equation is
FIG. 10
hence
y = a.r + (3 ,
r, = or
1
conditions are satisfied in our case provided that Xq' lies in the interval (XqXj),
and provided that we suppose that not only the derivatives mentioned on p. 55, but
also faa ■> faB ■ //3/3 ^""^ continuous in A (compare p. 54, footnote 3).
'This means: If we choose a positive quantity S arbitrarily but sufficiently
small, and denote by J/j and J/2 the points of Pj, whose abscissae are a\,~S and
.c +& then another positive quantity <r can be determined such that every extremal
Cr for which ', A- 1< <r meets Py at one and but one point P between Jtf j and Mo .
Compare p. 35, footnote 2.
If, on the contrary, j-j be any value in the interval (XyX,) for which
•fiyi-ro, y(,)*0 .
then two positive quantities S' and a can be determined such that no extremal C' for
which \k]<<T' meets ('0 between the points whose abscissae are .i-j - S' and X2 + S\
For in this case the difference
^ (X2 + h , y(, + l-) - <!> i-ro + h , yo) =Jc<t>y{x2 + h , y^ + ek) ,
where 0 < 9 < 1 is different from zero for all sufficiently small values of ' 7i 1 and 1 k I
64
Calculus of Variations
[Chap. II
and
A(r, x„) = x — x„ .
ExAMPT.E I (seep. 27): From the general sohition of Euler's
equation
?/ = tt cosh
we get
A {.r, Xo) = sinh c cosh t\, — sinh i\ cosh r + {v — i\) sink v sinh Vq ,
where ^r - /3„ _x„-(3n
V =
Ofl
Hence we obtain (if r^ =t= (^) i^v the determination of ;ro the tran-
scendental eqiiation
coth V — r = coth r„ — r^ . (32)
Since the function coth r — r decreases from + oo to -co as v
increases from — oo to 0, and from -\-cc to -co as v increases from
0 to + 00 , the equation (32) has, besides the trivial solution v = Vo ,
one other solution v^ , and Vo and fu have opposite signs.
Hence if i\> > 0 , /. e., if A lies on the ascending branch of the
catenary, there exists vo conjugate x>oint : A(x, o-o) =^ 0 for every
a- > .To . The same result follows for ro = 0 .
If, on the contrary, ro< 0, i. e., if A lies on the descending
branch of the catenary, there always exists a conjugate point A '
situated on the ascending branch. It can be determined geomet-
rically by the following property, discovered by Lindelof:^ The
tangents to the catenary at A and at A' meet on the x-axis.
For the abscissae of the points of intersection of these two
tangents with the £r-axis are
a-» — /3,i
X = .r„ — a^, coth
and
X' = x^ — Uu coth
')^(\ Pn
and they are equal on account
of (32).
ILindelOf-Moigno, loc. cit., p. 2t)9, and LindelOf, Mathematische Annulen,
Vol. IT (1S70), p. 160. Compare also tho references given on p. 28, footnote 1.
^16] Second Variation i;.")
4^1(). necessity of jacobl\s condition
It lias already been i)ointed out that the two theorems
of ^14 which constitute Jac obi's Criterion, thougfh ofiviuir
important information concerning the sign of the second
variation, contain neither a necessary nor a sufficient condi-
tion for a minimum or maximum.
But at least a necessary condition can be derived from
the first of the two theorems by a slio^ht modification of the
reasoning: If Xq < jr^, then B-J can be made not only zero
but even negative.
This was first proved by Weieesteass in his lectures :
the first published proof is due to Eedmann.' The fol-
lowing is essentially Erdmann's proof :
1 Zeitschrift fur Mathemat'ik und Physik, Vol. XXIII (1878) , p. 367. Scheeffer's
proof (Mathematische Annalen, Vol. XXV (1885), p. 548), is not esseutially difiFereiit
from Eedmaxx's.
Weiekstrass writes the second variation in the form
^^•^ = '0 I [(P + h)v^ + 2Qr,r,-+Er,2]dx-k i V^dw [ ,
A- beiuic a small positive constant, and applies to the first integral Jacob i's trans-
formation:
5 V = e- -] I >) * ^ ^ ) d.c -k \ v-d.r [ ,
where _ \ d
*W = {(P^k)-Q)v-^{Rv) .
Then he shows that there exist admissible functions tj which satisfy the differ-
ential equation * (ri) = 0. For such a function tj, &'J is evidently negative.
H. A. ScHWAEZ {_Lectures, 1898-99) uses the following function t) :
A(x,Xf)) + ku> in (xqXq-) ,
kuj in (.Cu'a-i) ,
where k is a small constant and lo is a function of class C which vanishes at j^, and .r,
but not at u.-^,'. The corresponding value of fi-J is of the form :
8^J = e2^2fc2e(a-o-)A(.ro-,.ro)a>(ro')-ffc2r| ,
which can be made negative by a proper choice of k. (Compare Sommerfeld,
Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. VIII (1900), p. 189.)
All these proofs presuppose. j"|j'<.C| ; for the case a'Q'= x, , so far as it is not cov-
ered by Erdmann's formula (2.3) for 6^J, compare Kneser, 3/a//ie«ta?/se?!eJn«aJ('>i,
Vol. L (1897), p. .50, and Osgood, Transactions of the American Mathematical Society,
Vol. II (1901), p. 166. This case will be treated in parameter-representation in
chap. V, §.3S.
,-S
66 Calculus of Variations [Chap. II
Take jcl so that
•x-o < JC.2 < .r, and A {x2 , .r„) 4= 0 ,
and ]mt
V = pA (.r , x.i ) ,
where /a = + 1 or — 1 ; ii and v are particular integrals of
(\}) and linearly independent; hence the relation (22) holds
and takes the following form for the differential equation (9):
Riiiv'- n'r) = K , (33)
K beine a constant different from zero.
We choose p so that i^ > 0 ; this is always possible, for,
if r is rei)laced by — r, K is changed into —A".
Further, since also ii and ii — v are linearly independent.
it follows from Sturm's theorem (see p. 58, footnote 2) that
ii — r vanishes for one value of x, say x = c, between d"oand
.i\l ; hence
a ((•) = V (c) .
Now define 77 as follows:
1' u in (x^ c ) ,
7]= < V in (c x-i ) ,
TIG. 12 ( 0 in (x.2 Xi ) .
This function rj fulfils the conditions under which the
formula (12a) for B'-J holds, and since '^{r))^=0 for each of
the three segments, formula (12a) becomes:
S'J = e"i? {ua' — vv') 'f ,
which may be written, since ?^(c) = r(c):
SV =: - €'R (uv' - u'v) f = - €-i^ ,
and this is negative according to ovy agreements concerning
the sign of r.
Thus we have proved the
Fundamental Theorem III: TJte ihird iiecessiirij con-
(lifionfor (( miuinuim (maxiiuum) Is tluit
i?ii5| Second A'ariatiun 01
A (.«•,. r„) + 0 (III)
fof 'ill rahics of .r in the ojx'ii nilcmil .r,) < .r < ./'i .
('(ii-()ll(iri] : The same condition may also be written
•^"i ^ •*■!' , or else .<•„' non-existent . (HI)
/. c, if ihc cii(l-})()liif B licst J)('//(>ii(l flic ('oiijnijdfc point A' ,
there is 'no miiiiiiiinn or iiKi.riiinnii.
We shall refer to this condition as Jacobis eoiulitioii.
.1r 1
CHAPTER III
SUFFICIENT CONDITIONS
§17. SUFFICIENT CONDITIONS FOR A "WEAK MINIMUM'
We suppose hencefortli that for our extremal ©^ the
conditions
R>0 (II')
A (jj, Xo) =1= 0 for .r„ < x ^ ir, ^ (III')
are fulfilled, and we ask: Are these conditions sufficient
for- a minimum?
a) It seems so, and until rather recently it was gener-
ally believed to be so : For the reasoning of § 1 1 shows that
after an admissible function y has been chosen, AJ will be
positive for all sufficiently small values of | e [ ; hence within
the set of curves with parameter e:
U = U + ^V (1)
the curve ©q does furnish a minimum. On the other hand,
every curve 6 niay be considered as an individual of such a
set, and therefore it seems as if we must actually have a
minimum.
But a closer analysis shows that the conclusion is
wrong. For all we have proved so far is this: After a
function r) has been selected we can assign a positive
quantity^ /3, such that A,/>0 for every |e|</)^. And if
1 Compare for this section Scheeffee, " Uebor die Bedeutimg: der Begriffc
Maximum uud Minimum in der Variationsrechnuug," Muthfiiiat.ische An7ialen, Vol.
XXVI (1886), p. 197. This paper has been of the greatest importance in clearing up
the fundamental conceptions in the Calculus of Variations.
2 Notice the equality sign which distinguishes (Til') from (III); for the case
^1 ~ *"o ' which we omit here, compare the references on p. 60, footnote,
3 The notation p^ indicates that p depends on the function i?; compare E, H.
MoOEE, Transactions of the American Mathematical Societi/, Vol. I (1900), p. .500.
68
§17]
Sufficient Conditions 69
Ave denote by iii,i the inaximnni of 77 iu (./'„.ri) aiul put
^^'n^fi'riPr,, we have
I ^// I < f^'r,
f(jr all curves of the set (1) for which | e < p^ ; and vice
rcrsci, if we draw in the neighborhood (k^) of ®o ^i^Y curve of
this particular set, the corresponding e satisfies the inequality
e\<C Pr, and therefore A.7> 0.
Now consider the totality of all admissible functions 77 :
the corresponding set of values A'^ has a lower limit ko^O.
If it could be proved that /.•o> (>, then we could infer that
AJ> 0 for every admissible variation Tj for which | A^ | < k^,
and we would actually have a minimum. But it cannot be
]n-oved that k^;,> 0 and therefore we cannot infer that ©q
minimizes J.
It is even a priori clear that the method which we
have followed so far can never lead to a proof of
the sufficiency of this or any other set of con-
ditions.^
For, if we apply Taylor's expansion (either infinite or
with the remainder term) to the difference
\F^F{x, y + \u, y'+ ^y') - F{x, y, y' )
and integrate, we can only draw conclusions concernig the
sign of A J" from the sign of the first terms, if not only \ Ay ']
hut also I Ay' | remains sufficiently small, or geometrically:
if for corresponding points of ©q ^^^^ 6 not only the distance
but also the difference of the directions of the tangents is
sufficiently small.
h) If there exists a positive quantity k such that AJ^O
for all admissible variations for which
\Ay\ <: k and I-^jy'l < k ,
Kneser (LeJirbucJi, §17) says that the curve ©0 furnishes a
'■ Weak Minimum,^'' from which he distinguishes the mini-
1 First emphasized by Weieestkass.
70 Calculus of Variations [Chap. Ill
muiii as we have defined' it according to Weierstrass, as
''Strong Minimum.'''' If a curve furnishes a strong minimum,
it alwavs furnishes a forfio)-/ also a weak minimum, but not
vice versa.
If we adopt temporarily this terminology, we can enun-
ciate the following
Theorem: An e.rfrciiKil (S\^ for irln'ch ihc c())i'lifi<)ns
R>0 (II')
A (j-, .»■„) dp 0 f„r X, < .r ^ .r, (III')
(t)'e fulfilled, furnishes <ii Icosi a '• h-<'(iI< inininuiiii" for llie
integral J.
The first proof of this theorem was given by Weierstrass
(Lectures, 1H1\}), the first published proof by Scheeffer
{Joe. cif., 1886). The following proof is due to Kneser:"
We return to equation (3) of ^11 which we write in the
form :
AJ = 1 f (Po)- + 2 (^coto' + B<o'') d.r + Jr ( ' {Loy + No,") d.r ,
where (o=^Aij, and L, X are infinitesimals in the following
sense: corresponding to e\ery positive quantity o- another
positive (juantity p^^ can be assigned such that:
\L\ < a , \N\ < a ill (jc,,Xi) ,
provided that
< Pff and |w'| < po- in (-Ar'f'i) •
w
By Legendre's transformation,^ the first integral may
be thrown into the form :
1 Compare §3, h).
^ Jahresberlcht der Deutschen Mafhematiker-Vereinipunf/, Vol. VI (1899), )>. 9.".
The theorem can also be proved by meaus of We iers trass's Theorem (§20) ; com-
pare Kneser, Lehrbuch, §§20-22.
3 Compare §11, b).
§1"] SUFFICIEXT CoN])ITI()NS 71
Since the conditions (II') and ('III') are fultilltHl. there
exist' sohitions of the ditferential equation
which are of cLass C in (■r(yt\); hence it follows" that, pro-
vided the constant c be taken sufficiently small, there also
exist integrals of the differential equation
which are of class C" in (.ro./'i): let ic be such an integral, and
introduce
instead of u>' . Then A./ takes the iorui
A./ = V n [(c- + A) w' + 2,x.oi + (R + v) r1 d.r ,
where \. fi, v are infinitesimals in the same sense as L and
A'. But this may be written
X7 = V,("'[.« + r> {i + --^^.,) + (,.' + X - ^J „■]
I..
and since X. ^i. v are infinitesimals, we can choose a positive
quantity />■ so that J? — t- >( ) and r- X — /x-/(R -~v)>{) in
(■i'cfX'i). and consequently A./X), provided that co < /,• and
ity'|<A-, Q. E. D.
RoiKirJ:: We have given this theorem cliiefl}' f(n" its
historical interest: It marks the farthest point which the
Calculus of Variations had reached before Weierstrass'k
iThis follows from the connection between Le«eu(ire"s and Jacubi's <lirt('ientiat
equations; see equation (8) in §11, b).
-'According to a theorem due to Poin'care (M^canique relcxte. Vol. I, ii. jS;
compare also E. II A, p. 205, and Picard, Trtiife, etc.. Vol. Ill, p. 157). .V similar
theorem was given by Weierstrass in his lectures in connection with his proof of
the necessitj- of J a c o b i ' s condition, see p. ti5, footnote.
72 Calculus of Variations [Chap. Ill
epoch-making discoveries concerning the sufficient condi-
tions for a "strong minimum."
After these discoveries, only a secondary importance
attaches itself to the "weak minimum;"" for the restriction
imposed upon the derivative in the "weak minimum" is
indeed a very artificial one, only suggested and justified by
the former inability of the Calculus of Variations to dis-
pense with it.
c) The terms "weak"' and "strong" are sometimes also
applied to the variations. A variation conf(iinin<i a 'parani-
cfci- e
Ay = w{x, e)
is called ircak if not only
L <D (x , e) = 0 Ijut also L Mj. (jc , e) = 0
€=0 " e — 0
uniformly in (ro.ri), strong if this condition is not satisfied.
The variations of the form
as well as the more general variations which we have men-
tioned in §4, f/), are weak variations.
Weieestrass gives the following example" of a strong
variation :
\y = €SMi\^ j ,
11. a positive integer; here the condition
/.A7/r=0
1 Especially if we think of geometrical problems, for instance, the problem of the
shortest curve on a given surface between two points.
For the more general problem, however, where higher derivatives occur under
the integral sign, such restrictions are of greater importance; compare Zermklo,
Dissertation, pp. 26-31.
2 The following modification of Weierstrass's example has the advantage of
vanishing at both end-points :
1 /(.r-a-,|)m"7r^
Aw= — sin
rn and n being positive integers.
/(.r-a-,i)m n\
§18] Sufficient Conditions IS
is satisfied, but not the condition
Z.A/y'=() .
Other examples of strong' variations will occur in ■^^^18
and '2-2.
i^lS. INSUFFICIENCY OF THE PRECEDING CONDITIONS FOR A
STRONG MINIMUM, AND FOURTH NECESSARY CONDITION
From the introductory remarks of the previous section,
it follows that we have no reason to expect that the con-
ditions (I), (11), (III) are sufficient for a minimum in
the sense in which we have defined it according to Weier-
STRASS (a "strong minimum'' in Kxeser's terminology).
(i) As a matter of fact fJte three conditions (I), {II' ) and
{III') ore NOT sufficient for a strong minimum, and it is
easv to construct examples' which prove this statement:
Example III- (see p. 39):
F=,r-{!/+\f .
Here ti\, is the straight line joiuiug the two g-iveu points A and
B, say
G„ : ii ~ iii.r + }i .
Further :
-R = 2 (G//r + i)m + 1) ,
A \X f Xij) ^^^ X Xq l
hence Xq' non-existent. Let m , , m2 be the two roots of the equation
6*//- + Cym + 1=0 , viz.,
m
■=K-'+r5) = -''-'"'-
iThe first example of this kind was the problem of the solid of revolution of
least resistance; already Legexdre had shown that the resistance can be made as
small as we please by a properly chosen zigzag line; see Legendee, Ioc. cit., p. 73, in
Stackel's translation, and Pascal, lor. cit., p. 113.
2Compare Bolza, " Some Instructive Examples in the Calculus of Variations,"
Bulletin of the American Mathematical Society (2), Vol. IX (1902), p. 3.
74
Calculus of Variations
[Chai). Ill
tih
K— 7l)-"
788^
then
i? > 0 if (;/ > J//, or }ii < iii-i ,
JR <^0 if III. < ;// < nil .
Ill the former case, the first three necessary conditions for a mini-
m u m , in the latter for a ni a x i m n m , are satisfied. Nevertheless, if
-1 < III < () ,
neither a maximum nor a minimum takes place. For, in this case.
if any neighborhood (p) of (E'.i l)e given, however small, we can
always join A and B hj a broken
line 6 made np of segments of
'^^ straight lines of slope 0 and
— 1, and contained in (p). But
for such a l)roken line J = (),
whereas for @„ the integral J
is positive. This proves that
Q„ cannot furnish a minimum.
That it cannot furnish a maxi-
mum will be seen later, in § 18, c).
FIG. 13
Example V: To minimize
'^0
the given end-points liaving tlin co-ordinates (.r„, //„) = (0, 0).
(.«•,,//,) = ( 1,0).
The extremals are straight lines, and (v,, is the segment (0 1) of
the j"-axis. Further,
K = 2 ,
A (x , x„) = X — .ro .
Hence the conditions (I), (11), III) for a minimum are satisfied.
Nevertheless A J can be made
negative. For, if we choose for
(S the broken line A Pi?, the co-
ordinates of P being (! — ;:>, q),
where 0 < p < 1 , and </ > 0, we
obtain
.2
P
<)
e.
FIG. 14
§18J SUFFICIEXT COXDITIONS (5
Any neigliljorhood (p) of (So being- given, choose q <Cp: then /i v;n\
always be taken so small that A,7< 0,
h) The insufficiency of the preceding three conditions
being thns established, further conditions must be added
before we can be certain that the curve Go minimizes the
integral J.
A fourth neccssarij condition was discovered by Weier-
STKASS in 1879 and derived by him in the following
manner :
Through an arbitrary ])oint 2 : (./■■), //j) of C?,, we draw
arbitrarily a curve (5 : !i^= /{■>).
of class C.
Denoting by 4: that point of
6 whose abscissa is ./-j — h . It
being a small positive quantity,
we draw, as in §8, a curve
(5. : ij ^= ij ~- er) of class C from -f
0 to 4 and replace the arc 02 of Gq ^^J tl^e curve 042. ^
By taking It sufficiently small we can make the curve
042 lie in the neighborhood (p) of ©q.
For this variation of @q we obtain in the notation of i^8:
A.7 = .7^„ + J,,-.7„, . (3)
But according to i^S, equation (•■iO). this is equal to
A J- = /( E (.r, . u, ; //; . 7/.; ) + // (h ) . (4)
where (A) denotes as usual an intinitesimal. and the E-
function is defined l)y
E(^, Z/ ; 1^,P) =F{x, y, p) —F(-r, //, j>) - (p -j>)F,J.r. //. ,>) .
Hence follows the
Fundamental Theorem IV: The foiirih ncccssdr/j con-
dition for (( inininiuni. {jnxwimnni) is th(d
76 Calculus of Variations [Chap. Ill
E(u-, //; !/',p)^0{^0) (IV)
along^ the curve ^ofor every fin He value of p.
We shall refer to this condition as Weiersteass's
condition.
c) Applying Taylor's formula to the difPerence
F{x.,y,p) - F{x, y,p) ,
we obtain the following important relation'^ between the E-
f unction and Fy^y-:
E(.r, y, P,p) = ^-^-^F,.A'^-, Z/,P*) (5)
where
This proves
Corollary I: Condition (IV) is always satisfied if for
every point {.r, y) on ©o and for every finite value of p
i^,„-(^-,2/,p)^0 . (Ila)
Furthermore, if we define the function'* Ei{x, y; j^, p)
by the equation
^i{x,y; p,p) = — (p-pf — ^^>
when p^p, and by
El (.r ,y;p,I>) = L El {x ,y;p,p) = \ F,r,r (•** ' y ' P) (6a)
when p = J) , we obtain
Corollary II: Condition (IV) is equivalent to the
condition
^i(x, u; y',p)^0 (IVa)
along Qq for every finite p.
d) Zermelo* has given the following geometrical
1 1, e., if (x , y) is any point of So and y' the slope of Go at (x ,y).
2 Due to Zermelo, loc. cit., p. 67,
3 Compare Zekmelo, Zoc. cjf., p. 60. *Loc. cit., p. 61. M
§18]
Sufficient Conditions
(7
FIG. 16
interpretation of the relation between the E-function
and Fyy'.
Let F[p) denote the function i^(.r, //, p) considered as a
function of p alone, x, y being regarded as constant, and
consider the curve
u = F{p) . (7)
Draw the tangent PqT at
the point Pq whose abscissa is
p = y' ; and let P and Q be the
points of intersection with the
line p=^p of the curve and of
the tangent PqT respectively.
Then
^{x,y; u', ~p) = F{p) - F{y') - [p - y') F {,/)
is represented by the vector QP , and the condition
^{x,y; y',p)^0 (IV)
means therefore geometrically that the curce (7) lies entirely
above — or at least not heJoiv — the tangent PqT.
In order that (IV) may hold it is therefore:
a) Necessary that the curve shall turn its convex side
downward at p=^y', i- e., that
F"{y')^() .
This is our old condition (II), which is consequently con-
tained in the new condition (IV).
/3) Sufficient that the curve shall everywhere turn
its convex side downward, /. e., that
F'{p)^0
for every p, which is the above condition (Ila).
But neither is the first condition sufficient, nor the
second necessary.
e) Example I (see p. 49):
F=yVr\^' ;
7'S Calculus of Vakiations [Chap. Til
lieuce y
Since ^>0 along the catenary, condition (Ila), and therefore also
(IV). is satisfied.
Example III (see \)Y>. 39, 73):
hence
(v„ is the straight line joining the two points 0 and 1 , say : y = w .r + u ;
hence along G',, , ij' = m .
The quadratic in p
I? + 2/> (y» + 1 ) + 3 y/r + 4 m + 1
is always positive if />/(;« +1)>0: it can change sign if /;;O»+l)<0;
and it reduces to a complete square if 7*/ (/u+l) = <••
Hence we obtain the result :
If m g: 0 or m. ^ — 1 , condition (IV) is satisfied; if — 1 < m < 0,
condition (IV) is not satisfied, and the line 01 furnishes no ex-
tremum, in accordance with the results of §18, a).
Example V (see p. 7-4):
^^ =//+// ,:
hence along the curve ©u : ^ = 0 we have
E(.r,//: //',i3)=pHl+I^) .
which can change sign at every point of ©„. Condition (IV) is
therefore not satisfied.
§19. EXISTENCE or A "FIELD OF EXTKEMALS"
Before we can take up the question of sufficient con-
ditions, w^e must introduce the important concept of a ''field
of extremals."
a) Drp'iiitioii of a ''fielcV
Consider any one-parameter set of extremals'
y = <f>{x, y) , (S)
1 Here the ;;ymbol 4> (.'•. y) is used in a more general ^piisp than in §15.
i
§19] Sufficient Conditions TU
in whk-li our extremal ©q is contained, say for 7 = 7,). Su})-
])ose (f>{r. 7), its first partial derivatives and the derivatives
<j>^._,.. (f),.y to be continuous functions of .r and 7 in the domain
^/o beini,^ a positive quantity and A'o, -AT^ having the same
signiiication as in §11. Let k denote a positive quantity less
than (/,), and ^^. the set of points (,/•, /y) furnished by (8) as
.r and 7 take all the values in the domain
HJa- : '^i ^ ^*' ^ -i'l , \y — yo ^ ^" •
^^. may also be defined as the strip of the ,r. //-plane swept out
by the extremals (S) as 7 increases from 7,, — /,■ to 7(, r A",
J- being restricted to the interval (/Vi).
Then ^;^ is called ' a ^[ficld of c.rf rentals aboiii flic arc
@o" if t1iroiir/Jt ercrif point (.r, //) o/^^ there passes but one
EXTREMAL of the Set (S) for irli ieh | 7 — 70 1 ^ /.■ .
This means analytically that there exists a single-valued
function , / \ ^
^n^^^that y = ct>(.v,^U,!j))\ ' ^^^
^"^^ i'/'(-r,^) -y„,^A'
for every (.r, //) in g*;^..
In addition to this princi[)al property we shall include in
the definition of a field the further conditions that the inverse
function yjr(.r, //) shall be of class C in ^j., and that it shall
be possible to choose a positive quantity p so small that the
domain ^^. contains the neighborhood {p) of the extremal 6\).
/>) With respect to tJie existence of a jichi the following
theorem holds:
WJie never
<^y (j-, y„) =}= 0 throughout (-foi^i) , (10)
1 According to Kxeser, Lelirbuch, §14; the notion of a field is due, in a more
special sense, to Weierstrass ; iu its most general sense to H. A. Schwarz, Werhe,
Vol. I, p. 225. Compare also Osgood, luc. cit., p. 112.
so Calculus of Variations [Chap. Ill
/r can be talxoi >«) ,<<W(il] Hint Hie e.rfreiiials (S) fiiniisJi a fiehJ
B^. about Qq.
Proof :^ From (10) it follows that 4>y{.t', 7o) — l)eiiig con-
tinuous in {.Tcpc-^) — cannot change sign in (.r(y)\). In order
to fix the ideas suppose that
^v('*", yo)> 0 ill {x,,a\) .
Then it follows, according to well-known theorems^ on con-
tinuous functions, that k can ])e taken so small that
^y{x, 7)>0 ill 1, . <11)
Hence if we give x any fixed value .r.j contained in (jt'o-^'i)
and let 7 increase from 70 — A" to 7o- A", (^(r-i, 7) increases
continually from ^('■o, 7o— A) to ^{-To, 70: A) and therefore
passes once and but once through every intermediate value.
Hence if 79 be any value of 7 in (70 — /.-, 7o+A') ^ii^tl we jmt
(f>{x2, J-?)^^!/-!^ then the equation lJi'=4^{-'C2, 'V) has in
(7o — A:,7o+Aj no other solution but 7 = 72, which means
geometrically that through the point (x2, Hi) — which is any
point of ^^. — there passes but one extremal of the set (8)
for which [7 — 7o 1 ^ A- .
The existence of the single- valued function 7=^ "^(-^S v)
being thus established, the existence and continuity of its
first partial derivatives follows from the theorem^ on implicit
fvmctions, since
</>yU-, y)=i=^' hi 1^. .
1 Another proof is given by Osgood, loc. cit., p. 113.
2 Viz., the theorems on "uniform continuity" and on the existence of a mini-
mum. Compare E. II A, pp. 18, 19, 49; J, I, Nos. 62, 63, 61, and P., Xos. 19 VI. VII, and
100 VI, VII.
3 See p. 35, footnote 2.
The values of these partial derivatives are obtained from (9) by the ordinary
rules for the differentiation of implicit functions:
In case the function </> (a- , v) is r e gu 1 a r in 25^. , also the function ^ (x , y) will be regu-
lar in g-^.; compare E. II B, p. 103, and Haekness and Moeley, Introduction, to the
Theory of Analytic Functions, No. ir)6.
§19]
Sufficient Conditions
81
FIG. 17
At the same time we see that the set of points ^;r. is
identical with the stri}) of the .r, //-plane bonnded by the
two non-intersecting: curves
// = (f> (.<■ , y„ — /.■) and // = <^ (.r , y„ + A-)
on the oiu' liand, and tht^ two lines .r = .r,| and ./■ .r^ on the
other hand.
Finally, a neighb<^rhood
ip) of the arc ©q can be
assigned which is wholly
contained in ^j^.
For each of the two
continuons functions
0 ( ■'% To n- /»■) — </> (■^" ' 7o) a iitl
4> [■^' , 7o) — <^ (■(' , 7o — /'■) lifis
a positive minimum value in (■ro.ri); hence if p l)e the smaller
of these two minimum values, the neighborhood {p) of (Jq is
entirely contained in ^^..
The region ^j^ has therefore the three characteristic prop-
erties of a "field," and the above theorem is proved.
CoroUar/j I: The slope at a point (.r, //) of the uni<pic
extremal of the field passing through (.r, ij) is likewise a
single-valued function of ./■, //, which we denote by ^^f./-, ij).
It is defined analytically by the two equations
p U, y) — ^x (■*- - y ) ' y = "A U - //) , (.13/
which show at the same time that j>{r. //) has continuous
tirst partial derivatives in S>i^..
In case 4> ("f" , 7) is regular in Mk , also j> { .r . // ) is regular in ^,^. .
Corollarij II: The slope p(.r, y) satisfies the fotloiriixj jHirfiol
differential equation of tJie first order : '
file arguments of tlie partial derirafirfs of F heiuy ,r, i/,j>{-i\ //)
iThis corollary forms part of Hilbert's proof of Wi- ic rs t rass 's theorem;
see below, §21, and the references there given.
82 Calculus of Variations [Chap. Ill
Proof : From (13) we obtain Ijy differentiation
hence if we make use of (12) we get
Px+PPy= ^xx ■
But since 0 (a*, 7) satisfies Euler's equation for every value of 7,
we have, for every vahie of x and 7.
^xx^ y y 1^ Vx-' y y \^ ^ y -f V -'
the arguments of the partial derivatives of F being x, <t>{x, 7),
0.,.(.r. 7). Hence, if we express 7 in terms of ;r, // l)y means of (9),
we obtain (14).
(•) ApplicdiioH to the set of extremals ttirouyii the
2)0 int^ A.
We can now establish the following
Theoi'em: If for the extremal Qq ttie conditions
R>0 , (XT')
A (.r , .r„) 4: 0 for ,r„ < x^Xi (HI')
(rre fulfilled, and if a point A be chosen on tlie continuatioir
of ©0 hejjond A, but snfficienihj near to A, then the set of
extremals tlirough A furnisJies afield about ©o-
It is only necessary to choose the point A (x-,, y^) so near
to A that
1 • Ay <! x^ <C x^, ,
2. A (a-, 0^5)4=0 in {XqXi) .
The possibility of such a choice of x^ has been established
in §14.
Under these circumstances, it follows by the method
employed in §15 that there exists a set of extremals
I
through ^4.
y^<f>{x,y), (15)
iThe introduction of the set of extremals through A instead of the set through
A, which considerably simplifies the proofs, is due to Zeemelo, Dissertation, pp. 87,
88; compare also Kxeser, Lehrbuch, §§14, 17 and Osgood, loc. ctt., p. 115.
^Compare the assumptions in §13 a).
§19J Sufficient Conditions 83
where' <f>{x, 7), its first partial derivatives and the derivatives
<^,, . <^^.^ are continuous in the domain
^i) < ^ < -^1 ) I y yo I < cto J
<Jq l)ein«^ a sufficiently small positive quantity.
Moreover
<l>y(x, y„) 4= 0 iu (.roa-,) ,
since, corresponding to ecjuation (30) of §15, we have in the
present case
<i>y(-r, yu) = C. A(a?, x^) ,
where C is a constant different from zero.
Hence the set of extremals through" A satisfies the con-
ditions of the lemma given under 6) and furnishes therefore
indeed a field about (So-
'Notice that in §15 the symbol 4>{.r, y) was used with a slightly different
meaning, viz., for the set of extremals through A.
-'To the set of extremals through the r>oint A itself the lemma cannot be applied,
since for this set </>y (-'o, yii) = 0. Nevertheless it can be proved that in this case
through every point of g>^., except the point A itself, a unique extremal of the set can
be drawn. For in the present case we have: <f>{.r„,y) = yQ for every y and therefore
<t>y {^t, 5 v)=0. Honco it follows that if we define
yj;(-'-0,Y) ,
when
x4=Xo ,
when
X = x„ ,
Xo5.
x^Xi,
the function x(^-y) •'^ continuous in the domain: Xq^o^^X,, Iy— Yo'^f'n- ^^'^
x(.c,Yf,) + 0 in (XqX,), also for x = Xi^, since <f>y^{X(f,y^) +0 according to equation
(:il) of §1."). We can therefore take k so small that x(<"> 7)+0 in the domain:
A'li^x^Xj, \y-yo\^k. Hence it follows that <^y(x, 7) has the same sign through-
out the domain: XqKx^Xi, ly-Volsfc. The further reasoning proceeds then as
under b).
It should also be noticed that in the present case it is impossible to inscribe
in ^^ a neighborhood (p) of %, since the width of »/. approaches zero as
X- approaches Xq.
We shall say that the set of extremals through A forms an improper field
about (^Q.
The inverse function v^ (.r , y) and the slope p (x , y) are in this case single-valued
and of class C in g>^, except at the point (Xf, , y^,) where they are indeterminate. Rut
if the point {x , y) aiiproaches the point (x„, t/,,) along a curve S of class C lying
entirely in g>j. , then both functions approach determinate finite limiting values. The
limit of tji (x , y) is the parameter y of that extremal of the set which is tangent to
'5 at (a-(| , ?/„) ; the limit of ;j {x ,y) is the slope of the curve «J at (x„ , y^) .
84 Calculus of Variations [Chap. Ill
§20. WEIERSTRASS'S THEOREM
We are now prepared to prove a fundamental theorem
whose discovery by Weierstrass in 1879 marks a turnini>-
point in the history of the Calculus of Variations. It gives
an expression for the total variation of the integral J in
terms of the E-function, from which sutficient conditions for
an extremum can be derived.
a) The gist of Weierstr ass's method can be best under-
stood from a simple example, in which the difficulties con-
cerning the existence of a tield, which complicate the proof
of Weierstrass's theorem in the general case, can be
entirely avoided.
Example VI: In order to prove that the straight line'
01 actually minimizes the integral
we draw from the point 0 to the point 1 any curve 6:
S: y=f{'X-) ,
not coinciding with the straight line 01. "We suppose for
simplicity that S is of class (".
Through an arbitrary point
- : [jco, >J>) of CS we can draw one
and but one extremal of the set
of extremals through the point
0, viz., the straight line 02.
We now consider the integral J taken from 0 along the
straight line 02 to 2 and from 2 along the curve 6 to 1, that
is, we form, in the notation of §2, /).
t/ii2 ~r «^-'i >
the stroke always indicating* integration along the curve (S.
1 For the notation compare §2, e).
2 Notation according to Weierstrass ; Kneser, on the contrary, uses the stroke
to indicate integration along au extremal.
§20j Sufficient Conditions 85
The valiTe of this integral is a single-valued function of
.r-., which will be denoted by S(.r-,), As the point 2 describes
the curve <2 from 0 to 1, SiJCo) varies continuously' from the
initial value _ _
S (xo) = J,, (along 6)
to the end value
/S (x,) = J,i (along G„) .
Hence the total variation
A t/ = J ill fJiii
is expressible in terms of the function S(x) in the form
AJ= - [s (.,■,) -5' (.*■;)] ;
and we shall have proved that AJ"^0 if we can show that
Si-vo) always decreases or at least does not increase as X2
increases from opq to iTj.
For this purpose we form the derivative of Sixo).
The integral J02 is the length of the straight line 02:
hence dJp^ _ (a-g - Xq) + {1/2 — Vo)/' {^2}
since ?/2=/(^2) •
If we denote the slopes of the straight line 02 and of the
curve (S at 2 respectively by 7)2 and />_,, /. e.,
the previous result may be written
dJpo _ 1 -\-p2P2
dXi ~ V T+pl '
On the other hand,
'X2
'See the explicit expressions for J^.^ and J.21 below.
86
Calculus op Variations
[Chap. Ill
and therefore
dJ.
21
dx.
= -Vl+p..
Hence we obtain the result
dS{x.^
dxr,
I+P2P2
( 1 l+i>n I+Pi
from which we easily infer that
dS(x,)\<0 i^Th^lh ,
dx2 ? = 0 if p,=lh •
The latter alternative cannot take place' all al()n<^ the
curve 6. Hence it follows that
AJ>0 .
The reasoning can easily be extended to the case in which
the curve 6 has a finite number of corners.
It is thus proved that the straight line 01 fiiniishcs a
proper^ absolute^ minimum for the integral ,/.
The preceding construction may be modified^ as follows:
On the continuation of the line ©q beyond the point 0
choose a point 5, and replace
in the preceding construc-
tion the line 02 by the line
52. Accordingly the func-
tion ^(0^2) is now defined by :
and therefore
'S' (^0) = "50 ~\~ Jill J
Hence we have aarain
' If p-i = P2 for every a-.y in (XqXj) it would follow that fix) satisfies the differ-
ential equation
fM-7/Q=(x-Xa)fix) ,
and therefore a must be a straight line through 0, which could be no other than the
line Sq , since e is to pass through 1.
=* Compare §3, o) and b). 3 Compare p. 82, footnote 1.
S{xj) = J51 = J-o + J,
01 •
I
§20] Sufficient Conditions ^7
For the derivative of S (.ro) we obtain the same expression
as before, if we let, in the present case, 2h denote the slope
of the extremal 52.
b) We now proceed to the general case. We suppose
that for the extremal @o the conditions (II') and (III ) are
fulfilled. Then we constrnct as in §19, (/) a field §>^. about
®o by means of the set of extremals (15) through the point
^4, chosen as indicated in §19, d) on the continuation of ©o
beyond ^4. Since the extremal ©g is supy)osed to lie in the
interior* of the region iJ, we can take k so small that ^^ is
entirely contained in U.
For our present purpose it will be convenient to use the
numbers 0, 1, 5 to denote the points A, B, A respectively.
Let now ^ be any curve of class C joining the two points
0 and 1 (see Fig. 19), and lying wholly in the field ^k, and
let 2 be an arbitrary point of 6. Through the point 2 we
can draw one and but one extremal of the field, /. e., one
extremal of the set (15) for which |7 — 7o|^/v; let it be
denoted by
We then consider the integral J taken from 5 to 2 along Q-,
and from 2 to 1 along 6, and denote its value by /^(j'oj-
S (a-a) = J,, + J21 = r ' -Fdx + f ' Fdx , (16)
,/X5 .'-Co
the arguments of F being
X , y = <^(.«, y.,) , y'= <f>^{x, 72) ,
those of F:
^ . 1/ =/(-^) , V — fi-r) ■
For X2=^Xq and X' = Xi, S^Xo) takes the values'
S (xo) = J50 + Jo. , S (x,) = J,, , (17)
1 See §11.
2 Properly sj leaking, SCu^') is not defiued for X2 = ^'i. Hut in order that S(.X2)
may be continuous also at ar2=a-i, we must define iS(a:j) =/S(a;j — 0) ; andSCxj— 0) is
easily seen to be equal to Jjj .
88 Calculus of Variations [Chap, ill
so that ^j ^ J _ _ ,^^ = -[s (.r,) — S (x,)] . (18)
The function >S'(j';.) /s contiiinoKs and aduiits in (.^V'l) ^'
(Icrirdfifc ir/iOf<(' Vdliie is
S'i-Ti) = -E (,r.,, y, ■ 2>2, P2) , (19)
where j)^ denotes the slope of 6, j)., that of Go, at the point 2.
Weierstrass' i-eaches these results in the following way:
Let 3 denote that point of 6 whose abscissa is ^"2 + //,
Ji Ixnng a small positive quantity; and let
l)e the unique extremal of the field which ])asses through the
point ;^. Then
5' (.n + // ) - >S U-,) = (J;, + e/„ ) - (./,, + /s, ) = Jra - (J,2 + J2,) ■
But this is precisely the difference which has been computed"'
in ^8, equation (30), the curves Go? ®35 ^ corresponding to
the curves there denoted by (5, (S, 6. Accordingly we
obtain
S U, + /O - S (x,) = -h[E (x, , u, ; 2h , Ih) + {h)\ , (20)
(It) denoting an infinitesimal.
Similarly, if 4 be that point of 6 whose abscissa is
iTo — /( , we obtain
S (ar'2 — h) — S (x^) = J54 + ^42 — ^52 ,
which, according to the lennna of §8, is eqnal to
S(x,- In - S{x.;} = + h\ E(x„ y,: P2,P2) + (h)] . (20a)
Hence the derivative of S exists and its value is indeed
given by (19).
^ Lectures, 1819; the proof here given isWeierstrass's original proof with tlie
necessary adaptations to the case where x is the independent variable, and with the
substitution of the set of extremals through 3 for the set through 0.
-In applying the lemma of §8 to the present case, we have to make use of the
remarks on p. 18 and p. 35. The variation
^!l = 4> (•'• , 7-2 + ') ■" * ^•'" ' ■>'2)
is indeed a variation of the typo [Tva) of §4, d).
i
§201 Sufficient Conditions 89
As the point 2 describes the curve Q. from 0 to 1, the
function E(.ro, t/o ; ih^ P-i) varies continuously. For, on the
one hand the E-function is a continuous function of its f(jur
arguments, provided that the point [x, y) remains in the
region S, and the field ^j. is contained in IS ; on the other
hand, //2=/(*"2) ^^^^ Pz^f (-^'i) ^^'^ continuous in (.rod",} and
the slope 7^2 "^ ®2 ^^ - i^' according to §18, />), a continuous
function of Xo, iJ->.
Integrating (19) between the limits .r,) and ir^, and
remembering (18), we obtain therefore for the total varia-
tion A J the expression^
A J" = I E (a-, , y, ; p^ , pi) dx^ . (21)
We shall refer to this important formula as "Weier-
STR ass's theorem."'
The theorem remains true for curves ^ of class /)'. For,
sup[)ose the curve (S to have a corner at the point 2. Then
(20) and (2C)a) still hold if we understand by Jh the progres-
sive and regressive derivatives of f[x^ respectively. The
function S{x) is therefore continuous at d^'o and admits a
progressive and a regressive derivative. Hence it follows"
that (21) still holds when (S has a finite number of corners.
c) Instead of first computing the increments S{x-2, ±h) —
S{xo), Kneser (Lehrhiich, §20) and Osgood (Joe. cit, p. 116)
compute directly the derivative S'ixo) by applying the
theorem on the differentiation of a definite integral irith
respect to a parameter. Supposing for sim})licity that 6 is
of class C, it follows from the properties of the function
^{x, 7) that S{x-2) is continuous and differentiable in the
1 The theorem remains true also for the " improper field " &^. formed by the set
of extremals through the point 0, and for a curve 5 which lies entirely in this field &^ .
For formula (19) holds also in this case at every point of i^ with the exception of the
point 0. Integrating (19) from x^-\-h to x^, and passing to the limit h = Q, we
obtain (21) since P2 approaches a determinate finite limit ; compare footnote 2, p. 83.
2 Compare E. II A, p. 100, and DiNi, §194.
90 Calculus of Variations [Chap. Ill
interval (ifV'!) ^i^d that the derivative can be obtained l)y
applying to the definite integrals J:^^ ^nd J-^x the ordinary
rules' for the differentiation of a definite intesfral with
respect to a parameter and with respect to the limits.
Accordingly we obtain in the first place
^=-F(.r,,^,,p,). (22)
In differentiating the integral
J-.X2
Fix, <^(.r, y,), <i>^{x, 72)) dx ,
we must remember that 7., is a function of Xo defined by the
equation <^(^2, V.) =/>2) , (28)
which expresses the fact that the curves ©2 and (S both pass
through the point 2.
Accordingly we obtain:
^^ = F{x„y,,p,)+fJ\F„cf>^+F^,<f>_,,)'^^dx ,
the arguments of (f)y, (f>j.y being ,r, 7.,.
From our assumptions concerning 4>(x, 7) it follows that
<^xy(a^> y2) = <l>yx 0^, 72) •
Applying then to the second term under the integral sign
the integration by parts of §4, and remembering that the
function y = (f)(x, 70) is an integral of Euler's differential
equation ,
F —--F. = 0
'■' dx " '
we obtain the result:
-^ = F(x,,y,, p,) + ^ [f,,. (x^ , 2/2 , P2) <f>y (a?2 , 72)
— Fy (^5 , y-. , P5) «^y(av. , 7i)] '
where jDr^ = (j>^{x-„ 70).
1 Compare E. II A, p. 102, and J. I, No. 83.
§211 Sufficient Conditions HI
But since the extremals of the set (15) all pass throiigh
the point 5 : (a--,, ^5), we have
y^=ct>{x^, y)
for every 7 ; hence
<^r(a^5, y) =0
for every 7, and therefore in particular
<f*y{X5, 72) =0 .
On the other hand, if we differentiate (23) with respect to
^2.. we get
therefore
at/ 52
CtJbo
I / \ dy2 —
't>y{i^2,y2)-^ = p2-P2 •>
F {X2 , 7/2 , 2h) + (p2 — P2) Fy {^2 , 2/2 , P2) ' (24)
Combining (22) and (24) we obtain again the fundamental
formula (19).
§21. hilbert's peoof of weierstrass's theorem
Weierstrass's theorem can be extended' to any set of
extremals constituting a field about the arc @o, /. r..
Whenever tlie extremal @o crtw he surrounded bij afield,
the total variation AJ^=J^ — J^, for any admissible eurve 6
lying wholly in the field, is expressible by Weierstrass's
formula:
E{x,y; p,p) dx ,
where {x, y) is a point of (S,p the slope of (S at (x, y), and
p the slope at {x, y) of the unique extremal of the field
passing thi'ough (x, y).
iThe extension seems to be due to H. A. Schwarz, who has siven the general-
ized theorem in a course of lectures in 1898-99.
92 Calculus of Variations [Chap. Ill
The following elegant proof of the generalized theorem
is due to HiLBEET.'
Suppose ^1^. is a field of extremals about our extremal Q^.
Ill §>;;. we draw any curve (S of class D' : ij ^=f(x), joining A
and B. Now let p{.r, Jj) be an arbitrary function of x, y
which is of class C in ^i^., and consider the integral
J*= j [f{x, y,p{3c,y))
+ (y'-p(''<',y))J^,j(^,y,p(x,y))] dx (25)
taken along the curve 6 from A to B. The value of ,7* will,
in general, depend upon the choice of the curve 6 ; we ask :
How must we choose the function p(.r, /y) in order that the
value of J* may be independent of the choice of the curve 6
and dependent only upon the position of the two end-points
.4 and B ?
Our integral J* is of the form
I \M{x, y) +N{x, y)y'^, dx ,
and it has been seen in ^7, d) that the necessary and suffi-
cient" condition that such an integral should be independent
of the path of integration is that
In the present case we have
M{x, y) = F{x,y,iS)—pFy.{x,y,p) ,
N{x, y) = F^.{x, y,p) ;
^^^^^ My = F,-p{F,,,^ + p,F„^) ,
N — F ■ 4- » F ■
iSee GSttinger Nachrichten, 1900, pp. 253-297, and Archiv der Mathematik und
Physik l3). Vol. I (1901), p. 231 ; also the English translation by Mrs. Xewson, in the
Bulletin of the American Mathematical Society (2), Vol. VIII (1902\ p. 473; further,
Osgood's presentation in the Annals of Mathematics (2), Vol. II (IWl), p. 121, and
Hedrick, Bulletin of the American Mathematical Society (2), Vol. IX (1902), p. 11.
2 Notice that the region gi^., to which the curves e are confined, is simply con-
nected.
§2n Sufficient Conditions 0;J
Hence, in order that the value of the intci/rol J* niaij he
Independent of the path of integration 6, it is necessorf/
and sufficie)d tlud the f miction j){.r, ij) sidixfjj the ])arfi<(l
differential eqnation
(p. + PPy) F,-y +pF,fy + F„,, - F,, = 0 , ( 2r,)
the arguments of the partial derivatives of F being
But this differential equation is identical with the ditfer-
ential equation (14) which is satisfied by the slope at fa", y) of
the extremal of the field passing through (.r, //). Hence the
value of J* will be independent of the choice of the curve (5,
if we select for the function p the slope just defined. In
the sequel p will have this special meaning.
The invariance of the integral J* being established, we
select for the curve (5 first the extremal @q ; then we have all
along (Sq :
y' = p{x, ij) ,
because (Sq is the unique extremal of the field which passes
through a point of ©q- Therefore (25) reduces to
J*=j 'f{x, ij, >/')d.r = J,.^^
On the other hand, if we select for (5 any curve 6 of class
D' , dilferent from (?o- ^^^^^ j'^i^^i^^g -^ ^^^^ •^^ ^^'^ K'^^
J*= j [F{x,7/,2^) + {p-p)F,j.{x,y,p)\dx ,
where p^ ij' denotes the slope of 6 at the point (.r, y). Both
values of J* being equal on account of the invariance of J*,
we obtain an expression for J,, in terms of a definite inte-
gral taken along Q. This expression we use in forming
the total variation
A t/ ^ t/^ t/ej, •
Then we obtain
94 Calculus of Variations fChap. Ill
- {P - P) Fy. (x, y,p)j dx ,
which is the desired extension of Weierstrass's theorem,
since the integrand is equal to E(j", 7/ ; jj, J>).
§22. SUFFICIENT CONDITIONS FOR A STRONG MINIMUM '
Weierstrass's theorem leads now immediately to suf-
ficient conditions for a strong minimum:
<i) Suppose there exists a field ^^. about ©y such that at
every point of ^^^
E(x,y; p{x,ij),p)^<> (27)
for every finite value of J), j) {x , ij) denoting again the slope
at (.r, y) of the extremal of the field passing through (x, y).
Then it follows from Weierstrass's theorem that
A./^O for every curve 6 of class D' drawn in ^^ from A to
B, and moreover that At/_>0 unless
^{x,y;p{x,y),Tj')=Q (28)
all along the curve 6.
From the definition of the E-function it follows that (28)
holds at a point {x^ y) of 6 whenever
y' = p{«^,y) ,
i. e., whenever the extremal through [x, y) is tangent to 6
at {j', y). This can, however, not take place at every
point of &, unless 6 completely coincides with ©q- For^ the
value of the parameter 7 of the extremal of the field passing
through that point of 6 whose abscissa is x, is determined
by the equation
f{3c)^^{x,y) ,
1 Compare for this section also Hedeick, Bulletinof the American Mathematicat
Society, Vol. IX (1901), p. 11.
2 This proof is due to Kntsser, Lehrhurh. §22 ; see also Osgood, loc. cit., p. 118.
§22] Sufficient Conditions 95
from which we derive by difPerentiation
fi^c) = <f>Jx, y) + <ky {x, y) ^ ,
or according to (13)
y'-p{sc, y) = <f>y{x, y)^ .
But according to (11), ^y{or, 7)4=0 ; if therefore y'=:p(oc, y)
at every point of (S, we should have
dy
— = 0 throughout {x^x^ ,
or 7 = const., i. e., 6 would itself be an extremal of the field,
which could be no other than Gg, since 6 passes through the
point (j"i, /yi) and ©q is the only extremal of the field which
passes through (;ri, 2/1).
Hence, if instead of (27) the stronger condition'
^,{a-,y; p{:x,y),p)>0 (29)
is satisfied at every point (a*, y) of ^;,. and for every finite j3, it
follows that A./>0 for every admissible curve (S drawn in
the field ^^,.
In the terminology of §3 we have therefore the result
that whenever [27) is satisjied, ^q furnishes a minimum for
the integral J; if moreover {28) is satisfied^ the 7ninimum is
a '^proper minimum.''''
Example III (see pp. 73, 78):
The set of straight lines
y = mx + y
parallel to the extremal AB furnishes evidently a field about @o,
and for this field
p{x, y) =m .
Therefore
1 Compare (6) and (6a).
2 It is even sufKcient that (27) and (29) be satisfied in a neighborhood (p) of e^
inscribed in g>^, ; the same remark applies later on to (lib').
96 Calculus of Variations [Chap, ill
When m > 0 or m < - 1 , condition (29) is fulfilled, and therefore
the straight line AB actually minimizes the integral
dx
in these two cases.
b) The sufficient conditions thus immediately following
from Weiers trass's theorem are, however, in general
inconvenient for applications, and it is therefore important
to remark that they can be replaced, under certain addi-
tional assumptions either concerning the curves 6 or con-
cerning the function F, by simpler conditions.
From the relation (5) between the E-function and F^y.,
it follows that both conditions (27) and (29) are always
satisfied when rr. /- „ rA -^ n /tti '\
^y V ('^' y^p) > U (lib )
at every point' (x, y) of ^^ and for every finite value of /;.
Hence if we remember the theorem concerning the exist-
ence of a field (§l-\ h)), we can state the following theorem:
Fundamental Theorem V:'- If the oxfremal (Sq.AB
does not contain tJte conjugate point to A, and If fln'ther
F,J■,,{.l■ , y, p) >() (lib')
at every point {x, y) of a certain neighborhood of Cr,, for
every p'nite value of p, then Gq ((ctually minimizes the
integral ^^\
J — i F{x, y, y')dx .
Corollary : The minimum is moreover a ^'■proper mini-
mum,'''' i. e., AJ>0 for every admissible variation of the
curve ©0 in ^ certain neighborhood of ©q-
1 It is even sufficient that (27) and (29) be satisfied in a neighborhood (p) of c^
inscribed in g>j, ; the same remark applies later on to (lib ).
2 See Osgood, loc. cif., p. 118; compare, however, below, the remark on p. 90,
footnote 1.
§22] , Sufficient Conditions 97
For a so-called rcf/itldr jtrohJem (compare §7, c)) it is
therefore sufficient for an extremum that the arc AB does
not contain the conjugate to the point A.
Example VII : '
F = g{.v,y)Vl + u" ,
g(x, y) being- a function of x and y aloQe, of class C" in a certain
region S . Here
g{x, y)
Hence every extremal A B which lies in the interior of S and which
does not contain the conjugate point to A, furnishes a minimum
provided that (j{x, y) > 0 along AB. For g{x, ?/),being continu-
ous in a certain neightorhood of yl-B and positive along AB, will
also be positive in a certain neighborhood oi AB, so that (lib) is
satisfied.
This covers the case of Examples I and VI, in which
<j{-i-, U) = y . and 1 (1)
respectively; and also the case of the "'brachistochrone'' in which
1
^(''■' ^) = ,, =rT •
All three functions are positive along the respective extremals.
On account of the extension of Weier stress's theorem
given in §21, Theorem Y may be replaced by the following :
If the extremal Gq can he surrounded hij a field and if
Condition (lib') is fulfilled, then ®q aetuallij minimizes the
integral J.
Frequently the existence of some particular field about
the arc Qq is geometrically evident ; in such cases the second
form of the theorem is more convenient.
1 Geometrical Interpretation (Erdmavn) : Let a straight line move perpendicu-
larly to the J-, y-plane along the curve y=f{x) from A to B, The area of thac por-
tion of the cylindric jiurface thus generated which lies between the ar, (/-plane and
the surface : z = g{x , ;/} is equal to
!i[-i--,y) 1 l-ry"^dx.
^0
98 Calculus of Variations [Chap. Ill
Example VIII : ' To minimize the integral
•^-r« y
the admissible curves being confiued to the upper half -plane {y > 0).
Here the extremals are semi-circles having their centers on the
a'-axis. If , / — 7 ; — ; — 5
is the particular semi-circle passing through the two given points,
the set of concentric circles
U = 1 — (jc - a^f + Y = <l>{.v, y)
evidently furnishes afield about (So. Moreover (lib) is fulfilled
throughout the upper half-plane. Hence the semi-circle thi'ough
the two given points actually minimizes the integral J.
.Remark: Though the above theorem is the one which is
most important for applications, it should be observed that
it assumes much more than is necessary. Indeed, the con-
dition {lib') is bij no means necessarij. not even the milder
condition ^^^,^, (^ ^ ^ ^ -^ = 0 (lla)
at crcrij point (./', /y) of ©0 and for every finite p.
This is illustrated by Example III (see pp. 73, 78, 95). For
^^'^ F„,. {x,y,p) = 2 (6F' + (Sp + 1)
can take negative as well as positive values at every point (x, 2/), and
nevertheless, as we have seen above, a minimum takes place when
m > 0 or »i < — 1 .
c) Question of necessary and sufficient conditions.
From Weierstrass's results concerning the sufficient
conditions for the problem in parameter-representation (see
§28), one is led to expect that the conditions" (I), (III'),
1 Given by Osgood, loc. cit., pp. 109, 11."), where also a geometrical interpretation
will be found.
2 The accent indicates the omission of the equality sign in conditions (III) and
(IVa); compare pp. 68, 76. (.II) may be omitted, since it is contained in (.IVa');
compare §18, equation (6a J.
§22] Sufficient Conditions 99
(IVa') are sufficient for a minimum. Leaving aside the
exceptional case when in one of the inequalities (III), (IVa)
the equality sign takes place, we should then have reached
a system of necessary and sufficient conditions.
The analogy of the problem in parameter-representation
is, however, misleading in this case. As a matter of fact
the three conditions (/), (III'), [IVa') are not' sufficient
for (I, minimum ivithout some additional assiimptio)is, not
even if [IVa') he replaced htj the stronger condition
l\,,,{x,y,p)>0 (Ila')
(d ererij point [.r, y) of (Sq for every finite value of p.
To prove this statement it suffices to construct a single
example in which the conditions in question are fulfilled and
in which, nevertheless, no minimum takes place. Such an
example is the following :
Example IX :^ To minimize the integral
J= { la if - -Lbuy'' + 'Ib.ri/"] dx ,
• 0
<i, h being two positive constants, with the initial conditions
y = 0 for x = 0, and ?/ = 0 for x = 1 .
Here liluler's equation reduces to
-.'/"^Vv = <>.
^^''''''' ■ F,,, = 2a - 246^/y ' + 24.bx>r •
The only extremal through the two given points -.1.(0, 0)
and B[l, 0) is the straight line :
do-. z/=0.
iThis statement seems to contradict directly the theorem given in Osgood's
article, loc. cit., p. 118. But it is to be remembered that Osgood makes (p. 108) the
assumption that F , ,{x ,y,p) +0 in a certain neiqhborhood of (*,). This assump-
tion, together with (Ila), is equivalent to (lib')-
2 See BOLZA, "Some Instructive Examples in the Calculus of Variations," Bulle-
tin of the American Mathematical Society (2), Vol. IX, p. 9.
100 Calculus of Variations [Chap. Ill
The set of extremals through A is the pencil of straight
lines through A; hence there exists no conjugate point, and
condition (III') is fulfilled.
Further
E,(a-, y; I/', p) = (a - ^byu'+ 6bxy")
-\-p(-iby + 4:bxy') + 2bu-p ■
hence along ©(,:
El {x , Mx) ; n {x), p) = a+ 2bxp' > 0 . (IVa')
TJie iliree conditions (I), {HI'), {IVa) are therefore mtis-
jied, even the stronger condition
F,y,- (x , /, (x), p) = 2a + 24.bxP > 0 . (Ila')
Nevertheless the line ©o '^^x'-"^ ^'^^ minimize the inte-
r - j^^zS ') fjral J.
For, if we replace the line
ABhj the broken line APE,
the co-ordinates of P being
FIG. 20 £r = /i>0 and y-=l\ the
total variation of J is easily found to be
A J = /.■•^[- ^V^ + « + ^^ + {h) ,
where {h) is an infinitesimal.
Now let /o > 0 be given, as small as we please, then choose
I /v |</3 and let h approach zero, keeping k fixed. Then since
6>0 it follows that A,/<0 for all sufficiently small values
of A, which proves that the line AB does not minimize the
integral J.
The complete solution of the general problem which we
have considered in these three chapters would require the
establishment of a system of necessary and sufficient condi-
tions. The above example shows that it will be necessary
to add a fifth necessary condition before the complete solu-
§22] Sufficient Conditions lOl
tion of the problem is reached. We have therefore to con-
clude this chapter with the statement of a gap in the theory
so far as it has been already developed.'
(/) We add a table of ihe various conditions which have
occurred in the problem to minimize the integral
J = 1 F(x, y, y')dx ,
the end-points being fixed:
1) The minimizing curve @o-^=/o('^) niust satisfy the
differential equation
n-^^V = 0. (I)
[Euler''s equation, p. 22 ; assumptions concerning its general
solution, p. 54.)
2) Fy.^. (x,Mx), /; (x)) ^ 0 , in {x,x,) . (II)
{Legendre''s condition, p. 47)
F,y,,.(x,fjx),p)^0 , (Ila)
in {x(^i) for every finite p (pp. 76 and V)8).
F^.,Jx,y,p)^0 , (lib)
ilf we modify the problem by the addition of a slope restriction, i. e., by sub-
jecting the admissible curves to the further condition that their slope shall not
exceed a finite fixed quantity, say
then the three conditions (I), (III), (IVa) are sufficient for a minimum.
For the function
Bi{x,<j>{x,y); <t>x(-e,y),p)
is continuous in the domain
and positive for y = yq .
Since the domain 33^, is cloned, it follows from the theorem on uniform continuity
that we can take k so small that
^l{x,<f>{x,y); <f>j:(-«,y),p)>0
throughout the domain B/^ , which proves the above statement.
102 Calculus of Vakiations [Chap. Ill
for every {x, y) in a certain neighborhood of ©q and for every
finite p (p. 9()).
3) ^.^^0 , (HI)
a-Q being the conjugate of .r^. [JacobPs condition, pp. 08,
59, ()7.)
4) E {x,Mx) ■ /: [x), p) ^ 0 , (IV)
in (xfyTi) for every finite J>. (Wcierstrass's cotKlifion, ]>. 70.)
E.{a-,/„(,r); /„'(a"),f>)^0 , (IVa)
in [xfyV^) for every finite p (p. 7(>).
The omission of the equality sign in (Il)-(IVa) is indi-
cated by an accent.
Conditions (I), (II), (III) are necessary, conditions (I),
(II'), (III') are sufficient, for a weak minimum.
Conditions (I), (II), (III), (IV) are necessary, conditions
(I), (lib'), (III') are sufficient, for a strong minimum.
^23. THE CASE OF VARL\BLE END-POINTS'
We have so far always supposed that the two end-points
1 Three essentially difPerent methods have been proposed for the discussion of
problems with variable end-points :
1. The method of the Calculus of Variations proper: It consists in computing
SJ and a- J either by means of Taylor's formula or by the method of differentia-
tion with respect to e, explained in §4, b) and d), and discussing the conditions
aj = 0, sV^O. The method was first used by Lagrange (1760) ; see Oeuvres, Vol. I,
pp. S.'JS, .343. He gives the general expression for SJ when the end-points are vari-
able, viz.:
U= C ' Sy (f^- £.F„) rfx + [fB. + F,^.Sy]l ,
and derives the conditions arising from 5J = 0.
The second variation for the case of variable end-points was first developed by
Eedmann {Zeitschrift fur Mathematik und Physik, Vol. XXIII (1878), p. 364). He finds
^2^^ r''"' j;(M8y'-it'8y)^to
.-11
+ ^FS'x + F,^.B'>j + 2F,/.rSy + 2F^^. SxSy+ g' Sx^-f (ir,^,„+ F,^.^. '^) &y- \^ ,
where u is an integral of .Jacobi's differential equation. By considering such spe-
§23] Sufficient Conditions 10:3
of the required curve are fixed. Tii this section we propose
to consider the modification of the problem in which one of
the end-points, say 0, is fixed, whilst the other, 1, is movable
on a given curve 6.
Suppose the curve
— which we suppose to be of class C and to lie in the inte-
rior of the region S — minimizes the integral J with these
cial variations for which Sy = Cu , he makes the integral vanish and thus reduces the
question to the discussion of the sign of the remaining function of the variations
Sx^., Sy.^ ^ Xp ^'.Vj. These variations are connected by relations which depend
upon the special nature of the initial conditions. For instance, for the initial con-
ditions considered in the text the expression for S J reduces to the expression (36)
for J"(.rj) multiplied by Sx^ .
For the general integral
x:
-^^••'•, .'/n 2/2' • • • ' Z/j, ' 2/1', Z/o', • . . ,.'/,/)t7.B
where t/j , 2/2 j/„ are connected by a number of finite or differential relations,
the second variation in the case of variable end-points was studied by A. Mayer,
Leipziger Berichte (1896), p. 4.36; for the integral in parameter-representation
f"
J= I F{,x,y,x\y')dt
by Bliss, Transact ious of the American Mathematical Society, Vol. Ill (1902), p. 1.32
(comijare §.30).
2. The method of Differential Calculun: This method is explained in a general
way in Dienger's Grundriss der Variationsrechnung (1867). It decomposes the
problem into two problems by first considering variations which leave the end-
points fixed, and then variations which vary the end-points, the neighboring curves
considered being themselves extremals. The second part of the problem reduces to
a problem of the theory of ordinary maxima and minima. This method has been
used by A. Mater in an earlier paper on the second variation in the case of variable
end-points for the general type of integrals mentioned above {Leipziger Berichte
(1884), p. 99). It is superior to the first metliod not only on account of its greater
simplicity and its more elementary character, but because — by utilizing the well-
known sufficient conditions for ordinary maxima and minima — it leads, in a certain
sense, to sufficient conditions if combined with Weierstrass's sufficient conditions
for the case of fixed end-points. For these reasons I have adopted this method in
the text.
3. Kneser's method: This metliod, which has been develop(>d by Kneser in his
Lehrbuch, is based ujwn an extension of certain well-known tlieorems on geodesies.
It leads in the simplest way to sufficient conditions, but must be supplemented by
one of the two preceding methods for an exhaustive treatment of the necessary con-
ditions. A detailed account of this method will be given in Chapter v.
104
Calculus of Variations
[Chap. Ill
initial conditions. Then we must have AJ^i) for every
curve (S of class D' which begins at the point 0 and ends at
a point of the curve 6 and which lies moreover in a certain
neighborhood' M of Qq.
a) Among the totality of these
"admissible curves" we consider in
the first place those which end at the
point 1. For these also the inequality
FIG. 21 ^j^() must hold, and therefore all
the conditions which we have found to be necessary in the
case of iixed end-points must be fultilled in the present
case.
The arc (Sq must fhcrcfurc be an ('.rircmal. L,c(jc)i(lvvs
condition
F,.:r^O (II)
y u
must he satisfied along Qq, and the rf>iijiif/ate point (>' ta 0
must -not lie between 0 ojid 1.
We suppose in the sequel that the arc Qq is an extremal,
that the condition
F^,,{'^,y,p)>0 (lib')
is fulfilled at every point (x, y) of a certain neighborhood of
(E'q for every finite value of p and that the arc Qq does not
contain the conjugate point 0' (Condition III').
b) Further necessary conditions are obtained by consid-
ering variations which do vary the end-point 1. Various
methods" have been proposed for this purpose. The follow-
ing elementary method reduces the further discussion to a
problem of ordinary maxima and minima:
If the extremal ©q minimizes the integral J in the sense
explained above, then ©g must, in particular, furnish a smaller
1 Compare §3, 6) ; we may for instance choose for H the special neighborhood (p)
used in the problem with fixed end-points (§3, c)), increased by a semi-circle of radius
p with the point 1 for center.
2Compare footnote 1, p. 102.
§23] Sufficient Conditions 105
value than (or at most the same value as) every extremal
which can be drawn from the point 0 to the curve 6 and
which lies in a certain neighborhood of ©q.
And since under the above assumptions (lib') and (III')
each of these extremals — (when its end-points are consid-
ered ac fixed) — minimizes the integral J^ it seems' self-
evident that also the converse is true.
Let then
y = cl>{x,y) (30)
represent the set of extremals through the point 0, and let
7o denote again the value of 7 which corresponds to @q.
From the above assumptions (lib') and (III') it follows that
this set furnishes for I7 — Jo\^k an (improper) field^ ^;^
about the arc @q if k is taken sufficiently small.
Hence, if 2 : (x2, 2/2) b© ^^J point of the curve 6 in a
certain vicinity of the point 1, then there passes one and
but one extremal
^2- y = <t>(x,y2)
of the field through the point 2. The parameter 7^ is a
single-valued function of x^, y-i of class C : 72 = '«/^(a"2, ,'72)-
If
y=f{x)
is the equation of the given curve, which we suppose to be of
class C" , then yo =f{x-^ and 72 = ''/^ (j"2 , fipc-i)) •
Hence the integral J taken along the extremal ©2 f I'om the
point 0 to the point 2 is a single-valued function of ^v, say
lit will be seen under e) how far this conclusion is correct.
2 Compare p. 83, footnote 2. lu the present case the fiold g-^. consists of all
points {x , y) furnished by (30), when .r, y are restricted to the domain
«(, S a; ^ Xj , I V - 7q I ^ A; ,
where Xj is some value greater than x^ ; fc is supposed to be taken so small that (lib')
holds throughout S"/^ and that "^^ (a:, y) +0 throughout the domain
aru<a-5Xi, ly-Vol^fc-
106 Calculus of Variations [Chap. Ill
J{x^) = I F(x, (t>{x, y.,), cf>_^{x, y2))dx .
And this function Jixy) must have a minimum for X2=^0Ci.
Therefore we must have
J'{x,) = 0, J"(x,)^0 . (31)
c) The derivative of the integral J^X2) has already been
computed' in §20, c) (equation (24)). Accordingly
J'ix^) = F{x2,y2, P2) + {Th — 2h) Fy. (.^2 > 2/2 , Pi) , (3'2)
where ])2^=4'x{^2j J2) is the slope of the extremal ©2, and
p2'^f'i'y2) the slope of the curve (S, at the point 2.
Hence we obtain the result :
The co-ordinates x, y of the movable end-point must
satisfy the condition^
F{x„y„y[)-\-{yl-y[)Fy.{x„y„yi) = 0 , (33)
where y[ and y[ refer to the extremal @o ^^^ to the curve S
respectively.
If this condition is satisfied we shall say that the curve 6
is TRANSVERSE^ to the extremal @o f*^ i^^^ point 1.
Equation (33) together with the two equations
/(a-o, a, IS) = y^ , f{x,, a, /?) =f(x,) ,
determine in general the two constants of integration a, /3 in
the general solution of Euler's differential equation, as well
as the abscissa Xi of the point 1.
We suppose in the sequel that condition (83) is fulfilled.
d) We next proceed to the computation oi J" {X2) . From
(32) we obtain
iWe suppose that the co-ordinates of the movable end-point do not occur
explicitly in the function F{x, y,y'); if they do occur, another term must be added
to the expression of J'(Xo). Compare for this case Kneser, Lehrhuch, §12. An
example of this exceptional case is the brachistochrone; compare LindelOf-
MoiGNO, Calcul des variations. No. 113, and the references given in Pascal, Varia-
tionsrechmuig, §.31.
2 In accordance vyith §8, end.
3In the use of the word "transverse" I follow Osgood, Ioc. cit., p. 112.
Kneseb, who first introduced the term {Lehrbuch, §10), used it with a slightly differ-
ent meaning; he says: the extremal (Sq is transverse to the curve 6 if (33) is satisfied.
§23] Sufficient Conditions 107
But
hence
^72
and - — is determined by
Substitutinij these values for
fly 2 dp.2 dp2
and remembering that on account of Euler's equation
we obtain for x=^Xi the following result:'
Let Ai and B^ denote the expi^essions :
A, = F,+ i2!j,-y;)F,^+yrF^+(V:-yiyF^y , (35)
-Si = {yl- yifFyy. ,
the arguments of the derivatives of F being x^, ?/i , y{ ; then
97 Ui, 7u)
For the further discussion of the inequality J"{xi)^0, we
leave aside the exceptional case where y\=y\, i. c, we sup-
pose that the extremal ©q cmd the curve 6 arc not tangent to
each other at the point 1. Then Pi>0, since we have
moreover already supposed that Fyy-^0.
'Given, in a slightly different form, by Bliss, Mathematische Annalen, Vol.
LVIII (190S 1, p. 77.
108 Calculus of Variations [Chap. HI
According to equation (30) of §15, we have in the nota-
tion of §§13 and 11:
and therefore
<l>yx{^i> yo) = c — ^ — .
Now let ii{xi, x) denote the function
H {x, , x) = AA (^. , ^0 + B, ^^^Q^'"^^ , (37)
then the expression for J"(xi) may be written
^ '^ A{x,, X,)
The function
A (.r, . x) = r, (x,) i\ {x) — v., (x,) ?•, (,r)
is an integral of Jacobi's differential equation and van-
ishes for ic = rf"! . The function H [x^ , x) is likewise an inte-
gral of Jacobi's differential equation, since it is linearly
expressible in terms of ri[x) and ro(x). Since 5i>0 and
r, (x,) r: (x,) - r, (x,) r/ (x,) ^ 0 (38)
(see pp. 57, 58),
H(^,,xO + 0. (39)
Hence if we denote by x^ the root of the equation
A (x, , a-) = 0 "
next smaller than x^ and by x'l the root next smaller than x^,
of the equation
H (j-, , x) = 0 ,
it follows from Sturm's theorem^ that
At x = Xi' , ii{xi, x) changes sign.
1 Compare p. 58, footnote 2. This remark is due to Bliss, Transactions, etc.
p. 138.
§23] Sufficient Conditions 109
Again from (38) it follows that
L {x, - x) ^ '' / A (o-i, x) = 1
x=Xi oXi I
and therefore
L{A,-{-B, ^^^^/A(a^i,a;)) = +oo .
a;=Xi-0 "'^1 /
Hence we infer that
r > 0 when a-," < cTq < a?i ,
J"{Xi) i = 0 when «*„ = xl' ,
( < 0 when a-,' < a-o < a*/' .
For reasons which will appear later on (under /), the
point of the extremal @o whose abscissa is a-/' is called, accord-
ing to Kneser,' the ^^ focal poinV of the curve 6 on the
extremal ©q.
We have therefore reached the theorem: For a minimum
it is necessary that the focal point of the curve 6 on the
extremal ©o shall not lie between the points 0 and 1.
e) It remains to consider the question of the sufficiency
of these conditions.
If in addition to (lib') and (33) the condition
xl' < A, (41)
is satisfied, then
J'{x,) = 0 , J"(.r,)>0 ,
and therefore the function t/(a'2) ^^^^ ^ minimum for £r2 = iCi.
Let now 6 be any curve of class D' which begins at the
point 0 and ends at some point 2 of 6, and which lies more-
over in the improper field ^j. about Gq defined under 6).
Let @2 be the extremal of the field from the point 0 to the
point 2 (see Fig. 21), then we have
'Tlie discovery of the focal poiut (" Brenupunkt ") is duo to Knesek, see Lehr-
hurh, §24. For the special case of the straight line, the focal point occurs already in
Erdmanx's paper referred to above. Bliss uses "critical point " for " Brennpunkt."
110 Calculus of Variations [Chap. Ill
On the other hand, since we have supposed k so small that
<^,(a-, 7)4=0 for
Xo < X :^ Xi , I y — 7u I < A; ,
the region ^^ is at the same time an (improper) field' about
the extremal ©9 find therefore since (lib') holds throughout ^i^.,
according to §22, ?>). Hence
The extremal ©o furnishes therefore a smaller value for the
integral J than any other curve of class D' which can be
drawn in the region ^j. from the point 0 to the curve ^, and
in this sense the exirenud (Sq minimizes' the intcf/iril J if the
conditions (lib'), (33) and (41) are fnlfillcd.
Example Via: To drair I he cnrrc of sliorfest lenyfli. from a
giro 11 point to a given ciirce.
Here : ^ = i/] -\- y" ;
hence we obtain for the condition of transversality
1 + //i' //]' = 0 ,
?. <\,the minimizing straight line must be normal to the curve S at
the point 1.
Further we get easily
H(.,,x) = 4£=,(x,-x)+ ^^^-^'^^
therefore ^^ ^ ^ y'gj^Y')
1 In the discussion concerning the construction of a field about (?„ in §19, we have
for simplicity restricted y to an interval (v,, - fc , 7o + '^') whose middle point is 7 = y,,.
We might just as well have taken an interval of the more general form ( y,) — A;, ^y^^ + lc^).
In the present case the term field must be understood in this slightly more general
sense.
2 It should, however, be observed that the region g-^ does not. strictly speaking,
constitute a neighborhood (see §.3, b)) of the arc (?„ since its width approaches zero
as X approaches the value .r^. The proof that (?„ minimizes the integral J is there-
fore not quite complete. Knesee's sufficiency proof, which will be given in chap, v
for the problem in parameter-representation, is not open to this objection.
§23] Sufficient Conditions 111
Hence it follows that ihe center of curvature 1" of the curve 6
at the point 1 must not lie between the point 0 and the jmint J.
Couvei'sely : If this condition is fulfilled and if moreover 1" does
not coincide with the point 0, then the straight line 01 actually
fiu'nishes a minimum.
Entirely analogous results are obtained in the case when
the point 1 is fixed and the point 0 movable on a given curve.
The condition of transversality must be satisfied at the point
0. Again, if ^4o, Bq have the same meaning for the point 0
as the constants A^, B^ for the point 1, and if ./",)" denotes
the root next' greater than jcq of the equation
H (a-o, x) = AA (^0, ^) + ^o--^r^^ = 0 , (42)
then Xq must not be less than x^.
/) Geometrical interj^refatioii of f lie focal point. Let us
consider the problem to construct through a point 2 of the
curve 6 in the vicinity of the point 1 an extremal which
shall be cut transversely at the point 2 by the curve 6. Let
y =f{x, a, /3)
be the required extremal. Then we have for the determina-
tion of a and /3 the two equations
M = f{x„a,/3)-f(x,) = 0 ,
N = F {x., , y-i , q.) + (P2 - ^2) ^y (^2 , ^2 , %) = 0 ,
where
Ui=fiXi) , Pi—f'{x^ , g2=/x(>^2, a, ^) •
The two equations (43) are satisfied for X2 = x-i, a=^aQ. ;S = /3^^,
since 6 is transverse to Qq at the point 1 ; the left-hand sides
of the two equations (43) are functions of x-y, a, /3 of class
C in the vicinity of X2 = Xi, a^=a^^, ^~-^o ^^^^^ their
Jacobian with respect to a and 6 is different from zero for
X2=^Xi, a = aQ, /3 ^ /3q, if ?// — yi^^O as we have supposed ;
for it reduces to
id'- Ui ) ^%i, { '-1 (-^i) 'V (-^"i) — ''2 U\) '•/ (■^■1)) •
1 Compare the Addenda at the end of the book.
112 Calculus of Variations [Chap. Ill
Hence the equations (43) admit, according to the theorem
on implicit functions/ a unique solution :
a = a (,r,) , /? = /3 (^^2) ,
which is of class (" in the vicinity of jcg^-'^i and satisfies
the initial conditions
a (a-,) =a„ , /3{.ri) =^0 •
If we denote
/(.r, a(.r,), f3(x2)) =g{x,X2) ,
the required extremal is therefore
y = y{x, 0C2) , (-t4)
and if we consider x-y as a variable parameter, this equation
represents a set of extremals each of which is cut transversely
by the curve 6 ; the extremal @o is itself contained in the
set and corresponds to X2 = x\.
The envelope ^- of the set (14) is defined by the two
equations
y = g(x, 0C2) , </^2 (^. ^"2) = 0 ,
and the abscissae of the points at which the extremal @o
meets this envelope are the roots of the equation
a:2=xi
To obtain this equation we compute the derivatives
da cJ£
(XQC^ CIOC2
from the two equations <12I/dx2 = 0, dN/dx., = ^, substitute
their values in the equation
and finally put X2 = Xi, a = aQ, fi = /3Q.
Carrying out this process, we are led to the three equa-
tions
1 Compare footnote 2, p. 35.
§23] Sufficient Conditions MS
r, (x) a' (.r,) + /'a (a') /3' (.r,) = 0 ,
'•i (■<'i) «' (■^•i) + >-2 (^i) /3' (J'l) = Tj! — U\ ,
from which, by eliminating «'(.'i). /3'{ji-^), we obtain the result
H(a-,, .r) = 0 , /. e..
The focal jioliit^ Is llic point of irhlcli flic ('.rlroiud (S'q )iu'cts
for the Jirsf Ihiic — couidiiKj fro))i flic point 1 toward the
point 0 — the eiirclope of the set of extremals which are cut
traiisrerselij hij the curve (S.
Example Via : The set (44) consists of the normals to the
curve 6 ; the envelope '^ is the evolute of the curve 6 .
(j) Case of tiro niorahlc end-points: We add a few
remarks concerning the case when the point 0 is movable
on a curve Sq and at the same time the point 1 movable on
a curve G^.
The consideration of special variations leads at once to
the result that the minimizing curve must be an extremal,
that the condition of transversality must hold at both end-
points, and that the inequalities
:= . " — /'
must be satisfied.
But still another condition must he added : If .r[ ' ' denotes
the root next greater than 'j\ of the equation
H(;r,,aO = 0 ,
then the fottowinii liiequaJiti) must Jje s(disfied .-'^
'This Keometrical interpretation of the focal point is due to Kxeser; see Lehr-
hiirh, §24.
^Tliis result is clue to Bliss; see ^fatheinafische Aitnalcn,Yol.'L\IIl (19031, p.
70. He also proves that for a regular problem the condition .Tj <;rj"<jy', together
with the two transversality conditions and the condition that the minimizing curve is
an extremal, are sufficient for a minimum. His proof is based upon Kneser's theory
of the problem with one variable end-point.
For the example of the curve of shortest length between two given curves, the
inequality (1.")) had already been given by Eedmann {loc. cit.). Another important
example with botli end-points variable (the special isoperimetric problem) has been
completely discussed by Kxeser {Mafhematische Annalen, Vol. LVI (1902), p. 169).
114
Calculus of Variations
[Chap. Ill
-^i< a'i'"<^u" • (45)
The problems on variable end-points which we have dis-
cussed in this section are special cases of the problem: To
minimize the integfral J when the co-ordinates of the two
end-points are connected by a number of relations : '
*..(.*•.,, Z/o. •*'i, y^) = ^^ ■
The "method of differential calculus'' used in this section
can be applied also to this case.
The number of independent relations cannot exceed four ;
if it is exactly equal to four, we have the case of fixed end-
points. If both end-points are perfectly unrestricted, the
vanishinir of the first variation leads to the four conditions
F
0 , F
= 0 ,
F..
= 0 ,
F,.
= 0 ,
which are in general incompatible.
I Compare Knesek, Lehrhuch, §10.
CHAPTER IV
WEIERSTRASS'S THEORY OP THE PROBLEM IN
PARAMETER -REPRESENTATION ^
§24. FORMULATION OF THE PROBLEM
Ix the previous chapters M'e have confined ourselves to
curves which are representable in the form ij^f{x), a
restriction of a very artificial character in all truly geomet-
rical problems. We are now going to remove this restriction
by assuming henceforth all curves expressed in parameter-
representation.
a) Generalities concerning curves in parameter-repre-
sentation.^
A "■continuons curve'''' S is defined by a system of two
equations
G: x = 4>{t) , y = ip{t) , t.^t^U , (1)
<j> and yjr being functions of /, defined and continuous in
(/o/i). As t increases from /q to Z^, the curve is described in
1 The treatment of the problems of the Calculus of Variations in ijarameter-repre-
sentation is entirely due to Weieesteass; he used it in his lectures at least as early
as 1872. In order to avoid repetitions, wo shall discuss in detail only those points in
which the new treatment differs essentially from the old one. For the rest, we shall
confine ourselves to an account of the results.
As regards the relative merits of the tiro methods, one is inclined to consider the
older method — in which x is taken for the independent variable — as antiquated and
imperfect when compared with Weierstrass's method; unjustly, however, for the
two methods deal with two clearly distinct problems, and which of the two deserves
the preference, depends upon the nature of the special problem under consideration.
Generally speaking one may say that in all truly geometrical problems the method
of parameter-representation is not only preferable, but is the only one which fur-
nishes a complete solution. On the other hand, the older method has to be applied
whenever a function of minimizing properties is to be determined (for instance,
DirichleVs problem).
For examples illustrating the relation between the two methods, see Bolza,
Bulletin of the American Mathematical Society (2), Vol. IX (.1903), p. 6.
2 Compare J. I, Xos. 96-113.
115
116 Calculus of Variations [Chap. IV
a certain sense, called the "))Ositive sense," from its origin,
say 0, to its end-point, say 1.
If we make the ^'- parameter-transformaiion'''' :
t = x{r), (2)
where %(t) is a continuous function of r which constantly
increases from /q to /j as t increases from Tq to r^, the equa-
tions (1) are changed into
X = <l>{xir))=^{r) , ^ = ^(x(T))=*(r) . (la)
Vice versa, the equations (la) are again transformed into
(1) by the inverse transformation
T = x-'(0- (2a)
We agree to consider the two curves defined by (1) and
(la) as identical, and conversely two curves will be consid-
ered as identical only' when their equations can be trans-
formed into each other by a parameter-transformation of the
above properties.
The curve (5 will be said to be of class C'{C") if the
parameter t can be so selected that <fi(f) and -^(f) have con-
tinuous first {and second) derivatives in (fofi), and if more-
over (f>' and yjr' do not vanish simultaneously in (fot^) so that
^'2 + ^"2^0 in (40 • (3)
A curve of class C has at every point a continuously
turning tangent; the amplitude 6 of its positive direction
is given by the equations
cos 6 = , *^ , sin 6 = '^ . (4)
Every curve of class C is rectifiable," and the length s of
the arc V is expressible by the definite integral
1 According to this agreement, a curve (more exactly '" path-curve," E. H. MooreI
is not simply the totality of points defined by (1) but the totality of these points
taken in the order defined hy {1).
2 Compare J. I, Nos. lOS-Ul.
§24] Weierstrass's Theoey 117
dt . (5)
By an '^ordinarjj curve''' will be understood a continuous
curve which is either of class C or else made up of a finite
number of arcs of class C. A point where two diflferent arcs
meet will be called a "corner" if the direction of the positive
tangent undergoes a discontinuity at that point. A curve
will be said to be regular at a point t = t' , if for sufficiently
small values of |f — 1'\, x and y are expansible into con-
vergent power-series :
x = ct + a,{f-i') + a,{t-tj^ ,
y^h + b,(t-t') + h,{f-t'f+... ,
and if moreover a^ and h^ are not both zero.
b) Iritegrals taken along a curve; conditions for their
invariance under a 2Kirameter-transforma.tion.
Let F(oc, y, x', ij') be a function of four independent
variables which is of class C" in a domain © which consists
of all points x, y, x' , y' for which a) x, y lies in a certain
region 1R of the x, v/-plane, 6) x' , y' are not both zero.
We suppose that the curve 6 defined by (1) lies entirely
in 2J, and select two points 2 and 3 (/2< 4) ^^ ^- Then we
consider the definite integral
J= I F(x, y, x', y')
dt
in which ,r, //, ,r', y' are replaced by ^(/), ir(t), (/>'(/), ^' (t)
respectively, and ask : Under what conditions will the value
of the integral J depend 07ily on the arc 23 and not on the
choice of the iJarameter t?
The simplest example of an integral which is independent
of the choice of the parameter is the length of the arc 23,
which is always expressed by the definite integral
X
Vx"'+y'^dt
2
lis Calculus of Variations [Chap. IV
no matter what quantity has been selected for the independ-
ent variable f, provided that ^9 < f-s, so that if we pass from
the parameter / to another parameter r by any admissible
transformation (2), we must have
Returning now to the general case, our question may be
formulated explicitly as follows :
Under what conditions is
with the understanding that this relation is to hold :
a) For every transformation / =%(t) of the properties
indicated above ;
/S) For all positions of the two points 2 and 8 on the
curve (S ;
7) For all possible curves (S of class C, lying in ?i ?
On account of /3) we may differentiate (6) with respect to
T;^ ; writing for brevity /, r instead of f^, T3, we obtain
„/ dx du\ (It , / dx dy\
dx _ dx dt dy _dy dt
dr ~ dt dr ' dr ~ dt dr '
(dx diAdt ^1 dx dt dy dt\ ,_.
On account of a) this must hold for the special trans-
formation
k being a positive constant. Hence
„/ , dx , dii\ , ^/ dx dy\
%2i] Weierstrass's Theory 110
But by properly choosing the curve (1) (see assumption 7) j
and the parameter /, we can give the four quantities
dx dy
^''^'di'di
anv arbitrary system of values in the domain 01, and there-
fore the relation
F{x, y, kx', ky') = kF{x, y, x, y') (8)
must hold identically for all values of the independent
variables x, y, x , ,?/' in (U and for all positive values of k, or
as we shall say: F{.r, y, x\ y') must he "■positively homo-
geneous'''' and of dimension one icitJi i-espect to x' , y' .
Vice verso, if this condition is satisfied, (7) holds since
we suppose
dt „
dr
and therefore also (6) , as follows by integrating (7) between
the limits Tg and Tg. This shows that the homogeneity con-
dition (8) is necessary and sufficient for the invariance of
the integral J}
We shall in the sequel always suppose that the function
F satisfies the homogeneity condition (8), and we shall
denote the value of the integral
£^' F{ct>{t),^{t),i>'{t),r{t))dt
indiiferently by J^ or Jqi, and call it the integral of the
function F(x, y, x, y') taken along the curve 6.
If we wish to reverse^ the direction of integration we
must first introduce a new parameter which increases as the
1 Weieesteass, Lectures; also Knesee, Lehrbtich, §3.
This lemma has been extended to the case where J' contains higher derivatives
of X and y by Zeemelo, Dissertation, pp. 2-23; to the case of double integrals by
KOBB, Acta Matfiematica, Vol. XVI (1892), p. 67.
2 Compare Knesee, Lehrbuch, p. 9.
120 Calculus of Vaeiations [Chap. IV
curve is described from the point 1 to the point 0, for
instance: ii = — /. The equations
g-': x = cji{—u) , u — ii,(—u), «fu^«^"i ,
where Uq^^ — /j, iii^^ — /q, represent the same totality of
points as (1), but the sense is reversed.
The integral of F{.r, y, ./•', ij' ) taken along 6~^ has the
value
r"' ^/ ^■^" '/'A ,
Jn,= I Fl.r, y, - , -h]da ,
= J„ ^'(*^<- '"' "Al- "), - <^'(- «)' -^'{-ti))du ,
If the relation (<S) holds also for negative values of /,•, as
happens, for instance, when i^ is a rational function of .r\ ij\
then
F{.r, ij, —y, — y')= — F{x, y, x' , y'),
and therefore : J^q = — Jqi •
But the relation (8) need not hold for negative values of
/>■ : tluTs in the example of the length we have for negative
values of k
F(.r, /y. k.r'. ky') = - kFU, y, x' , y') ;
hence in this case ^io = ^oi-
In other cases the relation is more complicated, for instance,
when „ , ' I \ , ~^~i — ^
F = xu - ,r y + A1 x -+y ' .
From the homogeneity condition (S) follow a niimber of
important relations hettrccn fJie partial dcriratircs of F.
Differentiating (8) with respect to k and then putting
A; = 1 , we get
x'F,. + y'F^, = F . (9)
Differentiating this relation with respect to x and y, we obtain
F, = x'F,., + y'F„, , F, = x'F,.,^ + y'F,,„ . (10)
§24] Weierstrass's Theory 121
Differentiating (9) with respect to x and y' we get
^'i^.x- + y'Fy.' = 0 , x'F.,;, + T/'i'V,. = 0 ;
hence if x and ij' are not both zero,
F,.,. : F,.,. : F^.^. = y" : - ^'t/' : x" ; (11)
there exists therefore a function F-^oi x, y, x' , y' such that
F.■.,^ = y"F, , F,.,, = - x'u'F, , F^.,, = x"F, . (11a)
The function F^ thus defined is of class C in the domain
OF, even when one of the two variables x\ y' is zero; but
Fi becomes in general infinite when x' and y' vanish simul-
taneously, even if F itself should remain finite and continu-
ous for x' ^0, y' = 0.
For instance :
F = y Vx" + ij" , F, = y—=J=^,.
{} x'-\-y-)
c) Definition of a Minimum:^ Two points A{.Vq, y^ and
B{xi, yi) being given in the region U, we consider the totality
m of all ordinary^ curves which can be drawn in iS from A
to B. Then a curve 6 of ilH is said to minimize the integral
J= I F(x, y, x', y')dt ,
if there exists a neighborhood II of 6 such that
^s^^e (12)
for every ordinary curve S which can be drawn in H from
A to B.
We may, without loss of generality, choose for H the
strip^ of the x, ^-plane swept over by a circle of constant
radius p whose center moves along the curve S from A to B.
This strip will be called "the neighborhood (p) of 6."
1 Compare §3. The definition is due to Weierstrass, Lectures, 1879; compare
also Zekmelo, Dissertation, pp. 2r)-29, and Kneser, Lehrbuch, §17.
2 An extension of the problem to a still more general class of curves will be con-
sidered in §.31.
3 In case different portions of the strip should overlap, the plane has to be
imagined as multiply covered in the manner of a Riemanu- surface (Weierstrass),
122 Calculus of Variations [Chap. IV
§25. THE riEST VARIATION
We suppose that we have found an ordmary curve
6: x = <f>(t) , y^^(t) , t„^t^U ,
contained in the interior of U, which minimizes the inte-
gral ./. We replace the curve 6 by a Jieighboring curve
6 : X = x-\-i , y = y + v >
where ^ and tj are arbitrary functions of / of class D' , which
vanish at /q ^nd /j :
${to) = 0, vito) = 0 ; ^(M=0, v{td = 0. (13)
The consideration of special variations of the form
i = €j) , v = ^a > (1^)
where e is a constant, and p and q are functions of t of class
D' , which are independent of e and vanish at /q and /j, leads
as in §4: to the result' that
AJ=Sj+£(e) , (15)
where (e) is an infinitesimal and
8J= r{Fj + F,^ri + F,.e+F^.r,')dt , (15a)
whence we infer again that Sj must vanish for all admis
sible functions ^, V-
Considering first special variations for which 77 = 0, and
secondly special variations for which f ^ 0 , we see that we
must have separately
CiFj + F^-ndt^O , C {F,y) + F^.rj')dt = 0 . (16)
iThe same results hold for variations of the more general type
where the functions l(<, «), ')(^, «), their first partial derivatives and the cross-
derivatives #f£ , fif^ are continuous in the domain ^q ^ f ^ fj , I « | 5 ^q , Eq being a suffi-
ciently small positive quantity. Moreover
f(^0,e)=0, l)(f„,e)=0,
1(^1, €)=0, »,«i,e)=0.
Compare §4, d).
§25] Weiersteass's Theory 123
To these two equations the methods of §'^^4-9 can be
applied with the following results :
a) Wcierst)riss''s form of Eider'' s equation: The func-
tions X and y must satisfy the two differential equations
(1 f1
F ——F =0 F —--F. — O ■ (M)
^^ dt "^ ' ^ dt " ' ^ ^
these two differential equations are however not independ-
ent ; for, if we carry out the differentiation with respect to /
and make use of the relations (10) and (11a) we obtain
F..-J^F^. = U'T , F,~-'j^F^.= -x'T (18)
where T= F^„ - F^^. + F,{x'y" - x"y') , (19)
oc" , y" denoting the second derivatives of x and y with
respect to /. Since x' and y' do not vanish simultaneously
(see §24, a)), the two differential equations (17) are equiva-
lent to the one differential equation
T=F.,,,-F,,. + F,{x'y"-x"y') = 0 . (I)
This is Weierstrass's fo7-m of Euler's differential equa-
tion.^ Every curve satisfying (I) will again be called an
extremal.
The same result can also be derived from a transforma-
tion' of 8 J which will be useful in the sequel.
If we perform in the expression (15a) for 8J the well-
known integration by parts, and make use of (18), we obtain
8j=\iF,.+ riF,T+ pTwdt , (15b)
where ic=^y'^ — x'r).
1 Weierstrass, Lectures; compare Zeemelo, Dissertation, p. 37.
If we introduce the curvature
1 _ x'y" — x"y'
the differential equation may also be written
X \l XII
1 F^.,-F^
F,{Vx'^+y')
3 • (la)
124 Calculus of Variations [Chap, iv
The differential equation (I) together with the initial
conditions determines the minimizing curve, but not the
functions x and /y of /. In order to determine the latter,
we must add a second equation or differential equation
between /, ,r, y. This additional relation (which is equiva-
lent to some definite choice of the parameter /) must be
such that X and y come out as single-valued functions of
i of class D' satisfying (3) ; otherwise it is arbitrary. The
best selection depends largely upon the nature of the par-
ticular example under consideration (see the examples in §2(j).
If we add to (I) a finite relation between /, ,r, y we
obtain as the general solution a pair of functions of / con-
taining two constants of integration :
a-=/(f,a,,J) , y = g{t,a,(3) . (20)
The constants a. /3 together with the unknown values /g
and /i have to be determined from the condition that the
curve must pass through the two given points :
a'o=/(/„, a, ^j , Z/u = ^(Ai. a, /?) ,
Xi=f(t,,a, (3) , y, = (7(/,. a, ^) .
b) Extremal through a given point in a given direction:
In order to construct an extremal through a given point
Oia, b) of S in a given direction of amplitude 7, we select
the arc of the curve measured from the given point for the
parameter / and have then to solve the simultaneous system
T = 0, x" + y"=l (22)
with the initial conditions
X = a , y = b , x' = cos y , y' = siu y
for /=^0. Differentiating the second differential equation
we obtain the new system
F,{y'x"- x'u") = F,^. - F,,. ,
XX +y y =0 .
§25] Weierstrass's Theory 125
Solving with respect to or" , i/" we obtain j-", y" expressed
as functions of x, y, x' , y' which are of class C in the
vicinity of ,r = a, y^^h, a?' =^ cos 7, y'^^siny provided that
Fi(a, 6, cos y, sin y) =1= 0 . (23)
Hence' there exists a unique solution
x = ^{t; a,b,y) , y = ^ (f ; a, b , y)
of the system (22a) satisfying the initial conditions and of
class C in the vicinity of t = 0.
This solution satisfies also the original system (22). For,
by integrating the second equation of (22a) we get :
x"- -\- y"- = const., and the value of this constant is found to
be 1 from the particular value / = 0. Thus we reach the
result:^
If Fi (a , b, cos y , sin y) 4= 0
one and l)iif qhc exiremal of cIcls.^ C rait, he drawn through
tlw point (rt, 6) in the direction 7.
Hence, if (23) is satisfied for every value of 7, a
unique extremal of class C can be drawn from O in every
direction.
If (23) is satisfied at every point [a, h) of the region |J
for every value of 7, the problem will be called a regular
problem (compare §7, c)).
c) ^^ Discontiniioiis solutions :'''' As in §1), oj we infer by
the method of partial variation that every "discontinuous
solution"^ must be made up of a finite number of arcs of
extremals of class C .
Furthermore, the method of §9, h) applied to the two
equations (16) leads to the result:^
1 Accordingr to Cauchy's existence-theorem ; compftre p. 28, footnote 4.
2See Kneser, Lehrbuch, §§27, 29.
3/. e., a solution which has a finite number of corners; compare §24, a).
*Weierstkass, Lectures; compare also Kneser, Lehrbuch, §43.
126 Calculus of Vaeiations [Chap. IV i
At a corner # = ^2 of the minimizing curve, the two con-
ditions
F ■
to—0
= F
lo+O
F ■
/9-0
F .
'2+0
(24)
must he satisfied, i. e., the two fiincfions F^^ and Fy nmst
remain continuous even at the coi'iicrs.
We add here the following corollary, though its proof
can be given only later (§ 28) :
At a corner {x^ , 2/2) of ^^^ minimizing curve, the function
Fi {x.2 , 2/2 > cos 6 , sin 0)
must vanish for some value of the angle 6.
Hence it follows : If (d everij point (.r, ij) of the region jR
F,{x, y, cos e, sin ^) =t= 0
for every value of 6 , no ^^discontinuous solutions''' are pos-
sible.
§26. EXAMPLES
In applications it is frequently convenient to use one of the two
equations (17) instead of (I), especially when F does not contain x
or y, in which case one of the two equations (17) yields at once a
first integral. It must, however, be borne in mind that each of
these two equations contains a foreign solution' (// = const, and
^ = const, respectively), and that only their combination is equiva- \
lent to (I).
a) Example X: To determine for a heavy particle the curve
of quickest descent hi a vertical plane between tivo given points
(" Brachistochrone "^) .
1 This happens, for instance, in Example I:
F=y^^x'^ + y'^ ,
where a first integral is obtained from (17) ;
yx'
l/ ,2 1 ,2
when a — 0 , y = 0 is such a foreign solution.
2Compare LindelOf-Moigno, loc. cit.. No. 112; Pascal, loc. cit., §31; Knesee,
Lehrbuch, p. 37.
§26] Weierstrass's Theory 127
If we take the positive //-axis vertically downward and denote
by g the constant of gravity, by ru the initial velocity, which we
suppose different from zero, we have to minimize the integral
^^ r'^Vx'' + y"dt
y u — u^^ + k
where
The curves are restricted to the region
S: u-y. + k->0.
Since i^^. = 0, we obtain the first integral
F^. = , ^^^= = a
X
(25)
The theorem on discontinuous solutions shows that the constant a
must have the same value all along the ctu've.
If a = 0, we obtain a' = const., which is the solution of the prob-
lem when the two given f)oints A and B lie in the same vertical line.
If a=|rO, we choose for the parameter i the amplitude of the
positive tangent to the curve ; then we have the additional relation
= cos t ,
V X ^-\- y
which reduces (25) to
y — ij^ + k = v{l + cos 2t) ,
where
1
la
Hence
2/'= — 2r sin 2t ,
and
cr'= z± -tr cos^^ .
If we finally make the substitution
2t = T-7r ,
we get the result
X — X(,-\- h ^ ± r[T — sin t) ,
y — yo + k= r (1 — cos t) ,
(26)
128 Calculus of Variations [Chap. IV
h being the second constant of integration. The extremals are
therefore cycloids^ generated by a circle of radius r rolling upon
the horizontal line y — y(,-{-k = 0.
Among this double infinitude of cycloids there exists^ one and
but one which passes through the two given points A and B and
has no cusp between A and B, provided only that the co-ordinates
of the two given points satisfy the inequalities
^•i =1= ^0 , yi — y,, + A.- ^ <^ •
b) Example XI: To determine the curiae of shortest length
tvhich can be drawn on a given surface between two given
points.
If the rectangular co-ordinates x, y, z of a point of the sui-face
are given as functions of two parameters u, v and the curves on
the surface are expressed in parameter-representation
n=<l>{t), r = ^(t), (27)
the problem is to minimize the integral
J= \^ Eu"-\-2Fu'v'-\-Gv'^dt ,
where
E = 'S.xl , F = 2 j->„ , G = 2.r^, ,
the summation sign referring to a cyclic i^ermutation ot x, y, z. .
The ciirves must be restricted to such a portion ^ of the surface |
that the correspondence between S> and its image 2J in the u , y-plane i
is a one-to-one correspondence. We further supi^ose that E, F, G ^
are of class C " in S and that g» is free from singular points, i. e.,
EG-F'>0 .
a) If we use Weierstrass' s form (I) of Euler's equation, and
denote by ^(F) the differential expression
iThis result is due to Johann Bernoulli (1696) ; see Ostwald's Klassiker, etc.,
No. 46, p. 3.
2See Heffter, "Zum Problem der Brachistochrone," Zeitschrift fur Mathe-
matik und Physik\\ol.XKXI\ (1889), p. 313; Bolza, "The Determination of the
Constants in the Problem of the Brachistochrone," Bulletin of the American
Mathematical Society (2), Vol. X (1904), p. 185; and E. H. Moore, "On Doubly Infinite
Systems of Directly Similar Convex Arches with Common Base Line," Bulletin of
the American Mathem,atical Society (2), Vol. X (1904), p. 337.
§26] Weierstrass's Theory 129
^(F) = F,,^. - F„,. + F.U'ij"- .r"ij') ,
we obtain easily
^(VEu"+2Fu'v'+Gv'-)= . ,, , , .vo(28)
^ ^ {v Eu'' + 2Fu'v'+Gv'^y '
where
V = {EG-F''){h'v"-u"v')
+ {Ea' + Fv') [(i^„ - 1 E,) ir- + G„u'v'+\ G„v"'] (29)
- {Fn'+ Gv')[^Ey' + E,u'o' + {F,-\G,)v"-'\ .
The extremals satisfy, therefore, the differential equation'
r = o . r29a)
This differential equation admits of a simple geometrical interpre-
tation :
The geodesic curvature of the cvirve (27) at the point t is given
by the expression -
1 r
- = -7 ,3 . (30)
Py \ EG-F'{\ Eir + ^Fu'v+Gv"") '
Hence the curve of shortest length has the characteristic property
that its geodesic curvature is constantly zero, i. e., it is a geodesic.
In passing we notice the relation
* ( \^Eu'' + 2Fu'v'-}-Gv") = ^ ^^~^^' , (28a)
Pi/
which will be useful in the sequel.
/3) If instead of (I) we use the two differential equations (17)
and, moreover, select the arc s for the parameter t , we obtain for
the extremals the two differential equations :*
iThat (29a) is the differential equation of the geodesies might be taken directly
from the treatises on ditt'erontial geometry: Knoblacch, Fldchentheorie, p. 140;
Bianchi-Ldkat, Different iabjeoinetrie, p. 154; Darboux, TMoric dcs Surfaces, Vol.
II, p. 403.
2 See Laurent. Traiti d' Analyse, Vol. VII, p. 132,
For an elementary proof see Bolza, " Concerning the Isoperimetric Problem on
a Given Surface," Decennial Publications of the University of Chicago, Vol. IX, i>. 13.
:* Compare Knoblauch, loc. cit., p. 142; Bianchi, loc. at., p. 153; Darboux, lo<\
cit., p. 405.
130 Calculus of Vartatioxs (Chap. IV
They have likewise a simple geometrical meaning : From the
definition oi E , F, G it follows that
dH f^clv _^ dx
^di( , ^dv 'sr^ d.r
ds ds ^-^ ds
Differentiating with respect to s we obtain
d'j-
hence on account of (31)
^.r„S = 0,
and similarly
d-JL
^.. — = 0.
" ds'
Therefore
Ci 7' flu fiZ
d7 ' li' ' d?^ (y»^. - yv^..) ■ i^u^r - ^v^u) ■ (■i\,2/r " -^r?/,,) • (32)
The geometrical meaning of this proportion is that at every point
of the curve the principcd vorwal coincides vith the vornial to
the s«r/ace, which is another characteristic property of the geodesic
lines.
§27. THE SECOND VARIATION
Let
X= fit, a„, /3„) = /(/) , . = , = , /qq.
y = g{t, (h,, P..) = g{f) ,
represent an extremal of class C" passing through the two
given points A and B , derived from the general solution (20 )
by giving the constants the particular values a = ao, ^ = /3,).
We suppose that the functions /(/, a, y3) and f/(/, a, /3),
their first partial derivatives and the following higher deriva-
tives,
ftt 5 fta - ftp ' ftta ' fitfi ; if 11 • Ula. ' 'J I ^ > Utla ' iltt^ >
§*27j Weierstrass's Theory 131
are coiitiimous in a domain
Z,^t^T,, ,a-a„|^r/, |y8-/?„|^d.
where T'o</(). Ty^f^, and d is a snfficieiitly small positive
quantity. >
Then we infer, as in §11, that in case of a minimum the
second variation of J must be positive or zero. The second
variation is defined by the integral
S'J= I 8'Fdt ,
where * '"
h'F = F,J^ + 2F,,Jri + i'V.r + '^K.'^^'+ 2F.,, VV'
the arguments of the partial derivatives of F being
J^=f{t) , y=--(j(f), y = f'{f) , i/'=y'(f) •
a) Weierstrars's Transformaiion of the second varia-
tion:^ This transformation proceeds by the following steps:
1. Express F, .-,.■, Fy,y, Fy y in terms of F^^ by means of
(11a) and introduce the abbreviations
iC = ii'^ — x'r] ,
L = F,,. - i,'y' 'F, , N = i';,, - r'x' ' F, , (35)
.1/ = F,„ + yy"F, = i^;,.. + ij'x'F, ;
the two expressions for M are equal since x and y satisfy
+he differential equation (I).
We thus obtain
^'F = F, ('jA'+ 2i:^r+ 2.1/(t>'+ 7?r) 4- ^N-q-n'
+ (F,,, - u"'F,) e + 2 {F^, + y'ij"F,) $r] + {F,^,^ - x"^F,) r,^ .
2. Observe that
2L$i'-\- 2M{$rj'-\-y]i'} + 2Nr,r)'
' Weierstrass, Lectures, at least as early as 1872.
132 Calculus op Variations [Cnap. IV
and introduce the abbreviations
M,^F^„ + y'u"F,-'^ , (36)
Then the above expression for B-F becomes
8'F = F, (^)"+ L,e + 2M,$v + i\W
8. The three functions L^, J/j, X^ have the important
property of being proportional to y"-, — r'//', jr'-.
Proof: From the definition of L, 31, N and the relations
(10) follows
Lx'+My'=F, , Mx'+Nu'=F„ .
Differentiating the first of these relations we get
dL , dM " I ir "
= F^^x' ^ F,,y + F,,.x" + F,, y" .
But
Lx"^My"=F_,,.x"+F^^,.y" ,
and from (I) it follows that
Fyx- — Ky = F, ix'y"- x"y') .
Substituting these values we obtain
L^x'-\- Miy'= 0 ;
similarly
M,x'+A\y'=0 ■
whence we infer that indeed
I-i : Ml : A'l = y'^ : — x'y' : x"^ .
There exists therefore a function Fo of / such that
A = ij"F, , M,= - x'y'F, , V, = x"F, . (37)
§27] Weierstkass's Theory 183
This reduces the expression for 8- J to the final f(jrm
+ [Le + 2M$ri + Nrj'T . (38)
*- A,
If, as we suppose for the present, the two end-points are
fixed, then f and rj vanish at /q and /j and the expression for
S'\T reduces to
«'''=X'[^''(f)+^«"']"'- ('">
This definite integral must then be ^0, for all functions iv
of class D' which vanish at both end-points.
From the assumptions made at the beginning of this
section with respect to the functions /(/, a, ^S), g{t,a,l3)
together with our assumptions concerning the function
i^(see §24, 6j), it follows that F^ and F.. are of class C in
the interval {TqTi) ; we suppose that they are not both
identically zero.
b) Weierstrass's form of Legendre's and Jacobi's
conditions: The second variation being now exactly of the
same form as in the previous problem (§11), we can directly
apply the results of Chapter II.
Accordingly we infer in the first place, as in § 1 1 :
The second necessari/ condition for a minimum (^maximum)
"'"*"' F.^O (F.^0) (II)
along the curve @o-
\Ye suppose in the sequel that this condition is satisfied
in the slightly stronger form
F,>0 , along©,. (II')
Again, Jacobi's differential equation (equation (D) of
§11) becomes
*W = ^,«-^,(i",^) = 0. (40)
134 Calculus of Variations [Chap. IV
Jacobi's then re III conceruing the integration of this dif-
ferential equation takes now a slightly different form. If
we substitute in the differential equation
^' -:>■'="
for jc and // the general solution
X=f{f,a, fi) , y =y{t,a, P)
and differentiate with respect to a we get
In this equation we express the second j)artial derivatives of
F in terms of L, M, X. F,, F. by means of (lla), (35), (36),
(37) and obtain, after some simple reductions,
.'"[^>-.M^'^)]-«'
where
<^ = Utfa — ftfja •
If we operate in the same manner upon the differential
equation ^j
we obtain
^■„-,„n. = o.
-/'[---^(-■'^)]=«
Therefore, since ft and (jf are not both zero, we find that
dt
(-^)-'-
An analoofous result is reached if we differentiate with
respect to ^. Finally, giving a, /S the particular values
«o- A)? we obtain Weierstrass's modification of Jacobi's
theorem :
§271 Weieesteass's Theoey 13e5
The (lijferential equoHon
*(«)iE^>-;;^(/.-*)=o
has flic firo pcnilciihir iiifiujrah
OAt)^gf{t)fp{t)-.f)(t)g^{t) ,
(41)
which are in general linearly independent.
Reasoning now as in §§14 and 16 we obtain the result:
Let
@{t,U = 0, (0 0, (A,) - e, (t) 0, {Q ■ (42)
then Jacobi's coiiditioji takes the following form :'
The third necessary condition for an extrennnn is fliat
®{t, t„)^() for f„<t<f, . (Ill)
If we denote by /q the zero next greater than /^ (jf the equa-
tion
©(^ /„) =0 ,
Condition (III) may also be written:
^1 < 'o ;
to is the parameter of the '•cnnjnuate point"" to the point A.
Example XMseep. 126).
We suppose that the two end-points A and B lie between the
two consecutive cusps t = 0 and t = 27r of the e^x'loid (26), so that
the vahies t= r^, and t = tj corresponding to A and B respectively,
satisfy the inequality
0 < To < Ti < 27r .
For the function Fi we obtain
1 1
i'\
1 V - Vo + h(\ x" + try 8 y 2 r' 1 ' r sin* ^
Hence Fi is indeed positive along the arc AB.
1 Weieestrass, Lectures; compare also Kneser, Lehrbuch, §31.
2L1XDELOF-M01GXO, loc. cit., p. 231.
IBP) Calculus of Vakiations [Chap, iv
Again, we obtain from (26)
© ( T , T„) = zh 4r^ sin ^ cos ^ sin -7^ cos -^
rT-2tan^-r„ + 2tan5l •
The parameter r^ of the conjugate point ^l ' is therefore determined
by the transcendental equation
T T
T — 2 tan ^ = T„ — 2 tan -^ ,
2
As T increases from 0 to tt and then from tt to 27r, the function
T
T — 2 tan n decreases continually from 0 to — co and then from +00
to +27r. Hence r = r,, is the only root of the equation between 0 and
27r. There exists, therefore, no conjugate 2)oint on the arc AB.
f) Kneser's foDii of JacobVs condition : As in §15 the exist-
ence of a set of extremals through the point A can be proved,' rep-
resentable in the form
x = <jy{t, a) , y ^^{t, a) , (43)
1 Weierstrass obtains the set of extremals through A as follows {Lectures, 1882) :
Let
represent the extremal passing- tliroiish ^4 and making at A a given small angle m
with the extremal
Let further t denote that valus of t which corresponds on (- to the point A. Then
we have for the determination ot t , a, ^ the three equations :
where the argument of J-', y' is ^q, that of x', y' : /", and where
a~(x " -\-y ) sin w .
The three equations are satisfied for t'^ = <„ , a = a,, , ^ = /S^^ ; the functions on the left-
hand sidfe are continuous and have continuous partial derivatives in the vicinity of
t = Jq , a = tty , /3 = ^Q , and their Jacobian with respect to i'^', a , ^ is different from zero
at this point, since it is equal to
ei(*0)«2'«0)-«2(^0)«l'«0)'
which is different from zero if, as we suppose, 0j (t) and ^2 (0 are linearly independent.
There exists, therefore, according to the theorem on implicit functions, a unique
solution t , a , p of the above equations, which leads to two functions <#> ( i , a ) , i// (f , a)
having the properties stated in the text.
§271 Weierstrass's Theory 137
where <t>(f , a) aud i/{t, a) are continuous with continiioxis partial
derivatives of the first and second orders with the possible excep-
tion of (t>aa, i^aa. — ill the doiiiaiii
t :^T^ , 1 a — c<u I < f/,
0 )
<i,< being the value of a which corresponds to the extremal (f'„ through
A and B, and </„ Ijeiug a suificienth' small i^ositive quantity.
Again, the Jacobian
d(i,a)
differs ' for a = a,, from the function ©(/ . A,) only by a constant factor:
A(^ a„) = C-©(/, f„) , (44)
where C'4=0.
Furthermore the value t^f which corresponds on the extremal
(43) to the point A , and which satisfies therefore the equations
a'„=c/>(f\«) , y, = ^(f,a) , (45)
is a function of a, which is, in the vicinity of a,,, of class C .
From (44) follows Ivxeser's-'/o/'^; of Jacobis conditio u :
\{t,a,)^() for f„<t<fi (III)
Further, if Ai denotes the value of t corresponding to the conju-
gate point A ' , we have
A (t; , cto) = 0 , (46)
and at the same time
A,(C,«o) + 0, (47)
provided that Fi , i^2 are of class C in the vicinity of U and Fi 4= 0
at tfl. The inequality (47) follows* from the fact that A(f, a,,) is an
integral of Jacobi's differential equation (40).
From this second form of Jacobi's condition it follows* easily
that the conjugate point A' has the same geometrical meaning as
in the simpler case of § 15.
' This follows either by direct computation from the equations ■which define
t , a, ^ as functions of a, or else from the fact that A (t, «y) and © {t , f^) are integrals
of Jacobi's differential equation and vanish for i = <Q.
2 See Knesee, Lehrbuch, §31.
3 Compare p. 58, footnote 2,
■♦See Knesee, Lehrbuch, §24, and the references given in E. Ill D, p. 48, foot-
note 117.
138 Calculus of Variations | Chap. IV
§28. THE FOURTH NECESSARY CONDITION AND SUFFICIENT
CONDITIONS
We suppose in the sequel that for our extremal (i"u the
conditions
F,>0 (ID
and
®{f,Q4.0 for t,<t^t, , (III')
are fulfilled.
a) These conditions are not yet sufficient for a (strong)
minimum; a fourth condition must be added.
Lot E(.r, // ; ,r', //' ; x' , Jj') be defined' as the following
function of six independent variables :
E{x,y; x', y' ; 7v\ 7)') = F{x, //, x , Ij')
- \x'F^.{x, y, x', y') -\-y'F,j.{x, y, .«•', y')^ , (48)
or, as we may write on account of (9),
E{x, y; x', y' ; x' , y') =
^'[f^(^, V, ~i-'- U') - F^ix, y, x', 7/') I
+ y'[F, {.X-, y,x',y') - F,,{x, y, x\ y')] . (48a)
Let further (x, //) be any point of the extremal Gq, j>, <]
the direction-cosines of the positive tangent to d'^ at {x . //).
and p, ~i the direction-cosines of any direction.
Then the fourth necessary condition for <i mini mum
[maximum) is that
'E.{x,y; p,q; p, q) ^ 0 (^ 0) (IV)
for everij point [x, y) of ©q and for every direction p, Tp
The proof follows^ immediately ivom.Weierstrass''slemma^
on a special class of varicdions:
Let I
iThis is Weieestrass's original definition; Kxesee writes — E instead of
Weierstrass's 4-E, Lehrbuch, p. To
2 Compare §18, b).
^The reasoning is the same as in §8; compare also §4, d).
§28j Weierstrass's Theory 139
be any extremal of class C" lying in the interior of the
region ®, and let 2: (/A,) ^^^ f^ii arlntrary point of ©.
Through the point 2 draw an arbitrary curve of class C :
6: .7- = <?(t) , ij = «/^(r) ,
the value of r~ r., corresponding to the point 2.
Let 3 : {x-i-r^z, !J->^rV-i) be the point of (^ corresponding
to T = T2 + A, where h is a sufficiently small positive quan-
tity. Finally, from a point 0: (/ = /o<^2) of ® ^^o the point
3 draw a curve (S representable in the form
6 : J. z= .r + ^ , T/ = y-^rj ,
where ^ and t; are functions of / and ]i which vanish identi-
cally for /i = 0, and which satisfy the following conditions':
1. 1,7/ themselves, their first partial derivatives and the
cross derivatives f^/, , ■}](,, , are continuous in the domain
JiQ being a sufficiently small positive quantity.
2. au,h) = 0 , r]{t, h) = ()
$(f,, h) =i, , rjiU, h) =ri,
for every O^A^/?o. Then the
difference"
has the following value :
J,:: - ( '/o2 + J^2:d = - h [e {x, , y, ; .r; , t/./ ; j; , y.^ ) + (A j] . (49)
Similarly, if we denote by -4 the point of ^ corresponding to
I Functions f , -q satisfying these conditions are, for instance, the followiui;:
if It, V are two functions of t of class f which vanish for t = /„ and an; equal Ui 1 for
t^t,.
'-F*>r the notation compare §§2,/), 24 a), and 8.
140 Calculus or Vaeiations [Chap. IV
T = T2 — h and draw a curve 6 from 0 to 4 of the same char-
acter as 6 , we obtain :
^04 + Jr2 — </o2 = + it [e i-^-i . 2/2 ; ■*'2 , Vi ; i?2' , Z/2' ) + (/i)] • (49a)
By the same method and under analogous assumptions
we further obtain the following
, results, which are sufficiently ex-
's'^'* plained by the adjoining dia-
FIG. 23 gram :
J23 + J-si — J-n = /' [e {Xi , 2/2 ; a-2 , yl ; ^2 , ^2 ) + (/o] . (50)
'lu — ('/« + ^21) = — /' [e (.^2 , 7/2 ; a-; , 7/2' ; x^ , ^2 ) + (^O] • (^^a)
From the relation (8) it foHows that
E(x,y; kx',ky'; kx', ky') = kE{x, y; x' , y' ; x',y') , (51)
if A^>0 and A;>0.
Hence if we set
P = =r=:z = COS 6 . q = =z==:r = slu 0 ,
} x"^y" Vx^ + y"
^ z ^ V
^= = cos tf y g = — -
(52)
we get
E{x, y; x', y';x',y') = Vx'^ + y"''E{x, y; p, q; P,q) , (53)
which reduces the second and the third pair of arguments of
the E-function to direction-cosines.
If we choose for the parameter r on the curve (S the arc,
we may replace in the above formulae x^, 2/2 and x^, yi by
the direction -cosines p2? Q2 and fu, q-z of the positive tan-
gents at 2 to @ and to 6 respectively,
6) Relatioyi between the 'E-function and the function i\ :
If the angles 6 and o are defined by (52), we have, accord-
ing to (48),
§28] Weierstrass's Theory 141
E(.«', y; P,q; p, q)
= COS O^F^. {x, y, cos 6, siu 0} — F^(x, y, cos 6, sin ^)1
+ siu 6 yFy. (x, y, cos ^, sin 6) — i^^^,. (x, y , cos ^ , sin ^)1 .
But
^xi"^'} y > cos 6 , sin ^) — Fy..{x, y , cos ^. sin ^)
= i'^ 't-F^.{,v, y,cos(^ + T), sin(^ + r)).^r ,
where (o = 6 — 6: and an analogous formula holds ior Fy.
If we perform the differentiation with respect to t and
then make use of the relations (11a), we get
E(a', y; p, q; p. q)
= 1 F^(x, y, cos {6 + t), sin ( ^ + t) ) sin (w — t) cIt .
By adding to 6 a proper multiple of 27r, we can alwaj's
cause (o to lie in the interval
TT <C to ^ TT ,
so that sin (to — t) does not change sign between the limits
of integration. We may then apply the first mean-value
theorem and obtain the following relation^ between the
E-f unction and the function F^:
E {x , y ; cos 6 , sin 6 ; cos d , sin 6)
= (l - cos (6 - ej) F, {x , y,cos 6*, siu 6*) , (51)
where 6^ is a mean value between 6 and 6.
From this theorem follow a number of important conse-
quences :
1. If we let 0 approach 6, we obtain
Ejx, y; p, q; p, q) , ..^.
-'- — ~^ 7s — 7r~ = ^' (^^, !J-}>, q) • ('>'^)
0 = e 1 — cosf6' — (9)
Hence it follows that Condition {IT) is contained in Con-
dition (IV).
1 Weieestkass, Lecturer, 1882.
142 Calculuk of \ariations [Chap. IV
2. Condition (IV) is always satisfied when
F^ {.X- , // , cos y , sin y) ^ 0 (Ila)
for every })oint (.r, >/) on ©q ^^^^ ^'^^' every value of 7.
3. The E-function vanishes whenever 6 = 6 ("ordinary
vanishing")^; for a value ^=|=^ it can only vanish' (''extra-
ordinary vanishing") if J^\{r, //, C0S7, sin7) vanishes for
some value y^=6* between 6 and 6.
c) Example XII :^ To minimize the integral
y
The value of the E-functiou is (>asily foinid to be
E(.T, //; p, q; p, q) = (^ + g-)- (j7^ + r)
= y sin' (0-6) sin (2^ + ^) .
Apart from the exceptional case when both end-points lie on m
the .r-axis, E can ])e made negative as well as positive by choos- "
ing d suitably; and therefore no minimum can take place.
More generally, whenever the homogeneity condition (8) "
holds not only for positive but also for negative values of /.•,
as happens, for instance, when F is a rational function of
x' , ij\ no extremum can — in general — take place.
For in this case (51) holds also for negative values of A', so
that
^{x, y; p, q; — p, —q) = —E(.r, y; p, q; -\-p,+q)
.Condition (lY) can therefore be fulfilled only if
E(.^7, y;p,q: p, q) = 0
1 Kneser's terminology, Lehrbuch, p. 78.
2 Hence follows the corollary on discontinuous solutions stated on p. 126. For
from (24) follows
E(j-, y; p, q; p,q)=0.
■'To this definite integral leads Newton's celebrated ijroblem : To determine the
solid of revolution of minimum i-esistance. Compare Pascal, loc. cit., p. Ill ; Knesee,
Lehrbuch, §§11, 18, 26; the above expression for E was given by Weiersteass (1882).
i
§28] Weieesteass's Theoey 143
along Ci'o for every directiou J>, q, which, on account of (S-t),
is possible only in the exceptional case when jPi = 0 along G^q-
d) Sufficicncjj of the four preccdin<] condift'oiis:^ The
four conditions which so far have been shown to be iicccs-
sary for a minimum of the integral ./. are — apart from cer-
tain exceptional cases' — also snfficicid.
Let us suppose
1. That ©0 (or AB) is an arc of an extremal of
class C" without multiple points, lying wholly in the (I')
interior of the region^ IS ;
2. F^{x, u,p,q)>0 along* Q^ \ (H')
3. The arc (Sq does not contain the conjugate point
.4' of the point .4. (Ill')
-t. E(.r, II : p, q: p. ri)>0 along* G, (IV)
for every direction p, q different from tlie direction j). <[ of
the positive tangent to @o at [x^ y).
Moreover we retain the assumptions made in §27 con-
cerning the general integral of Euler's differential equa-
tion.
We propose to prove that under these circumstances the
extremal (Eo actually minimizes the integral
J= I F{x, ij, x , u )dt
•y '0
From the assumptions (III) follows the existence of a field
of extremals about the arc Gy. /. c, there exists^ a neighbor-
1 Weieesteass, Lectures, 1879 and 1882; Zeemelo, Dissertation, pp. 77-ii4; and
KxESEE, Lehrbxich, §20.
-The exceptional cases are
1. I'll has multiple points or corners, or meets the boundary of iR;
2. i^j — 0 at certain ijoiuts of P„ :
3. A' coincides with B; this case will be considered in §.38.
i. E = 0 at points of (r^ for certain directions j7, q not coinciding with />,</.
3 Compare §21, h).
*That is, for every point (x, y) of t'^, p , q denotiuy the directiou-cosiues of the
positive tangent to e^ at (x, y).
^Compare §19. A sharper ff)rmulation and a detailed proof of these statements
will be given in §34 in connection with Kneser's theory.
144 Calculus of Variations [Chap. IV
hood [p) of ©0 such that to every point P of (p) there can be
drawn from the point ^ A a uniquely defined extremal which
varies continuously with the position of the point P and
coincides with ©q when P coincides with B.
Let now
be any ordinary curve drawn from ^4 to B and lying wholly
in the neighborhood {p) of ©o, -s' denoting the arc of the
curve 6 measured from some fixed point of (S, and let A J
denote the total variation
Then a reasoning' analogous to that employed in §20 leads
to the folio wing expression for A,/ (Weier stress's Theorem):
r'l _ _ _ _
Ae/= I E{x, y; p. (i: p, (j)d.s , (56)
where (j-. Tj) denotes a point of (S, /j, 7^ the direction-cosines
of the positive tangent to 6 at (j-, //), and p, q the direction-
cosines of the positive tangent to the unique extremal of the
field passing through (.r, ij).
It now only remains to show that, as a conseqvience of
our assumptions (II') and (IV), the integrand in {■)(')) is
never negative* along the curve (S.
Let (x, y) be any 2:>oint of the above defined neighborhood
(p) of ©0 ^ncl let, as before, p, q denote the direction-cosines
of the positive tangent at {x, y) to the unique extremal of the
field passing through (x, ?/), and p, q the direction-cosines of
any direction o, and define
1 Or better from a point A in the vicinity of A on the continuation of i'„ beyond
A, as in §19, c).
2 The lemma of §8 must be replaced by the lemma of §28, a). Other proofs of
Weiers trass's theorem will be given in §37 in connection with Kneser's theory.
3 It is in this last conclusion that the problem in parameter-representation differs
essentially from the problem with x as independent variable ; compare §22, c).
i
§28] Weiersteass\s Theory 14.")
Ei(.r, yi p, q; p, q)
( E{x,y; p, q; p,q) . / ~ i ~ , ^ ,\
\ , ~ I :^T— > when 1 -(^>7. +yr/)^() ,
( i'\ (a:, ?/, p, g) , when 1 - ( j^Ji + qq) = 0 ,
/. ('., p =p , q =q .
The direction -cosines p, q are single-valned and continuous'
functions of a-, ij in the neighborhood {p) of S,,. Hence it
follows, on account of (54), that Ej is a continuous function
of .r, u, 6 in the domain
{x,y) in {p) , 0^0 ^2,
iTT
and since, according to our assumptions (II') and (IV), E^
is positive along ©o for every value of 6, it follows from
general theorems on continuous functions that Ej is positive
throuijhout the domain
{X,!J) in (p) , 0^6^^ 27r ,
provided that p has been taken sufficiently small.
The integrand of (56) is therefore positive at all points
of 6 at which the direction J), q does not coincide with the
direction p, q, and zero where these two directions do coin-
cide. Hence AJ">0 unless it should happen that j> ^p, q—q
all along 6, in which case we should have At/=0.
But the latter alternative is impossible" unless 6 be iden-
tical with ©Q. This proves that the arc @o cictually minimizes
the integral J if the four conditions enumerated at the
beginning of ^^6', d) arefidfiUed.
Example VII (see p. 97) :
F = g{.r,y)Vx" + y" .
Here
Ei(^, y; p, q; p> '/) = f/(-^"> u) ,
1 Compare §34, Corollary 4.
-The proof is similar to that given in §22, a) ; for the details compare Kneser.
ie/ir/mc7i, §22.
14r> Calculus of Variations [Chap. IV
and therefore Condition (IV' ) is satisfied if
along- (£•„.
This shows that in the prolilem of the l)rachistochrone an arc
AB oi the cycloid (26) actually furnishes a mininuini if it contains
no cusp (compare p. 136) .
Corollary : If the condition
F]^{x, y , cos y, sin y) > 0 (H'l )
is satisfied for every point [x, y) of @o ^^^ fa'' ct'c^'H ruliic
of 7, then (II') and (IV') are a fortiori satisfied, the latter
on account of (e54).
Example XI (see p. 128) : Tlie Geodesies.
Here
EG-F^
F,=
{VEir' + 2Fu'v'+Gv"'y'
Hence under the assumptions made on p. 128 concerning the nature
of the jjortion of the surface to which the geodesies are restricted,
Condition (Ila ) is always satisfied.
e) Existence of a minimum ^^im Kleinen'': We add here an
important theorem which has been used, without proof, by several
authors' in various investigations of the Calculus of Variations,
viz., the theorem that under certain conditions two points can
always be joined by a minimizing extremal, provided only that the
two points are sufficiently near to each other. An exact formula-
tion and a proof of this theorem have first been given by Bliss.-
His results are as follows :
We suppose that in addition to our assumptions concerning the
function F (see §24, bj) the condition
Fi{x, y, cos y , sin y) > 0 (58)
1 Weibrstrass (Lectures, 1879) in his extension of the sufficiency proof to curves
without a tangent, see §31: Hilbert in his existence proof (see the references siveii
in chap, vii); Osgood in his proof of the identity of Weierstrass 's and Hil-
bert's extension of the meaning of the definite integral J to curves without a tan-
gent (Transactions of the American Mathematical Society, Vol. II (1901), p. 29.>).
^Transactions of the American Mathematical Society, Vol. V (1904), p. 113. His
proof is based upon an extension of a theorem of Picaed's concerning the exist-
ence of an integral of a diiferential equation of the second order, taking for two
given values of the independent variable two arbitrarily prescribed values {Traits
df Analyse, Vol. Ill, p. 91).
§?8i Weiebstrass's Theory 147
is fulfilled for every point (x,y)in a finite closed region So con-
tained in the interior of S, and for every value of 7.
Since Fi(x, y, cost, sin 7) is continuous at every point (x, y) of
S and for every value of 7, a finite closed region, S, , contained in S
and containing So in its interior, can be determined such that the
inequality (58) still holds for every point {x, y) of S^ and for every
value of 7.
Under these circumstances, if a positive quantity e be assigned
arbitrarily, a second positive quantity p^ can be determined such
that from every point Piixi, ?/,) of So to every point Piix^, yi) in
the circle (Pi. p), where 0<p^pe, an extremal of class C can be
drawn which lies entirely in the circle (Pi, p), and which has the
property that at every one of its points the slope with respect to
the direction P1P2 is numerically less than e. Moreover the circle
(Pi, p) lies entirely in the region Sj .
This extremal is at the same time the only extremal of class C
which can l)e drawn from P, to Pj and which lies entirely in the
circle (Pi . p ) .
Let this extremal be represented by
y = ^(t; Xi, yr, *2. Z/2) >
0 ^ f ^ f.
Then there exists a positive quantity ?, independent of Xi, y^, X2, y-z,
such that the fvmctions *, ^, *f, ^t iire continuoiis and have con-
tiniious first partial derivatives with respect to t, Xi, yi, x-2, yi
throughout the domain
^ 1 < T; { -^'i . Ui) ill SL. ; 0 < 1 ' {x., — x,f + {y. — y,f < p
Finally also the value t = t-i which corresponds to the point Pt
is a continuous function with continuous first partial derivatives of
./•i. //i, Xi, yi for all iDOsitious of the two points Pi, Pi here consid-
ered.
For the parameter t of a point P of the extremal we may choose
the projection of the vector PiP upon the vector P1P2.
This unique extremal P1P2 furnislies for the integral J a
.'^waller value than any other ordinary curve (£ n-Jtich can be
drawn from Pi to Po and u-ltich lies entirely in. tlic circle (Pi, p).
If in addition to the inequality (lib ) the further condition
F{x, y , cos y, sin y) > 0
148
Calculus of Variations
[Chap. IV
is fulfilled for every point (x, y) of the region S,, and for every value
of 7, and if both points Pi and P2 lie in ffiu, then the unique
extremal P1P2 furnishes for the integral J even a smaller value
than any ordinary curve, different from the extremal P1P2, which
can l)e drawn from P, to Pi^nd which lies entirely in So, provided
that IP1P2' 5aj, where Po is a certain positive quantity less than p
and independent of the position of Pi and P2 .
,,>i
§29. BOUNDARY CONDITIONS
(i) Condition along a scr/ment of ihe honndary: If the
minimizing curve 0231 has a segment 23 in common with
the boundary of the region S to which the admissible
curves are confined (see Fig. 7), we obtain the condition
which must hold along the boundary
as follows :
In order to fix the ideas, we sup-
pose that as we go along the boun-
dary 6 from 2 to 3, i. e., in the
positive direction of the minimizing
curve, the region U lies to our left.
Let the curve (S be represented by
6: x = ^{s), y = (f{s),
s denoting the arc, and suppose that the first and second
derivatives of ^(s) and •^(s) are continuous along 23.
Then if we construct at a point (.r, 7/) of 23 a vector of
leno-th u , normal to 23 and directed toward the interior of
S, the co-ordinates of its end-points are
X = X + $ , y = y + 7] ,
uy' iix'
FIG. 7
where
i =
V =
Vx' + y"' • vx^ + y"
Hence if we substitute for u a function of s of the form
u = tp ,
1 Due to "Weieestrass, Lectures, 1S79; compare §10 and Knesee, Lehrbuch. §44.
§29] Weiersteass's Theory 149
where e is a positive constant and p a function of s of class
D' which is ^0 in (.'*2^3) and vanishes at So and S3, the pre-
ceding formulae represent for sufficiently small values of e a
curve wdiicli remains in the reijion 1R and which is therefore
an admissible variation of the arc 23.
For this variation we obtain, if we apply (15a), for A,/
the expression
A J = e [- £ fp l/r^+r ds + (e) J , (59)
from which we infer, by the method of ^5, that in case of <(
minimum ire )iiiisf have
f^O along 23 , (60)
where T is the expression (19) in which x, y are replaced
by J-, y.
If Fi is positive not only along the arcs 02 and 31 but
also along 23, the preceding condition admits of a simple
geometrical interpretaiion :^ For, if we introduce in the
expression for T the curvature 1/r of 6 at a point P, and
denote by 1/r the curvature at the same point P of the
extremal which passes through P and is tangent to S at P,
then (60) may be written, according to equation (la) of p, 123,
footnote 1,
^^l . (61)
r r
Hence if r>0, i. c, if the vector from the point P to the
center of curvature 71/ of 6 lies to the left of the positive
tangent to 6 at P, also r must be positive and the center of
curvature M of the extremal must lie between P and M or
coincide with M.
If, on the contrary, r<0, i. e., if the vector PM lies to
the right of the positive tangent, M must lie either on the
iThis is an extension of the results given for the special case F=Vx +yby
Kneser, Lehrbuch, p. 178.
l-jO Calculus of Vaeiations [Chap. IV
opposite side of the tangent to M (when ?• > 0) , or else on
the same side as, but beyond, M (or coincide with M).
If, as we go along the boundary from 2 to 3, the region
U lies to the right, the condition becomes:
r^O along 23 (60a)
or
-^i . (61a)
r r ^
h) Conditioiii^ (if ihe poiiifx. of ffansition: An additional
condition must hold at the point 2 where the minimizing
curve meets the boundary, and likewise at the point 3 where
it leaves the boundary. To obtain the first, let h be a posi-
tive infinitesimal and let 4 be the point of 6 whose parameter
is s = S2 + //; join the points 0 and 4 by a curve 6 of the
type defined in §28, a), and consider the variation 0431 of
the minimizing curve. For this variation we obtain, accord-
ing to (41)) and (53) :
A J = J„i — ( J„2 + J-u) = — /i [e (.r, , y.2 ; 2h , (h > Pi > <i2) + (^O] ,
where _/92, Q2 ^^^ Pij Q.2 ^i'® the direction-cosines of the posi-
tive tangents at 2 to the curves 02 and 23 respectively.
Similarly, if we join the point 5 (s = S2 — h) of 6 with
the point 0 by a curve 6, we get, according to (49a),
A J" = ,7„5 -f J^2 — ^u2 = + /i [e (,r. , 2/2 ; 2>2 , Qi-, P2, q^) + ('O] »
whence we infer in the usual manner that at the pomt 2 the
folloLving condition must he satisfied:
E Ca , y. ; 2h , q-i ; pi , ^2) = 0 . (62)
Applying similar reasoning to the point 3 and making
use of (50) and (50a), we reach the result that at the point 3
the analogous condition
E (x-i , y., ; Pi , 0-3 ; P3,qi) = 0 (63)
must he satisfied, where ^^3, q^ and p^, q^ are the direction-
cosines of the positive tangents at 3 to 31 and 23 respectively.
§29]
Weierstrass's Theory
151
The two conditions (62) and (63), together with the con-
dition that the minimizing curve must pass through the
given points 0 and 1, determine in general the constants of
inteirration of the two extremals 02 and 31.
If the problem is a "regular"' one, /. c, if the condition
Fi{x, y , cos y , sin y ) =t= 0
is satisfied at every point (-r, y) of the region B and for
every value of 7, it follows from (51) that (62) and (63) can
only be satisfied if
i>2 = j>2 , g2 = (i2 ; 75s = p3 , <ii = qs ■
This means geometrically that fite arcs 02 (tiid 31 niiisf
iouch the bouii(J((rij of flic points 2 and 3 in such a manner
that their positive tangents coincide with the positive
tangents of the boundary.
c) Case ichere the minimizing cnrve tias onlij one point
in common icitJi the boiinda)-!): Sup-
pose that the minimizing curve has
only the point 2 in common with
the boundary 6. Then the arcs 02
and 21 must be extremals. To find
the point 2, let 3 be the point of 6
whose parameter is s = S2 + /^ and
consider a variation 031 of the curve 021 (see Fig. 24).
For this variation we obtain
which, according to (49) and (19a), is equal to:
A J = /i [E(a'2, Ui ; Pi, ^2 ; Pt, §2)
— E U\ , iji ; Jh, ^2 ; p2, Qi) + (^0] »
where p2, q..; po, (h\ Pi^ 92 are the direction-cosines of the
positive tangents to the arcs 02, 21, 23 respectively at the
Doint 2.
FIG. 24
152 Calculus of Variations
[Chap. IV
Similarly, if 4 be the point o'f 6 whose parameter is
,s = ,s'2 — //, and we consider a variation 041 of the curve 021,
we obtain
A J = [ Jo4 - J^n + Ji'l J + ['^4, - (^42 + ^2.)]
= - h [E {x.2 , y.2 ; ih , q-i ; ]h , q^
— E (,r2 , //, ; Pi,q2; fh , 52) + {h)~\ •
Hence we infer that (if llic jtoiitt 2 the condiiioti
_ + +
E {.r^ , Vi ; Pi , qi ; Ih , ^2) = E {.r., , ij., ; p., , q., ; p., , ^2) (64)
uiiisl he satisJiciL
d) Example VI' (see p. 84) :
F = Vx'^+tr .
Suppose the region 2J to be the whole plane with the exception
of the interior of a simply closed curve of class C", and suppose
that the straight line joining 0 and 1 passes through the excluded
region.
The minimizing curve must be com-
posed of segments of straight lines and
segments of the boundary, the latter
3 turning their convex side outward
since in this case 1/r = 0 and therefore
or
0
FIG. 2.5
according as 23 is described positively
or negatively with respect to 15 . The
lines 02 and 31 must touch the arc 23
positively at 2 and 3 since F-iix , y, cost, sin 7) = 1.
Again,
E {x, y ; cos 0, sin 0 ; cos 6 , sin ^) = 1 — cos (9 — 6)
Hence if the minimizing curve is to
have one point 2 in common with the
boimdary, the condition
cos {O2 — 62) = cos (6.2 — 62)
must be satisfied at 2. This means
that the lines 02 and 21 must make
equal angles with the tangent to the
boundary at 2 .
1 Compare Knesee, Lehrbuch, p. 178.
FIG. 26
§301
Weieestkass's Theory
153
e) Example I (see p. 1):
F = y\ .r'-+y"' >
the region S is the upper half-phrue :
The extremals are here
a) The catenaries
X — t ,
P) The straight lines
1 ^-l^
y =z a cosh ;
X = a.
Since the catenaries never meet the ,<'-axis,
the only possil^le solution containing a seg-
ment of the boundary consists of the ordi- /^
nates of the two given points :
/y» .-v»
and
X = X
FIG. 27
1 )
together with the segment 23 of the a^-axis between them.
Since along the j:'-axis
T=-l .
condition (60) is satisfied along 23 ; and since
^ {x, y ; cos 6 , sin 6 ; cos d , sin ^) = ^1 — cos {d
conditions (62) and (63) are satisfied at 2 and 3.
^})u ,
§30. the case of variable end-points
The methods explained in §23, slightly modified, can be
applied to the case when all curves considered are expressed
in parameter-representation. In one respect the treatment
of the problem in parameter-representation is even consid-
erably simpler, viz.: the variation of the limits of the inte-
gral J can be completely avoided. For let
@o: x = cl>{t) , y = il^{t) , t.^t^t, , (65)
be the minimizing curve, and
g : X = <^(t) , 7/ = iJ/(t) , To ^ T ^ T, , (66)
154 Calculus of Variations [Chap. IV
a neighboring curve. If we then apply to 6 the "parameter-
transformation'' (see §24, (())
■ ,^ (/■-4)(r-r.)
we obtain for 6 a representation in terms of the parameter
t for which the end- values are /q and f^, the same as for ©q-
We consider briefly the case where the point 1 is fixed
and the point 0 movable on a given curve of class C ' :
6; I- = <?(a) , .(/ = «A(«) • (67)
The minimizing curve (65) must again be an extremal; it
begins at a point 0 of the curve (S whose parameter on 6 we
denote by (Iq. Let 2 : [fi ^^ Oq -\ e) be a point of (l in the
vicinity of 0, d"o + ^o? Z/o + ^o i^s co-ordinates; then
,^=e[^'(«.) + (^)] ' '?" = " [f(«o) + (e)] ■
An admissible variation CS of sufficient generality which
^ — -^ passes through 2 and 1, can easily
§-. ^ — ~>i I ^® constructed analytically in the
/^K'''^^^'^^'^'^ y form
vl,-'^ where
7^ FIG. 28 • . .
u, V being two arbitrary functions of / of class C which
vanish for t=^ti and are equal to 1 for /^/q.
For this variation of the curve @j we obtain, according
to (15b),
Substituting the values of f , ?; at /q and f^ and remembering
that ^=0 along the extremal ©o? we get^
where
1 Weierstkass, Lectures, 1882.
§30] Weiersteass's Theory 155
^, dx ^, dy
da da
We obtain, therefore, the condition of fransversali'fjj in
the form
x'F,.{x, y, x, y') + ~y'F,,{,v, y, x' , y') f = 0 (68)
I
where x , y' refer to the extremal @o? ^' ■> D' to the given
curve 6.
Example XI (see j). 128) : The Geodesies. The condition of
transversa lity is
u(Eu'+Fv')-^d'{Fu'-\-Gv')=0 ; (69)
its geometrical meaning' is that the geodesic must ])e orthogonal
to the given curve.
The focal point is determined by the following formulae :^
Let Aq and Bq denote the following two constants
_ x"F^.+ y"F,j,-{-Lx'^+2Mx'Ti'+ Ny"
_ (x'y'-y'xyF, "
0 — ;^'2 _j_ r.'-i
(70)
x' + y
where the arguments of i^_,.., Fy-, F^ are a'o, //o, ■'"o, //o ^^^f^
iv, JjT, iV" are defined by (35). Bq is different from zero if
we suppose, as in §23, that @o ^^^^ ^ ^^^ ^^t tangent to each
other at the point 0. Let further
H {t„ , t) = A,® {U , t) + B, ^-^j^ , (71)
the function © being defined by (42). Then the parameter
/q' of the focal point is given by the equation
H(f„0=0. (72)
If
x = (i>{t, a) , y = ij,(t, a)
1 Compare Bianchi-(Lukat), Differentlalgeometrie, p. 65.
2 See Bliss, Transactions of the American Mathematical Society, Vol. Ill (1902)
p. 136.
156 Calculus of Vaeiations [Chap. IV
is the extremal which passes throu<Th the point a of the
curve 6 and is cut transversely by 6 at that point, and if
A(/, a) denotes the Jacobian of the two functions ^, i/^ with
respect to /, a, then'
A(/,a) = CH(A,,/) (73)
which proves the geometrical meaning of the focal point.
The question of sufficient conditions will be discussed in
detail in connection with Kneser's theory in chap. v.
§31. WEIERSTRASS'S EXTENSION OF THE MEANING OF THE
DEFINITE INTEGRAL
I Fix, y, ,r',u')
dt
We have confined- oiurselves in all the preceding investigations
to "ordinary" curves. This limitation was indeed necessary for
most of our proofs, but it is not implied in the nature of the
problem .
The most general class of curves for which the problem has a
meaning would be the totality of cui'ves for which the integral
e/= I F(x, y, x', y')
dt
'0
is finite and determinate.
In many problems of a geometrical origin, however, a still
further generalization is desirable.
a) Example of the lengtli of a curve : Thus, for instance, the
problem to determine the curve of shortest length between two
given points A and B, is not exactly equivalent to the problem to
minimize the integral
J = f ' Vx" + tj" dt ,
because the length of a cmwe cannot in all cases be expressed by
this integral.
The length of a continuous ciu've
iSee Bliss, loc. cit., p. 140.
2 Compare §24, a) and c).
§31] Weierstrass's Theory 157
£: jc = <b{t) , u = ^{f) , U^t^t, (74)
is defined ' as folio us :
Consider any partition n of the interval (Vi) into n subintervals
by points of division t,, tj, . . ., t„_,, where
to < Tl < T.^ • ■ ■ < T„_, < f, ,
and denote by A, Pi, P2, ■ ■ -, P„_i, B the corresponding- points
of 2, by A'o, To; A,^i; ^'2,^2; • • •; .r„_i, ?/„_,; A',, F, their
co-ordinates. Then the length of the polygon '^n inscribed in the
cui've 6 whose successive vertices are these points, is
I' =11
where ^
AXy = Xp_)_i J',. , A //,, =: .//,._{_, — ?/^ .
If Sn approaches a determinate finite limit'* J as all the differ-
ences (t,,_|_i — T^) approach zero :
J ^ L Su ,
the curve 2 is said to have a finite length whose value is J.
If the first derivatives 0 (/),'/''(/) exist and are continuous in
(foti), the above limit always exists and can be expressed by the
definite integral*
-■&'
f"
T x''-\- y'-df
b) Extension of the meaning of the general integral .• In an
entirely analogous manner Weierstrass'' has generalized the mean-
ing of the definite integral
1 See Jordan, Cotirs iV Analyse, Vol. I, Nos. lOo-lll. This is the definition which
is most convenient for our present purpose ; compare also §44, n), end.
2With the understanding that T|| = f||, .rf|=XQ, j/q^Yq and 'r„=ti , a'„ = X] , y„-^Y■^.
3That is, corresponding to every positive e, another positive quantity S^ can
be assigned such that
for all partitions n in which all the difPerences {^r^,\■^ — t^) are less than S^.
■'Compare Jordax, loc. cif.. No. Ill, and Stolz, Transactions of the American
Mathematical Society, Vol. Ill (1902), pp. 28 and 303.
^Lccturex, 1879; compare also Osgood, Transactions of the American Mathemat-
ical Society, Vol. II (1901), pp. 275 and 293.
158 Calculus of Variations [Chap. IV
J = 1 F{x, ij, y, !i')dt ,
taken along a continuous curve S (defined ]>y (74)) which lies
entirely in the interior of the region S of § 24, b).
Consider as before a partition n of the interval {Utx) and denote
by TFii the sum
Then, if the curve 2 is of class ^ C , this sum Wu approaches a
determinate finite limit as all the differences (t„_j_i — t,,) approach
zero, viz., the definite integral^ Ji{AB):
LWn= r F{x,u,y,v')dt . (76)
This remains true when S has a finite number of corners.
We now agree to define the definite integral
I F{.v, y, x', y')dt ,
iThis implies tliat (/>'" (?) + >/'^(<) +0 in («(,<,); compare §24, a).
2 For the definite integral may be written
» — 1 H — 1
where t|, is some intermediate value between T^, and ■>■,,_[_ j . On the other hand
where r^ and t]'' are again intermediate values between t^ and Tj, , j. Hence we
have, on account of the homogeneity of F,
From the theorem on uniform continuity applied to the function F (x, y, x', y') on
the one hand, and to the functions <i> (t), 'I' (t) and their derivatives on the other hand,
it follows that corresponding to every positive quantity e another positive quantity
6j. can be determined such that
I F{<f, (t„), ^|J (tJ, </.-(t-), ^'(r;')) -f(<i, K), ^ (t;,), 4>\t[,), 4>-{tI,)) \ <€
fori' = 0,l,2, • • •,?! — 1, provided that all the differences (Tj, , j — t^) are less than
6^ . Hence
which proves our statement.
§31] Weieestrass's Theory 159
takt'ii along' the curve 2, as the limit of Wn in all cases in which
this limit exists and is finite : and we denote its value by J*(AB) :
J*{AB)=LWn . (77)
This is a natural extension of the definition of the definite inte-
gral since it coincides with the ordinary definition for all "ordi-
nary" ciu'ves.
c) First modification of Weirrstrass's definition: Various
modifications of this definition will be of importance in the sequel:
Since the curve 8 is supposed to lie in the interior of the region
S, the rectilinear polygon whose vertices are the points A, Pi, P2,
■ ■ ■ , P,i-i, B will likewise lie in the interior of S, provided that
the differences (t^4_i — r^,) have been taken sufficiently small. Let
Vn denote the value of the integral J taken along this polygon
fi'om A to jB.
If, then, the curve 2 is rectifiable, and if one of the two sums
Fii and TFn approaches * for LAt = 0 a determinate finite limit, the
other approaches the same limit,^ so that we may also define
Jf{AB) = LVn . (78)
d.) Second modification of Weierstr ass's definition: If the
curve 2 is rectifiable and lies in a finite closed region So {con-
tained in the interior of the region S) in luhich the condition
Fi {x , 7/ , cos y , sin y ) > 0 (58)
is fulfilled for every value of 7, then the preceding extension of
the meaning of the definite integral J may be modified as follows :
Let a positive quantity e be chosen arbitrarily. Then deter-
mine for the region %, the quantity Pe defined in § 28, e) and choose
a positive quantity p ^ Pe arbitrarily. Further select, according to
1 See Osgood, Transactions of the American Mathematical Society, \ol. II (1901),
p. 293. If Z^^] and y^^_i denote the length and the amplitude of the vector P^P^ i j ,
the difference ^u~^^^n "^^y be written in the form
»— 1
^U - ^'^H = ^ ^ I [-F'(-?v-)-i , ^iz+i , cos 7,,_j.i , sin Vj.-fi)
where x^_,_ j =.c^ + s cos 7,.4-i , y,,.i^i- 'Jy + s sin y^_^^ .
The above statement follows, then, from the theorem on uniform continuity
applied to the function F{x, y, x', y).
liiO Calculus of Variations [Chap. IV
the theorem on uniform continuity, another positive quantity 5 so
small that
!</.(r)-c^(r')i<p//2 , \4;{t')~^{t")\<p/^ 2
for every two values /', /" of the interval (/,/i) for which
\t" -f'\<S .
Finally choose the partition II so that
T„
— T,, < 8
for >'=0, 1,2, . ■, H -1.
Then the distance | P„P„+il is less than p. and therefore we can,
according to §28, e), inscribe in the cnvve 'il a unique polygon of
minimizing exirevials with the points A, Pi, P^, • • ■, P»-i, B for
vertices, i. e., we can draw from P,. to P^+i a unique extremal (5', -i
of class C which lies entirely in the circle (P„, p) and which fur-
nishes for the integral J a smaller value than any other ordinary
curve which can be drawn from P„ to P^+i and which lies entirely
in the circle (Pv , p) , Moreover, at every point of Q^+i the slope
with respect to the direction P„P,.+i is less than c
We denote by L^n the value of the integral J taken along this
j)olygon of extremals, /. e.,
Un=^J.,.^,(P.P.+,) . (79)
Then if we pass, as before, to the limit Z. Ar = 0, and if one of the
iivo sums Un cmd Wn approaches a finite and determinate tim.it,
theotlier approaches ttie same limit,^ so tliat we may also define
1 First remarked by Osgood, Transact ions of the American Mathematical Societi/,
Vol. II (1901), p. 293. The statement can be proved as follows:
Let the extremal <?;,^_i be represented by
where, as in §28, e), the parameter < of a point P of e,._^i is the projection P^Q of the
vector P^P upon the vector P^P^_|_, , and lv-\-i is again the distance | Pi,Pi,_^^ | . If
we denote by y^r j the amplitude of the vector P^P^^i and by u the perpendicular
QP with the sign + or — according as the point P lies to the left or to the right of
the vector P,,P^ i i , then we have
*i.-l-i it) = .tv+« cos 7;,_,_i - u sin Vr+i ' 'Pi'+i W = !/v + t sin 7,.+i + " cos v^_j_, ,
*»>+! (0 = cos Vy_,_] - tr sin 7i,_Li , >^;,_|.i (0 = sin 7,,_|_j + m' cos y^_j.i .
Hence if we write
§31] Weierstrass's Theory 161
Jf{AB) = L r„. (80)
We shall call the totality of rectifiable curves for which the sum
Wn approaches a determinate finite limit, '* the class (K).^'
e) Extension of the sufficiencij i^roof to curves of class {K):
After these preliminaries, let (S,, denote an extremal of class C
dratni from A to B and lying icliolhj in. the interior of the region
S. We suppose that Qn does not contain the conjugate A' to the
point A, and that for every point {x, y) of @,j and for every
value of y the condition
we have for every / in the interval (01^ , j)
! f V I 5 P , ! 'J^ ! 5 P ,
since (?^ i j lies in the circle (P^, p) ; and
KJ < e , I 0 J < e ,
since the slope u' of (?j, , , at the point P with respect to the directiou P»'Pp4_] is
numerically less than e .
Apnlying now to the integral J,? the first mean-value theorem we obtain
v-\-l
J^(-^^,(Pr-P^-fO= h'^iFi-i-y + i^. 2/^-f ?^,cos7^_{_l + r,,,sin v^_(_i+5'y) ,
where the argument of ^y.Vy, 4V . ^^ is some value of t between 0 and 1^^^ .
On the other hand, we have on account of the homogeneity of F,
P(.<V, 2/^. A.C,,, A 2/ J,) - l^,_^Fi.r^,. :v. COS 7v_j_i, sin y^_|.i) .
The extremal of p,._^] — though it need not lie entirely in the region S^ — certainly
lies in the larger region S) defined in §28, e).
Further, the function Fix, y , .<•', 2/) is uniformly continuous in the domain;
(.-■,//) in Si , \-a^Vx-'^ + /^^\ + a ,
wherr- a is any positive quantity less than 1.
Hence if a positive quantity <t be assigned arbitrarily, the (quantities «, p and S
can be chosen so small that
lPUV-r€''l/.' + '^ri cos v^^i + r,,, sin 7,,_;_, +5"^)
- F (.IV . Hi, , cos y^_^j , sin Y;,_|_, ) I< o- ,
f)r>' = 0,l, • ■ •, /I — 1 , and therefore ,
II — I
ICn-Tr-n'<<r^?^+, .
r=(i
But if, as we suppose, the curve v lias a finite length /, we have
H-l
s
and therefore
which proves the above statement.
1 Without multiple points.
162 Calculus of Variations [Chap. IV
F,(.r, ?/, cos y, sin y) > 0 (Ha')
is fnlfiUed.
Then we can construct, according to § 28, d) and § 34:, about the
extremal @o a field ^^ which lies in the interior of S ; and if we take
k sufficiently small the inequality (Ha') will be satisfied throug-h-
out the region ^^ •
Now let S he any curve of class (K), not coinciding with (^o,
beginning at A and ending at B, and lying entirely in the inte-
rior of S'i ; let it be represented by (74), We 2>ropose to }>rore
that
J.,<Jf, (Si)
Jf hexng defined as in b).
Proof .-^ We may apply to the cm've S the results of d), the field
B,, taking the place of the region there denoted by S,, .
Accordingly we can choose a partition n of the inteiTal (Ai/i),
whose points of division P.. do not all lie on Go, so that the distance
P.P.+,!<p/3 , (v = 0,l,--- ,n-r) ,
and that at the same time the arc P, P^+i of id lies entirely in the
circle (Pv, p/3), where p has the same signification as in d), and is,
moreover, chosen so small that the circle (Pv , p) lies entirely in the
interior of Bj, .
We may then, on the one hand, inscribe in S a polygon of mini-
mizing extremals with the vertices A, Pi, P2, • • -, P„_i, B. This
polygon is an ordinary curve; it lies entirely in the interior of ^t,
and it does not coincide with ©,, . Hence we have, according to
§28, d),
Un > t/(?„ >
say
Un-J,,=P>0 . (82)
On the other hand, let n be a partition derived from 11 by subdivi-
sion of the intervals, and so chosen that
\Un-Jf\<l^ , (83)
which is always possible on account of (80). Let Qi, Q2, • • •, ^,„_i
be the points of division interpolated between the points Pv and
• The outlines of this proof were given by Weiersteass in his Lectures, 1879_
Another proof has been given by Osgood, Traivsactions of the American Mathemat-
ical Society, Vol. II (1901), p. 292, by means of the theorem given in §36, c).
§•^11 Weierstrass's Theory 163
Pv+i of the partition n . These points lie in the circle {Pv , p/S)
and therefore
I Q, (^,^, ; ^ 2p/8 , (i = 0,\,-..,m-l; Q, = P,, Q,„ = P,^,) .
Hence the minimizing extremal from Qj to ^,+1 lies in the circle
( {), , 2p/3) aud therefore also in the circle (Py , p) . Hence it follows,
according to d), that the minimizing extremal from Pv to Pv+\ f^^^'-
nishes for the integral J a smaller value than the polygon of min-
imizing extremals P^QiQo ■ • • Q,„-iPv+i, or at most the same
value.' Therefore
Un' ^ Uu . (84)
But from (82), (83) and (84) follows (81), since we may write
Jf - 'h„ = iJf - Uu) + (Un - Un) + (Uu - J>.j •
iViz., when the two curves are identical.
CHAPTER V
KXESER'S THEORY
§32. gauss's theoeems on geodesics
Kneser has given, in his '^LchrhiicJi <lcr V(in'((t(0)is-
rccliniing'"' a new theory of the extremum of the integral
J'= ( F{x, y, X , y' ,)dt ,
essentially different from Weierstrass's theory and reach-
ing farther in its results, inasmuch as it furnishes sufficient
conditions also for the case when one end-point is movable
on a given curve.
Kneser 's theory is based upon an extension of certain
well-known theorems on geodesics, of which we give — by
way of introduction — a brief account in this section.
(i) Suppose on a surface there is given a curve @o whose
points are determined by a parameter v. At a point M{v)
of Sq we construct the geodesic @ normal to (Sq and lay off
on (S an arc MP =u? The position of the end-point P is
uniquely determined by the two
_j (juantities ^f , r.
If we restrict ourselves to such
FIG. 29 a region ^ of the surface that also
conversely P determines uniquely
the values of u and v, these two quantities may be intro-
duced as curvilinear co-ordinates on the surface ("geodesic
parallel-co-ordinates''). According to a well-known theorem
due to Gauss,^ the lines u=^ const, are ortliogoncd to the geo-
desies v^= const.
1 1, e., the length of the arc is | w | , its direction is determined by the sign of u.
2 Gauss, Disquisltioues genet-ales circa superficies curvas, art. 16.
164
\
§32] Knesee's Theory 1<)o
li) Hence it follows that the square of the line element
takes, for this special system of co-ordinates, the form'
ds^ = dit^-j- m^dv^ .
We consider now a particular geodesic, ©q? o^ ^^^^ set
vr:=r const., saj v = Vq, and on it two points 0 : (^fo^ ^'o) ^^^^
1 : (?^i, /•„), where ?/o< '^i-
We join the points 0 and 1 by an arbitrary curve
g : u=u (t) , V = v (t) , (t„ ^t^t^) .
nite integral
©0 is
Then the length of the arc 01 of 6 is given by the defi-
On the other hand, the length of the arc 01 of the geodesic
J = «i — Wo •
r^^du
and therefore the total variation becomes^
This may be written
J= I ^dr
A J= J- J
The integrand is never negative, and can be zero throughout
the whole interval {tqT^ only when 6 coincides with ©q-
Hence it follows that among all curves which can be drawn
in ^ between the two points 0 and 1, the geodesic @o has the
sho}i<:sf U'lujth.^
It should be noticed that the assumption that the geo-
desic ©0 belongs to a set of geodesies satisfying the condi-
1 Gauss, loc. cH., art. 19.
2 Compare Darbotjx, ThSorie des surfaces, Vol. II, No. 521.
3 The conclusion can easily be extended to the case where the point 0, instead of
being fixed, is movable on a given curve orthogonal to the set of geodesies.
166 Calculus of Variations [Chap. V
tions imposed upon the region ^, is equivalent to Jac obi's
condition.
c) The necessity of Jacobi's condition follows from a
well-known^ theorem on the envelope of a set of geodesies:
If the set of geodesies through the point 0 has an envelope
%, and 02 and 03 are two geo-
desies of the set touchinof the
envelope at the points 2 and 3,
then
arc 02 + arc 23 = arc 03 .
The point 3 is the conjugate to 0 on the geodesic 03. Now.
if 2 be taken sufficiently near to 3 on the envelope %, the
compound arc 023 is an admissible variation of 03 for which
AJ=0. And since the envelope % is never itself a geo-
desic/ the arc 23 can be replaced by a shorter arc 23, and
therefore A. J can even be made negative.
Hence the arc 03 does not* furnish a minimum, still less
an arc 01 of the same geodesic whose end-point 1 lies beyond
the conjugate point 3.
The method whose outlines have just been given applies
with only slight modifications to the case where only one of
the two end-points is given, while the other is movable on a
given curve on the surface.
§33. kneser's theorem on transversals and the theorem
ON the envelope of a set of extremals
We consider in this section Kneser's extension to any
set of extremals of the two fundamental theorems on sets of
geodesies given in the preceding section.
1 Darboux, Theorie rfes surfaces. Vol. II, No. 526, aud Vol. Ill, No. 622.
2 See Daeboux, loc. cit.. Vol. Ill, p. 88.
3 Apart from a certain exceptional cas3; see §38.
I
§33] Kneser's Theory 107
(() Construction of a iransversal to a set of extremals:
Let
x = (fi{t, a) , U = ^{f, a) (1)
be a set of extremals for the integral
J= f -F{.r, y, X-', i/)dt ,
containing the particular extremal
@o: x = <f>{t, tto) , y = ^{t, a,)) , t^f^ti ,
whose minimizing properties are to be investigated. A and
B are again the end-points of @o-
We suppose that the functions </>(/, a) and "^(t, <i} are of
class C" in the domain
1 : 3; — £ ^ f ^ T, + e , I a — a„ , ^ d ,
where /q — Tq, T^ — t^, e and d are positive quantities.
We suppose further that for the extremal ©q
4>]{t, a,) + ^U^ «u) + 0 iu {tj,) . (2)
It follows, then,, from the continuity of 0;(/, o) and ■^i{t a),
that the quantities ^o — ^O) ^i — hi ^i <^ can be chosen so
small that also
<^H^a) + .A?(^") + 0 (2a)
throughout the domain IS.
We denote by U^ the rectangle
in the /, r/-plane, and by ^i^. its image in the ,r. //-plane
defined by the transformation ( 1) .
To every point (/, a) of 1E;u corresponds a unique point
[x, y) of ^i- which we shall call "the point [/, «].'" To a
continuous curve
in iSfc corresponds a unique curve in ^i^ :
108 Calculu!^ of Variations [Chap. V
■ ~ x = <l>(g{T),ii{T)) = <k(T) .
which we calP the curve [/^.^C^), a=^h{r)].
The point t of 6 coincides with the point / = r/ (t) of the
extremal a = h{T) of the set (1). If for every value of r
the curve 6 is transverse^ to the extremal a=^h{T) at their
point of intersection, we shall say that 6 is a trdiisversal
to ihe set of extremals [^1).
We write for brevity
F{<l>{t,a), i(f{t,a^, <}>,(f,a), il;,{f, a)) =¥{t, a) , (3)
and use the analagons notation for the partial derivatives of
F and the function F^. Then the condition of transversality
may be written
But
dx . df , , da dJi , dt , , da
hence, remembering the relation (D) of §24:, we get
F(f,«)^+[F,,.(«,a)<^„(f,a) + F,.(^«)^.,(/,a)]^ = 0 . (5)
This differential equation for the functions t and (< of t is
the necessary and sufficient condition that the curve ^ may
be a transversal to the set (1).
We now introduce the further restricting assumption*
that
F(f,a„)^0 iu (foO • (6)
1 For the deductions of this section it is not necessary to assume that also
conversely to every point (x, y) of ^j. corresponds a unique point (t , a) of S^. , pro-
vided that we consider the points and curves of ^;i only in so far as they are the
images of definite points and curves of 1&^., and this is what our notation is to indi-
cate. Accordingly two points [r, a ] and [f , a"] of g-^. are considered as distinct-
even if they should have the same co-ordinates .r, 2/— if the points (t\ a ) and (r, a")
of i&i^ are distinct.
■i Compare §30. 3 W'e shall free ourselves from this restriction in §37, r).
§33] Kneser\s Theory 100
It follows, then, from the continnity of F(7. d). that wo can
take To, T^ so near to /q, ^i and /,• so small that
F(^a)4=0 (6a)
throughout the region S/..
If the condition [6a) is satisfied, it follows from Cauchy's
existence theorem' on differential equations that throiujli
every point [/', a' J of tlie region ^^. a uniquehf defined
transversal of the set (1) of extremals can be drawn, rep-
resentable in tlie form
y = ijj{f, a) j
X(<'i) being single- valued and of class C" in the vicinity of
a = a', and taking for a^^a the prescribed value t = t' .
The curve S may degeyieraie^ into a point, viz., when the
functions <^(t), ^(t) reduce to constants, say oc^, if. For
such a degenerate curve the condition of transversality (4)
is evidently always satisfied.
Conversely, if any point (x^, if) in the interior of the
region S of §24, b) is given for which
i^:(x-°, /, cosy, siny) + 0
for every 7, and if we construct by the method of §§15 and
27, c) the set of extremals through the point [x^, y^), this
point may always be considered as a degenerate transversal
to the set of extremals. For there exists, according to
§27, c), a function t^[a) of class C , such that for every a
within certain limits
the point {x^, //^) is therefore indeed the image of the curve
t^^t^(a) in the /, rt-plane.
1 Compare p. 28, footnote 4.
2Compare footnote 1, p. 1G8. 3 See Kneser, Lchrhuch, p. 47.
170
Calculus of Variations
[Chap. V
b) The function ii(f, a): Let A^ be a point on the con-
tinuation of @o beyond A, corresponding to an arbitrary
value t = f^ between Tq and /„, and let'
/ = f{a)
be the transversal %^ passing through the point [/«, ciq].
We suppose k taken so small that in the interval
(aQ — h\ (iQ^k) the function /o(a) is of class C and
To<t^{a)<Ti. The curve / = ^"(o), interpreted in the
/, (/-plane, divides the rectangle Sfc into two regions ; we denote
by Sfc that one for which
and by ^^ its image'' in the
X, //-i)lane.
We consider now any point
F:[{,(i]ol^^.. Theextremal
of the set (1) which passes through P, meets the curve %^
at the point PO; [/«, a].
Now denote by u or n (/, a) the value of the definite integral
a = a„ + fc
a ^ la — k
u — 1 ^ F(f, a) dt = n{t, a) .
0)
The function v{f, a) is single-valued and of class C in
the domain S^ ; moreover it represents,** in Ea:, the value of
our integral
J= j F{.r, y,x',y')clt
taken along the extremal © from the point P^ to the point P:
u{t,a)=J^iP'P) .
iWhen the transversal z" shrinks to a point, the function ^(a) becomes iden-
tical with the function so denoted at the end of a).
2In Fig. 31 S>'^. is the non-shaded part of §;_,.
3 Only in S;^., since we always suppose that the lower limit of the integral J is
less than the upper limit; compare §24, b).
§33] Knesee's Theory 171
The partial derivatives of u{t, a) are :
^| = F(^a), (8)
9w ^/.o ...df , r'9F(f,a)^^
But
^~^j~^ = F. <^„ + F, ^„ + F,. <^,„ + F,. ^,,
= I [F.. <^„ + F,. ^„] + <!>„ [f,. - I F,..] + V'., [f„ - g^ F„ ] ,
since <j>ta = 4>at , ^ta = ^at ■ ^OW
F,- g-^F,. = 0 and F, - g^F,. = U ,
since ^(/, a) and i/r(/, a) satisfy Euler 's differential equation.
Hence we obtain
= (F,, «^„ + F,. ^„) - (f ^ + F,. <^„ + F,. tj I .
du
da
But the second term disappears since / = /°(r() represents a
transversal and therefore satisfies the differential equation (5) .
Thus we finally obtain
^ = F,. {t , a) ct>^, if , a) + F,, (/ , a) t. if , o) . (9)
If the point P : [/, «] moves along a curve (S defined by^
t =g{T) , a = // (t) , /. e.,
U = <^(</(T),/i(T))=^(r) ,
71 becomes a function of r whose derivative is, according to
(8) and (9) :
1 The functions g (t) and h (t) are supposed to be of class C ' and to furnish points
{t, a) in a^. so long as t is restricted to a certain interval (tt") to which we confine
ourselves in the following discussion.
172 Calculus of Variations [Chap. V
^ = F(^ a) ^ + [F. (^ c,)<l>,,{t, a) + F,.(/, a) ^„(f , «) 1 '1^ ,
cIt ^ (It L J clT
The extensions of the two theorems on geodesies of §32
follow immediately from this formula by specializing the
curve 6.
c) Kneser's Tlieoix'm on Transversals: In the first place
we suppose that the curve 6 is a transversal to the set (1).
Then it follows from (4) and (10) that
dr
and therefore it = const.
Thus we obtain the
Theorem I : Two iransversals %^ and %^ to fhe same set
of extremals intercept on the extremals arcs along ivhich the
intef/ral J has a constant value.
More explicitly: If ©' and S" are two extremals of the
set (1) meeting the transversals %^, %^ at the points Pq, P{
I and Pq , P[' respectively, then
J^.{P',P[) = J,..{P','P[') . (11)
i
Conversehj: If along the curve
^ ^.^ _ l. ^^ the function u{t. a) is constant,
1° FIG. 32 ^ V / '
then %^ is a transversal of the set (1) .
In the special case of the geodesies, transversality is iden-
tical with orthogonality/ and therefore Kneser's theorem
is indeed a generalization of Gauss's theorem on geodesic
parallels.
The theorem remains true if one or both of the two
transversals shrink to a point ;^ thus we obtain the following
corollaries :
1 Compare §30, a). 2 Compare the remark at the end of a).
Pi^ e- ^ — '^.
p; \ a- -J ^i"
§3.31
Knesek's Theory
173
Corollru'ij /.' If X^ is a transversal to the set of extrem-
als through a point Pq, then the integral J has the same
value if taken along the different extremals from the point
Pq to the curve %^, and vice versa.
Corollarij II: If 'X^ is a transversal to a set of extremals
passing through a point Pj , then the integral J has the same
value if taken along the different extremals from the curve
S^to the point Pi.
CoroUarij III : If the extremals passing through a })oint
Pq all pass through a second point P^, then the integral J
has the same value if taken along the different extremals
from Pq to P^.
(/) Theorem, on the envelope of a set of extremals: In
the second place, we suppose that the curve 6 is tangent to
all the extremals of the set (1), and therefore is the envelope
of the set.
More explicitly : As it has been remarked before, the
point T of 6 coincides with the point t^=g (t) of the extremal
a = li (t) of the set (1 ) ; we suppose that for every value of
T, at least in a certain interval (t't") in which
m^m*'
the curve CS and the corresponding extremal are tangent to
each other at this common point, so that
dx
dy
dr
4>t
= 0
It follows, then, that there exists a fTinction ni of r such that
dy
dx
i/'f = m.
dr
1 Applied to geodesies, this is Gauss's theorem on geodesic polar co-ordinates,
Gauss, loc. cit., art. 15.
174 Calculus of Yariatioxs [Chap. V
w is continuous in (t't") and can not change sign.^ We
may without loss of generality" suppose that
m > 0 in (t't") ,
/. c, that the positive directions of tlie tdiu/cnfs io the tiro
curves coincide.
From the homogeneity properties of F it follows, then, that
^"^^ . r, /- ~ dx dy\
and therefore, according to (10),
du , /_ _ dx d])\
d-r=^V''d^^ dr) ■
Hence, integrating from t = t' to t=-t" (t < t" ) and
remembering the meaning of v{t, a), we obtain the
Theorem II :^ Let %^ he a transversal to the set of
extremals (1) and % tlie envelojje of the set; let, furtlter,
P' Q' , P" Q" he two extremals
of the set starting from the
points P' , P" of '^^^ and touch-
ing % at tJie points Q' , Q'\ then*
J,..{P"Q")=J,{P'Q')
+ JMQ") , (12)
iThis follows from (2a) and the assumption that
(lf) + (f)'*» "■"■'■)
2 If m is negative, introduce a new parameter
T = — <r on 6 .
3 The theorem in the special case when 2 shrinks to a point is due to Zeemelo,
who proves it by means of Weierstrass's expression for A J in terms of the
E-function (Dissertation, p. 96). The theorem in its general form and the above
proof are due to Knesee; see Knesee, Lehrbuch, §25, and also idem, Mathe-
matische Annalen, Tol. L (1898), p. 27. The simplest case of the theorem is the
theorem on the evolute of a plane curve.
*By a limiting process it can be shown that the theorem remains true if the
assumption
§3-l( Kneser's Theory 175
with the understanding that the positive direction Q' Q" <>n
% has been chosen as indicated above.
The theorem remains true if the transversal X^ shrinks
to a point, in which case we obtain tf;,___^ /
the corollary: /^"^''^t %
PqQ\ PqQ" being two extremals of the set through Fq, and
% the envelope of the set.'
§34. CONSTRUCTION OF A FIELD
Before we can extend to the general case of extremals the
results given in §32, b) concerning geodesic parallel co-ordi-
nates, it is necessary to impose upon the set of extremals (1)
such further conditions that the correspondence between the
two regions H^ and ^;^ defined in §33, o) becomes a one-to-
(ff)>(fr*»
ceases to be satisfied at Q", i.e., if the curve Tv ha> a "cmp" at Q', provided that
there exists a positive quantity m such that
- / (r -r)>^ and - / (r - x)
approach, for Lt = t"—0, finite determinate limiting values not both zero (a condi-
tion which is, for instance, always fulfilled if ST and ff are regular in the vicinity
of t"). The proof follows immediately from the homogeneity property of the func-
tion F; see §24, (8).
iThe two theorems on sets of extremals proved in this section can be derived
by still a different method indicated for the case of the geodesies by Daeboux
(Theorie des Surfaces, Vol. II, No. 536). Let
be a particular extremal derived from the general solution of Euler's equation,
and let M^f{.t=%, x = a^,y = h^;) and ^I■^(t=t■^, x = a^,y = b^) be two points on (?„
which are not conjugate in the more general sense that 0 {.t^ , <„) + 0 . Then it follows
from the theorem on implicit functions that if we take two points -PoC-iV,, ^„) and
■Pi (•'"i 1 2/i) sufficiently near to J/q and M^ respectively, a uniquely defined extremal
can be drawn through Pq and Pj :
g: x=f{t,<t,^) , y = fj(t,a,p) .
The constants a , 3 , the two values of / which correspond on if to the two points
176 Calculus of Variations [Chap, v
one correspondence, or in other words that the set of extrem-
als (1) furnishes a field about the arc ©q.
The proof of the existence of a field is based ui)oii the
following
Tlu'orcm : Let
x = MUa) , y = ^(f,a) (15)
be a one-parameter-set of curves satisfying the following
conditions :
A) The functions cf) and ■yfr are of class C in the domain
T„ — c ^ t ^ Ti -\- € , \a — a„ \^d ,
€ and (I being two positive quantities.
B) The particular curve
x = <j>(f,a,) , y^^{f,a„) (16)
has no multiple points for Tq — e^/^ Tj +e.
C) If we denote by A(/, a) the Jacobian
then ^(^«)
A(f.a„)^0 m (To-£, T, + e) . (17)
P(i and Pj, and consequently also the value of the integral .J taken from Pj, to Pj
along- (f are single-valued functions of a-,, , j/q, Xj , y^ which are continuous and have
continuous partial derivatives in the vicinity of a^, b,,. a, , b^. We denote this inte-
gral J(i; ( P(, P[) considered as a function of x^ , 2/,, , .»•] , (/, , by
it is a generalization of the .greorfes/cd/stonce6c<(t'eewt;('oiJo/»ifs (see Dakboux, loc.cit.).
The total differential of this function can be obtained by precisely the same
method as that which Darbodx applies to the geodesic distance, and the result is
dJUi,, 2/0 ' •«'i ' I/O = ^x'^-^i ' Vi ' ^1'' -Vi') dXi + Fy.{Xj^ , 2/1 , x{, y{) dy^
- l^x'^^u ' Vo ' -^o'' yo') rf-'"o - ^y(-^'Q ' //(I ' -'Vm 2/0 ' ^^0 ' (14)
the derivatives x^', j/q' and x{, y^' referring to the extremal c.
Now suppose that Pq and Pj move along two curves Pf, and ly, whose co-ordinate ;
are expressed in terms of the same parameter t. Then the extremals joining corre-
sponding points of P,) and (;, form a set of extremals with the parameter t, and
"^(^"oi ^05 -''i ! Vi) changes into a function of t whose derivative is obtained immedi-
ately from (14). By specializing the curves iS^ and P, the two theorems I and II are
obtained.
iKneser's proof {Lehrbuch, §14) must be supplemented by a lemma such as
that given below under a) and 6). Compare also Osgood, Transactions of the AmeV'
ican Mathematical Society, Vol. II (1901), p. 277, and Bolza, ibid., Vol. II (1901), p. 424.
§34]
Kneser's Theory
177
Umh'i- tJiese circumsf<inces a positive quant if ij k<Cd can he
taken so small tliat the transformation [15) establisties a
one-to-one corre>ipoii(1eiice hetireen tlie domaiii
*A--
7; ^ /
1 )
a — a,.
k
ill file f . a-phine, and its ima(/e ^^. in tlie x, y-plane.
£.
n„ — k
T.t,
FIG. 35
PIG. 36
Proof: We suppose it were not so ; that is, we siippose
that however small A: may be taken, there always exists in
^y. at least one pair of distinct points (/', a'), (/", a") whose
images coincide at a point {x^ y) of ^j^, and we show that
this hypothesis leads to a contradiction to our assumptions.
a) We first select a sequence of decreasing positive quan-
tities
k> lc,> k^> • • • Av > • • • > 0 ,
beginning with /v and approaching the limit zero, subject to
the following rule : After A'j has been chosen, we select in
the rectangle ^f. a pair of distinct points P[{t[, a[) and
P['{t,[' a[' ) whose images coincide; this is always possible
according to our hypothesis. According to B), a^ and a['
cannot both be equal to ^'^g ; we may therefore choose A^
smaller than at least one of the two quantities ja^ — ao|,
\a[' — ao|' so that at least one of the two points P[, P^' lies
outside of Sl^- •
Next we select in IS^- a pair of distinct points Pq (/j, a-i)
and Po {t'l , a'z ) whose images coincide. As before, we can
178 Calculus of Variations [Chai). V
choose fcs smaller than at least one of the two quantities
\a2 — ao| > I (h —f'o\^ etc., etc.
Proceeding in this manner, we obtain corresponding to
the sequence \k,,\ an infinite sequence of distinct pairs
of points
p:{t:, a:) , p:.'{t:\ <) , v = i, 2. • ■ • oo ;
the two points P^', Fl' lie in S^^, and their images coincide
at a point {x^, JJ^) of ^j,.
We consider now the set of points
in the four-dimensional space (/', a'; /", a"). The set Z
contains an infinitude of distinct points all lying in the finite
domain
5 : fo < ^ < j^i ; — A; < a a,i < « ;
it has therefore at Icdst one accnmiiUdioii point^
I = (t', u'; t", a") ,
which belongs itself to 1 since 1 is closed ("abgeschlossen").
6) We are going to prove that
Out of the sequence \z^\ we can select'-^ a 'subsequence \z^,\
(/ = 1, 2, • • • X ; i^J4-i>i'0 such that
L z,.. = ^ , i. e.,
Z.C = t', /.a;. = a', Lt'' = r", Lal' = a".
( = 30
But since L l\. = 0 and
it follows that
1 Compare E. I A, p. 185, aud II A, p. « ; J. I, No. 27. 2 See J. I, No. 28.
§34] Knesee's Theory 179
a'= tto , a." — a^, ;
besides t' and r" are contained in (ToT,).
On the other hand, let D{f', a' ; /", ((") denote the dis-
tance between the two points (.r', //' ) and (./", y") corre-
sponding to (/', d') and (/", d"). Then we have
i;(/J. al ; /,'/, a'p') = 0 .
Bnt since D{f', a'; /", a") is a continuous function of
its four arguments, we have
D{t', a,,; t", a,)) = L B(fl., «,'. ; tl'., a'/.) = 0 ,
that is, the images (f, v' ) fiiid (|", v") «f the two points
(r', Oq) and (t", Oq) coincide. According to B), this is only
possible if
r ft
T =T , say
There e.rists therefore a point (t, Oq) in ISa-, '" erer/j vicitiifi/
of which pairs of distinct poinis (/', <i'), (/", o") con be
found whose images /u the ./■, y-plane coincide.
c) The theorem on implicit functions' leads now immedi-
ately to a contradiction. For, let (^, v) denote the image of
the point (t, (l^^) ; take (x, y) in the vicinity of (f, ij) and
consider the problem of solving the system of equations
jc = cf>{t, a) , y = ^(f, a)
with respect to {f , a). Since A(t, «o)=^^* it follows from
the theorem on implicit functions that after a positive quan-
tity € has been chosen arbitrarily but sufficiently small, a
second positive quantity S, can be determined such that, if
(.r, //) be taken in the vicinity (8,) of (|, ??), the above two
equations have one and but one solution (t, a) in the vicinity
(e) of (t, Oo) .
Further, we can determine, on account of the continuity
of (f) and -v/r, a positive quantity e'^e such that the image
1 Compare p. 35, footnote 2.
180 Calculus of Variations [Chap. V
of every point (/, (i) in the vicinity (e') of (t, a,)) lies in the
vicinity (SJ of (|, ?;). Hence if (/', ((') and (/", a") are
any two distinct points in the vicinity (e') of (t, Oq), their
images {x , y') and (.r", //") must lie in the vicinity (S,) of
(|, 77) and can therefore not coincide, according to the defi-
nition of S^.
But this is contrary to the result reached under }>) ; the
hypothesis from which we started must therefore be wrong
and our theorem is proved.
Corolhiries: 1. From the continuity of the functions
4>{f, (i), "^{f, o) and the one-to-one correspondence between
iS;t and ^;^, it follows that the image S' of the boundary ? of
the' rectangle ^j. is a continuous closed curve without mul-
tiple points (a so-called ''^Jordan-curve''') . It divides, there-
fore,^ the oc, ?/-plane into an interior and an exterior.
According to a theorem due to Schoenfliess" the set of
points §>j. is identical ivitli the interior of 2' together witli
the houndarji 2'.
2. Let /q, iy be two values of / satisfying the inequality
and let @o denote the arc of the curve (10) corresponding to
the interval (/o, ti). Since the line: 0.^:0^, to^t^t^ lies
in the interior of iSj^., its image @o li^s in the interior of ^^.
and has, therefore, no point in common with the boundary
2'. The two curves ®o ^^^^ ^' being continuous, it follows,'"
therefore, that a neighborhood (/o) of the arc @o c^in he con-
structed which is entirely contained in ^f..
3. Since A(/, Oo)=t=0 in {TqTi) and A(/, a) is continuous
in iSfc, it follows from the theorem on uniform continuity*
that k can he taken so smcdl that
1 Compare J. I, No. 102. The interior as well as the exterior is a " continuum."
"^ Gottinger Narhrichten,1899, p. 282; compare also Osgood, ihid., 1900, p. 94; and
Bernstein, ibid., 1900, p. 98.
ii Compare p. 13, footnote 4.
* Compare E. II A, pp. 18 and 49; P., Nos. 21 and 100; J. I, No. 62.
§35] Kneser's Theory 181
A(f,a)^0 iu iB, . (18)
We suppose in the sequel that A- has been selected so small
that IS;;, and ^^. are in a one-to-one correspondence, and
that at the same time (18) is satisfied. Under these cir-
cumstances the region ^j^ is called a field about the arc Qq,
formed by the set of curves (15).
1. The one-to-one correspondence (15) between S^. and
^^. defines / and ft, as single-valued functions of x and //
which are of class C throughout §>i.; we denote these
inverse functions by
t = f{x,y), a = a{x,y). (19)
Their derivatives are obtained by the ordinary rules for the
differentiation of implicit functions, according to which
df da , ^^ I , ^^
(20)
§35. kneser's curvilinear co-ordinates^
Our next object is to extend to the general case the
results given in §32, 6) concerning the introduction of geo-
desic parallel co-ordinates.
a) Curvilinear co-ordinates in general: Let us intro-
duce, instead of the rectangular co-ordinates x, y, any sys-
tem of curvilinear co-ordinates
xt=U{x,!j) , v= V(x,y) (21)
where the functions Uix, ij) and T (,r, //) are of class C" in
a region ^ contained in the region iR of §21, h) ; in the
same region their Jacobian is supposed to be different from
zero.
We interpret u, v as the rectangular co-ordinates of a
1 Compare Kneser, Lehrbuch, gl6.
182 Calculus of Variations [Chap. V
point in a u, r-plane and denote by QI the image in the
u, r-plane of the region B. "NVe suppose, further, that the
correspondence established by (21) between # and 51 is a
one-to-one correspondence. The inverse functions
x = X{u,v) , y = Y(n,v) (22)
will then likewise be single-valued and of class C" in the
region ® and moreover their Jacobian
We consider now the integral
taken along an ordinary curve
Q: X ^ <t>{T) , y = iI/{t)
from a point A{tq) to a point B{t^), the curve 6 being sup-
posed to lie in the interior of the region ^.
If we introduce the new co-ordinates u, v into the inte-
gral J, it will be changed into
f 1 / da dr\ ,^^^
the function G of the four arguments ii , v, n' , v' being
defined by
G{u, V, u', v') = F {X, Y, X„a'+ X,v', r„H'+ Y,.v') . (26)
The inteorral J' is taken along the image d' of 6 in the
u , t'-plane :
g' : u = U (> (t) , il; (r)) , f = y (<^ (r) , ^ ( r))
from the point A' (image of A) to the point B' (image of B).
From the equality
J'=J (27)
it follows that if the curve 6 minimizes* the integral J, its
1 With the understanding that only such curves are admitted as lie in the regions
S and 0; respectively.
S">"'] Kneser's Theory 183
image 6' necessarily minimizes ,/', and vice versci. Hence
the problem to minimize the integral J and the problem to
minimize the integral J' may be called equivalent problems.
The following properties of the function 0(u, r, fi\ v')
can immediately be derived from its definition (26) :
1. Gin, V, n', r') is positively homogeneous' of dimen-
sion 1 in ii', r' .
2. By differentiation we get
G .= F X -\- F . Y
(t,,.= FyX„ + F,j. 1',, .
Hence if
y ^= X^fli' -\- X^x>' , .r = Xji -\- Xj' ,
y'= ^^n«'+ ^^vv' , y = yj' + i'.^' >
the following identity holds :
uG„.{n, r, u', r') + vG^iii, v, u' , r')
= }-F^.{,r, y, x', y') + yh\f{x, y, x' , y') , (28)
from which we infer that the E-function is an absolute
invariant for the transformation (21), /'. r., if we denote the
new E-function by E'(/f . ?' ; u' , v' ; u, r) we have
E'(", r; u', v'; h, t-) = ^{x, y ; x' , y' • x, y) . (29)
3. Also Fi is an invariant ; if we denote the correspond-
ing function derived from G hj G^, we obtain easily
G, = D'F, , (30)
where D is defined by (23),
4. Also the left-hand side of Euler's equation is an
invariant ; after an easy computation, we obtain
Guv — G'hu+ Giitt'r"— n"v')
= D \f,,, - F_,„+F,{x'y"- x"y')\ . (31)
The image of an extremal of the old problem is therefore an
extremal for the new problem ; and the same relation holds
for the transversals, as follows from (28).
ICoinparc §24, equation (8).
lS-1- Calculus of Variations [Chap. V
All these results are in accordance with, and can partly
be derived (( 'priori from, the equivalence of the two prob-
lems.
h) Dcjiiiifioii of Kncsei'\'^ curvilinear co-ordinates: To
the assumptions concerning the set of extremals (1) enumer-
ated in ^38, a), we add the further assumption that
A(f,«.,)4=0 in (tj,) , (32)
where A(/, a) denotes again the Jacobian
d{t,a) ■
It follows, then, from the continuity of A(/, a), that the
quantities /q — ^o? ^^i — ^i- ^^' ^^^^ ^^^ taken so small that
A(/,f/)=^0 (33)
throughout the region ^,..
According to §34, the correspondence between the
domains jR^ and ^j^ defined by (1) is then a one-to-one
correspondence, and the inverse functions
f = H-^''U) ' a = a(x,u) (34)
are single-valued and of class C" in the domain §>^..
We now combine with the transformation (34) the trans-
formation
H =^ u{f, a) , f = a (35)
between the /, rt-plane and the u, f- plane, u[f, a) being
defined by (7).
Since, according to (Oa) and (S),
g| = F(f,a)=#0 in Sfc ,
it follows that the correspondence between the region jS;^ and
its image (Ua- in the u, t;-plane, defined by (35), is a one-to-
one correspondence and moreover that the Jacobian
9 (v , v)
d{t, a)
§3.)] Knesee's Theory 1R5
Hence, if we combine the two transformations (35) and (34),
we obtain a transformation of the form (21) which estab-
lishes a one-to-one correspondence between the region ^j. in
the ,r, //-plane and the region ©a- in the n, r-plane, and
which satisfies all the conditions imposed under a) upon tlie
transformation (21). For every point [x, y) in the region
^'k defined in §33, h), the function u= U{x, y) represents,
according to the definition of 7({f, a) given in §33, the value
of the integral J taken along the unique extremal of the set
(1) passing through the point {.r, y), from the transversal of
reference X^ to the point (jc , y) .
c) Properties of Kneser^s curvilinear co-ordinates: For
Kneser's curvilinear co-ordinates, the images of the
extremals are the lines v = const.; the images of the
transversals^ the lines u^ const. Moreover, tlie function
G{u, r, u' , v') Jias the following characteristic properties :
G(u, V, u', 0) = u' ,
(ob)
(?„.(«, V, u', 0) = 1 , G,..{h, V, u', 0) = 0 ,
which hold for every u, r and for every u' which has the
same sign" as F(/, a).
For the proof of these statements it is convenient to rep-
resent a curve 6 in the region ^,^ of the x, //-plane in the
form
x = (i>(t, a) , ) t — g (r) ,
y — ij/ {t, a) , \ a= h (t) ,
which is always possible on account of the one-to-one corre-
spondence between iS^. and ^j^. The image 6' of (5 in the
u, r-plane is then represented by
u = «(f, a , ] t =g{T) ,
V = a ,]« = /< (t) ,
and on account of ( 20) the following identity holds :
1 Again with the restriction that the transversal must lie in the region &^.,
-Since F(i, a) =f=0 and is ronf inuous in Sj. , it has a constant sifjn in S^. .
18() Calculus of Variations [Chap. V
If 6 is an extremal of the set (I), it can be defined by the
equations
t = T . <(=(('.
a constant.' Hence the above formula becomes:
F(t, a') = g(u(t, a'), <i\ )ir(T, d') , O) ,
and therefore, on account of (8) :
iirir, a') ~ G {uir, a'), u' . Hjir , <i'). 0) .
Since r and <i' are arbitrary and, moreover,
G{u, i\ pii\ 0) = pG{n. V, n\ 0)
for every positive p. the iirst of the three equations (-it)) is
proved.
The second follows immediately by means of the identity
ti'G^-\-v'G,.= G .
To prove the third, let
define a transversal ; then, according to §38, <■) :
n {g{cr) , 0-) = const.
Hence the condition of transversality, which must be sat-
isfied at the point of intersection of this transversal with the
extremal t^r, (i^=a', reduces to
— G,{u(t, a'), a', Urir, a'), 0) = 0 ,
from which we infer the third of the equations (36), since
da-
iJts image is the line Q' : u = u{t, a), v = a' and the angle S' which the positive
direction of e' makes with the positive w-axis is 0 or tt, according as the constant
sign of F (i, a) is -j- or — .
§36] Knesee's Theoey 1S7
The relations (8(5) lead to two important consequences :
In the first place, we obtain immediately from the defiiu-
tion of the E'- function on a])plying (3()):
E'(», r ; n\ 0; h . r) = G{u , r . h . r) — 7i . (37)
In the second place, we get by Taylor's theorem :
G{u, i\ II , v) — G (u . i\ u' . i))
= {ii — u')G„.(>(, r, u'. 0) -\-cG,A". r. u\ 0)
+ 1 [(" - >'y(l..r + 2 (7, - >,') rG,,,, + l^G,..,. ] ,
where the arguments of (^r^^ „ , etc., are
i(,c, n'=H'+e{h-ii') , V'^dr , and 0 < ^ < 1 .
If we simplify the remainder-term liy the introduction of
Gi, and make use of (3(>), w^e obtain:
G{n, 17, h, v) — ?t — ^-u'^rtri . (38)
From the preceding equation we see that whenever (/^ and
If are both positive (negative), also G{ii, r, a, v) is positive
(negative). Hence, if for a given point {u , ?•), the functions
G[u, V, u , r) and Gi{u, v, u , r) are difPerent from zero (and
therefore do not change sign) for all values of u, v (except
possibly u^O, v = 0), they must both have the same sign.
Remembering now the relations (26) and (30), we obtain
the following result,' which will be useful in the sequel :
If at a poini (.r, y) ihc functions F(x, ij, cos 7, sin 7)
(uid i^i(-f, y, cos 7, sin 7) (O'c, both different from zero for
all values of y, tJten tlunj must both Jiave ilie same sign.
§3(). SUFFICIENT CONDITIONS FOE A MINIMUM IN THE CASE
OF ONE MOVABLE END-POINT
The introduction of Kneser's curvilinear co-ordinates
leads to a number of important consequences :
a) Kneser''s snfficieid conditions : Through the point ^4
ISee Kneser, Lehrhuch, p. 53.
188 Calculus of Vakiations [Chap, v
(xq, yo) of the extremal ©q (compare Fig. 31, p. 170) we con-
struct the unique transversaP ^: [/ = %('^')]; and from an
arbitrary point A of X we draw any ordinary curve 6, join-
ing the points A and B and remaining in the region ^l. :
The image of C"^ in the ii , r-plane is the line v = ((q; the
images of Xq and X are the lines u = 0 and ii = Uq = U(xq, ijq);
the imao^e of the curve 6 is an ordinarv curve 6' :
The abscissae Uq and //^ of the images A' and B' of ^1 and
i? are
and according to the defi-
nition^ of TJ{jc, y) we have
J,. (AB) = III — »„ .
FIG. 37 On the other hand
But since* ^('^o) ^ "o? ^i'^i) = "i- ^'^ have
'1 dn
1
i
i
i
e'
\
/
7
/^
Al
1
(J.-
",
^
tl
i
i
= 0 U = i
/
dr := t<i — i(„ ,
0 ar
and therefore the total variation
^J=J^ (AB) - J-c,^(A B)
may be written :
The relation (38), together with (30), leads now to the
following result :
1 The arc of 2 corresponding to the interval (a^ — fc, a^+k) of a lies entirely in
the interior of B^. ; for A lies in ©^ since tf,> /", and t and S** do not intersect in #^.
The image ©^, of gij^. is that part of ST^ in which m 5 0 or m §0 according as the con-
stant sign of F^f , o) is + or — .
2Compare §3"), b). 3Compare, for this important artifice, §32, 6).
§36] Kneser's Theory 189
If flic ('()ll(lffi<)))S
nw sdfisjicd foi- /o^/^/i. 'iiid if. iiiorcorcr,
Fi {x , //, cos y, sin y) > 0 (Ha')
a loin/ flic crfrcmal ®ofoi' crcrij nihic of 7, then the extremal
@o furnishes for the integral J a smaller value than every
other ordinary curve which can be drawn in ^l from the
transversal X to the point B, provided that A" be taken sutfi-
ciently small; and therefore ilw exfronal (Eq mininiizes^ flic
infegi'dl J if fhc ciid-poiiif B is to rcnidiii fixed irliilc f/ic
ofJicr end-point /s niorahle on the curve %.
Ji) Wcierstrass' s fftcoreiii, for flic case of one roriohte
end-point: Still another important conclusion can be de-
rived from (39). On account of (37) we obtain from (3',l)
AJ= I E (u, r; v',0: - , ~-)<It ,
'^^u \ dr (It/
where n' is any quantity having the same sign as F(/, o).
We may therefore " write the last equation :
I 'e'ITi. v\ cos $', sin 0' , -- , '-rhlr . (40)
•Ai \ f/r dr/ '
where 0' is the angle detined on p. 186, footnote 1, and whose
value is 0 or tt. But since the E-function is, according
to (29), an absolute invariant for the transformation (21),
we obtain, by returning to the original variables ./•, /y, the
extension of Weicrsfrosss theorem to the case of one
movable end-point :
AJ= ( 'e (x,1j; y,!j'; y.Tj')dT , (41)
•""0
iTo make the connection with the problem: To minimize the integral ./ by a
curve joining a given curve ~ with the point B, the I'ollowius remark is necessary:
After an extremal i\) of class C has been found which passes througli B, is cut trans-
versely by "iT at A, not touched by is at 4, then it is always possible, according to §23,/)
and §30, to determine a set of extremals which has the properties assumed in §33 of
the set (1) and to which the curve ~ is a transversal. The transversal 1 of the pr&.
ceding theory will then coincide with the given curve IT.
2 Compare §28, equation (.">!).
AJ
190 Calculus of Yaeiations [Chap. V
where (ir, y) is a point of the curve 6: Ic' .y' refer to the
curve 6; x' , y' to the unique extremal of the set (1)
passing through the point {x , y).
Reasoning now as in §28, (/), we infer that in the al)ove
enumeration of sufficient conditions flic condifloii (I la')
may he replaced by the ui'ddev condition
E(.r, y; p,q: T>.Tj)> 0 along ©„ , (IV)
understood in the same sense as in §28, f/)..
c) Osgood's tlieoreni concevniiui a characteristic prop-
erty of a strong minimnin: The introduction of Kneser's
curvilinear co-ordinates leads to a theorem due to Osgood'
concerning the character of the minimum of the integral ./,
in case the stronger condition (Ila') is satisfied.
If we denote by 0 the angle which the positive tangent
to ^' at the point (u, /;) makes with the positive 7f-axis, and
introduce on 6' instead of the parameter t the arc 8 of 6',
we may write (10) in the form"
{•«i _ _ _
E'{u, v; cos ^'. sin 6': cos 6 , siu d)d>< .
- II
Applying the theorem^ on the connection between the
E-function and Fx to E' and Gj. we get
E'(«, r; cos^', siu^'; cos ^, siu ^)
= (1 - cos {0 - 0')) G, {Ti, V, cos 6*, siu 6*) ,
where 6* is some intermediate value between 6' and 6.
Since ^' = 0 or tt. the first factor on the right is
Iq^cos^.
But if we suppose that (Ha') is satisfied, we can always
take 1: so small that
Fi {x, y, cos 7 , sin y) > 0
for every x, y in ^^ and for every 7.
'See Transactions of the American Mathematical Society, Vol. II (1901), p. 273.
For the following proof see Bolz.a., ibid., Vol. II (1901), p. 422.
2Coinpare §28, equation (ol). sCompare §28, equation (U).
1
§36] Kneser's Theory 101
From the relation (30) between F^ and G^, and from the
continuity of G^, it follows, then, that a positive quantity m
can be assigned such that
6r, {ii , V , cos (D, sin w) ^ m
for every ii, v in (3^. and for every oo. Accordingly we obtain
A t/ ^ w I (1 =p cos 6) ds ,
or, since (/7,
cos y = --- ,
A J ^ 7?i [/ =P (»i — tto)] ,
Z beint; the leno^th of the curve (£' from A' to i?'.
Now suppose that the curve S in the a-, ?/-plane passes
through a point P of the extremal a^^^ciQ-^-h of the set (1),
where
0 < 1 /i I < Ar . g]-~--^^C^ ^ " = "■ + "
(i' will then pass through a i
point P' whose ordinate is ^i s'
^ , " = «. FIG. 38
r = do + /' •
Let Q' he the foot of the perpendicular from P' upon
the line ii^^iiq. Then
l^\Q'P'\ + \P'B'\^\Q'B'\ ,
that is. , _ / , ., , , r^
and therefore
A J ^ m \^^lr + {>,,- n,r -T ('<. - ",)] > 0 . (42)
Hence, if we use the symbol ^j[ in the sense analogous to
that of ^/., we may formulate the result as follows:
Under oti)- present assiiimjtioiis concerning the extremal
©0 (t'^d the functions F <ind F^, it is aiivdijs jyossible to
determine, corresponding to evcrij positive quantity h
numericnlly less than k. a })ositive quantity e^^ such tJiat
AJ=.J^ (AB) - J. {A B) ^ £;. (43)
102 Calculus of Yariatioxs [Chap, v
for evcrij ordinarfj cin-rc (S which joins ike transversal H
with the point B, and reinains within B[. but not wholly
IN THE INTERIOR OF ^/'.
Osgood' derives from his theorem a sim[)le proof of
Weierstrass's extension" of the sufficiency proof to curves
without a tangent :
Let, in the notation and terminology of §31, d),
2 : .r = <^ ( t) , U — ^ '"t) , Tu^r^T^ ,
be a curve of class {K), not coinciding with Gq, joining the
points A and B, and lying wholly in the interior of the
region ^j.. Let 11 be a partition of the interval {JqTx) whose
subintervals are chosen so small that the corresponding rec-
tilinear polygon %n, inscribed in ^', lies in the interior of ^^..
The polygon being an ordinary curve, we have, if Kneser's
sufficient conditions of §36, a) are fulfilled for the extremal ©q.
Vn > J,
0
if T'n denotes, as in §31, c), the value of the integral ./ taken
along the polygon -^n.
Hence if we pass to the limit and remember equation
(78) of §31, we obtain
It remains to show that the equality sign cannot take place.
Let Q be any point of S not situated on the extremal ®o,
and denote by Oq + ^' the value of the parameter a of the
extremal of the field passing through Q. Then : 0 < j // j < A*.
Now consider in the above limiting process only such parti-
tions n for which Q is one of the points of division. There
exists, then, according to Osgood's theorem, a positive quan-
tity €}^ such that
'^Loc. cit., p. 292. 2 Compare §31, e).
§37] Kneser's Theory 193
Hence if we ])ass to the limit.
and therefore
Jf > J.^, , Q E. D.
^dl. VARIOUS PROOFS OF WEIERSTRASS's THEOREM.
THE ASSUMPTION F(/, o)4=0
The function
u= U{x,y)
introduced in §35, h) was derived from ii(f, a) by substitut-
ing for / and a the inverse functions (34) :
f = t(x, y) , a = a (.r , y) .
Hence flic partial (lo-icatircs of U{r, y) with respect to ./•
and y are, on account of (8) and {'.•) :
8Z7 dt , da
Remembering that
and that l)y detinition
4> (/ i-f , y) , fi (•^' , u)) = -^^ > ^{t P' ' u) ' « i-^' , y))^y ^
we obtain the important result :^
^ =. F, = P (.r , //) : 1^- = F,, = Q{x,y) , ( 44)
where P{r, y) and Q(.r , y) denote those functions of x and
ij into which F^. (/, a) and F^.(/, a) are transformed when
the variables /, a are replaced by their expressions in terms
of X, y.
From these expressions of the partial derivatives of U
Kneser, Lehrhuch, p, 47; compare also p. 175, footnote 1.
v.n
Calculus of Variations
[Chap. V
FIG. 39
two further proofs of Weierstrass's theorem for the case
of one variable end-point, can be derived.
a) K)i('S(')-'s proof :^ We repeat the construction of
§36, (t). denoting, however, the points Aq, A, A, B hj num-
bers : 5, 0, 0, 1 respectively.
Then we ap[)ly Wcicrstrass^s
ronstrtiction' slightly modified:
Through an arbitrary point
2(t = T2) of 6 we draw the
unique extremal of the set (1).
It meets the transversal X° at
a unique point, 7. Now we consider the integral J taken
from 7 along the extremal 72 to 2, and from 2 along the
curve 6 to 1, and call its value S'It.,) :
using the same notation as in §§20 and 28.
In particular we have (see Fig. 39):
But according to Kneser's theorem (§33, c))
hence
A J = jj, - j,„ = - [s (T,) - s{n)\ .
According to the definition of the function U(x, y) given in
§35, />). we have
on the other hand
<^2i— I F{J-,lj, x\ y')dr .
Hence, making use of (^ii), we get as in the case of fixed
end-points :
1 Kneser, Lehrhuch, §20.
-'Compare §§20 and 28.
§37] Kneser's Theory 11)5
- y-^ = — E U.,, //,; .r, , y, ; u-, , U^) • i +->)
(I To
Integrating with respect to T;, from r,, to r, . wt^ obtain
IFr'iersfras.s's flicoron (4:1).
The above deduction leads to the following f/co metrical
interpretation of the E-fnnction, due to Kneser :
Let 3 be the point of 6 corresponding to T^^To-j-Jt, and
draw the extremal 83 through the point 3, and the transversal
•J 4- through the point 2 (see Fig. 40) . Then
S(t2 + // ) - S{t.,) = J Si + ^4:) - ^72 " ^2:i i
and since
"Si = " 72 >
Sir, + //) — N(t,) = J,:; - J,, .
Hence we obtain, on account of (45), the result:'
J23 - ^« = h[E( X, , Ji, ; .r,' , 7/,' ; J-,/ , y/,' ) + (70] . (40 )
h) Proof htj means of IIiibert''s invariant integral: The
important formula (44) leads immediately to Hilbert"s
invariant integral^' for the case of parameter-representation.
The integral
J* r= ( ' I P {x , 7j) x' + (^ (J- , u) y'\<lr , (47)
taken along 6 from 0 to 1 is, according to (44), equal to
J*= ~yV(.r, uXir ;
lience
J*z= f/(,r, , //,) — U{7v^, '//„) ,
J-Q, //,) denoting the co-ordinates of the point 0.
The value of the integral J"* is therefore independent of
tJie curve 6 and depends only upon the position of the end-
I KxESER, Lchrbuch, p. 79 ; compare footnote 1, p. 138.
^Tompare §21. Another proof of the invariauce of the integral J*, followiuy:
more closely the reasoning of Hilbeet's original proof, is given by Bliss, Transac-
tioiMofthe American Mathematical Society, Vol. V (1904), p. 121.
196 Calculus of Variations [Chap, v
points; it even remains inr/in'diif wltcii Hie point 0 ))wves
along the transversal 2:, since U{x, ^)=^ const, along every
transversal.
Hence, by letting 0 coincide with 0 and (S with ©o "^'^
obtain
The integral J^i can therefore be expressed by an integral
taken along the curve (5. viz.,
Joi= 1 \^F^.(7v.Ti. .v\ y')7v'+ Fy.{x,y, x', y')y\<iT .
Substituting this value of Jq^ in the difference : A ./:= Jq^ — ./^„
we obtain immediately Weierstr ass's theorem.
(•) The assumption F(/, 0)4=0: It is important to notice
that in the preceding two proofs of Wei erst r ass's theorem
no use has been made of the assumption (0) that F(/, f/o)=l=**
at all points of the interval (tJi), but only of the two special
assumptions^
F(/;;,rg^O , ¥{f,,rQdp() (6b)
which, according to §33, a), are necessary for the construc-
tion of the two transversals %^ and 2^.
Hence, also in the sufficient conditions derived from
Weierstrass's theorem, the condition (6) may be replaced
by the milder condition (6b), whereas, in the former deduc-
tion of sufficient conditions by means of Kneser's curvi-
linear co-ordinates, the assumption (6) was essential.
This apparent discrepancy" between the two methods can
be removed as follows :
iThe first of these may be replaced by F(<, a,,) ^0, because for t^ any value of t
between T^ and ^q may be chosen. Only in very exceptional cases can J^ vanish all
along an extremal, since the differential equation J^=0 is, in general, incompatible
with Euler's differential equation.
2The discrepancy is still more striking in Kneser's own presentation, since he
makes, instead of (6), the stronger assumption
F(x, y, siny, cosyT + 0
along Py for every y (compare Lehrbuch, pp. 49 and 53).
§37j Kneser's Theory r.t7
Compare tlie two ])roblems :
(I) To minimize the integral
J = F{.r, u, y, !J )dt ,
and
(II) To minimize the integral
where '"
F'\x,y,x',y') = F{x,y,y,y')
+ ^M,y)y-{-^,U,y)u' , (48)
<J> (.r , u) being a function of ,r , ^ alone, of class C in g*/^. . Since
J<»' = J + $ (o-i , ^i) - ^ (.r, , ^„) , (49)
we obtain
A J*"' = A J
for all variations which leave the end-points fixed.
If, on the other hand, the integrals are to be minimized
with one end-point, say (.Tj , y^), fixed, while {xq, ^/o) is movable
on a given curve %, the same result holds, provided that
<!>(./• , y) remains constant along this curve.
With this condition imposed upon <I>, the lira prohlems
are equivalent; that is, every solution of the one is also a
solution of the other. Hence it follows that every extremal
for the one is also an extremal for the other.' In particular,
our set of curves
x = <i>{t,a) , y = i{;(f,a) (1)
is a set of extremals also for J'^^^
We now suppose that the function F satisfies the two
conditions (Gl)), but not (6), and we propose to show that it
is alivays 2^ossible so to select the function ^(.r, y) that
F'°'(^ a)>0
throughout the region M/,. defined in §33, a).
iThc analogous statemeut for transversals is, in ff(Micral, not. true.
108 Calculus of Variations [Chap. V
Let )ii be the minimum of F(/, a) in the region S^, and
let 3£ be a positive constant greater than |m|.
Further let, as before,
t — t (x , I/) , a^a (.r , y)
denote the inverse functions defined in 'j^35, ecjuation (34).
1. Case of fixed end-jwinfs: In this case we select
^{x,y) = Mt{.x-, y) . (:>())
Then g
F""(/,a) = F(/, a) + M^^t[<t>{f, a). ^ {f , u)) .
But by the definition of the inverse functions we have
hence
F'°MY. a) = F(/. a)-\-M ,
which is positive in |&^..
2. Case of one variahle end-point : Suppose (.rj. ^i) fixed
and (.ro, iJq) movable along the curve 2 . which is a transversal
of the set (1) for the problem (I) and represented, as in
§36, a), in the form
x = <i>{t,a) ,)
y = ^p{f, a) . I
In this case we select
^{x,y) = M [t (.(• , // ) - X (" (•<• > y))] ; (51)
then ^{.r, y) = 0 along 2', and
^{cl>(t,a),if{t,a))=M{t-x(aj) .
Hence we obtain, as before,
F'"' (f , a) = F it , a) +M>0 in S,. .
It follows, further, that % is a transversal of the set (i) also
for prohlem {II). For
§38j Knesek's Theory 199
The first term on the right vanishes for / ~%(<f), since X is
a transversal of the set (1) for problem (I) ; the second term
vanishes likewise for /^=%(rt), and therefore also the left-
hand side, which proves our statement.
The assumption (G), upon which the introduction of
Kneser's curvilinear co-ordinates depends, may therefore
be made without loss of generality ; for, if it should not be
satisfied, we can always replace the given problem l)y an
equivalent problem for which it is satisfied.
i^3S. THE FOCAL POINTS
The assumption
A(f,a,)z^O in (fj,) (32)
was indispensable in the previous sufficiency proofs for the
construction of a field ; but our deductions give no indica-
tion whether it is at the same time a necessary condition for
a minimum.
We are going to prove, according to Kne8ER.' that at
least in the milder form
A(f, a,)^0 for f„<t<f, , (32a)
which corresponds to Jacobi's condition in the case of
fixed end-points, the condition is indeed necessary for a
minimum.
We retain all the assumptions of §33 concerning the set
of extremals (1), and we suppose moreover that, in the nota-
tion of §33, a),
■FAt,a,)>0 in (fj,) ; (52)
iKneser, Mathcmatische Annalen, Vol. L, p. 27, and Lehrbucli, ^%'H, 25.
200 Calculus of Variations [Chap. V
hut we drop the assumption (82) and suppose, on the con-
trarv, that , ^ ,^ /--ox
A(/.:,a,) = 0 , (53)
where ^o< ^o < ''i' ^^^'^^^ moreover, that /„ is the smallest value
of /, greater than to, for which (53) takes place. The corre-
sponding point A'i-TQ, ijo) of @o is then the focal point' of
the transversal 'X on the extremal ©y
a) Existence of the envelope: We propose to find all
points^ [/, a] of the x, 2/-plane in the vicinity of [/q, Oq] for
which / ^ /^ i\
A{t,a)=0 . (o4)
For this purpose we notice in the first place that the function
A(/, Oq) is an integral of Jacobi's differential equation
dt\ dtf
This is proved exactly as the similar statement in §27 1)
and c) by substituting in Euler's differential equation
x^^(f>{t, a), y^'^it, (i), differentiating with respect to a
and then putting ci^aQ.
Since i^i;=Fi(^, ciq) is continuous in the vicinity of t^^t^,
and. according to (52), different from zero for / = /o, it fol-
lows that^ , ^ __,
Hence it follows, according to the theorem* on implicit
functions, that there exists a unique solution
t = t{a)
of (54) which is of class C in the vicinity of a = ao, and
takes for a = ciq the value t^^to.
The curve^ [t^=t{a)] in the x, ^-plane, /. e., the curve
1 Compare ^23 and 30. If Z shrinks to the point .4, the focal point A' becomes
the "conjugate" point to A.
2 For the notation compare §33, a). * Compare p. 35, footnote 2.
3 Compare p. 58, footnote 2. SFor the notation, see §33, a).
§381 Kneser'r Theory '201
;y : x = cl> (t{a) , a) = ^ (a) , // = '/' (^(«) , ^0 = "^ ('')
is the envelope^ of the set of extremals (1).
For, since
dx , dt dJi (If
da da da da
it follows that
^ ^, - ''^ <^, = - A ( r(a) , a ) = 0 . (r.(i )
aa c/a
This shows, apart from the points at which
that the curve ^- touches all the extremals of the set (1) for
which (I is sufficiently near to Uq, and therefore % is indeed
the envelope of the set,
h) Application of flic theorem on oivelopes: We must
now distinguish two cases :
Case I : The envelope % does not degenerate into a point,
i. e., «^(o) and ■^(a) do not both reduce to constants.
Let us suppose that the functions 4^ {a) and "(/^(o) are of
class C^''> in the vicinity of a=^ao, that for a.^=(io their
derivatives up to the order r — 1 vanish, but that the r^^
derivatives do not both vanish. Then we obtain by Tay-
lor's formula
'E = (a - a,)'-' L^ + "] ' ? = (« - «o)'-' lB + 13] , (57)
da da
where A and B are constants which are not both zero, and
a and 0 approach zero as a approaches Oq.
Substituting these values in (56) we get
A = n(f>f{t',, ao) , B= }ixpt(f^, ftu) , (58)
where n is a factor of proportionality which is different
from zero.
1 Compare E., Ill D, p. 47, footnote 117.
202
Calculus of Variations
[Chap. V
tiiipt
(59)
We now introduce on J a new parameter t by the trans-
formation a — a„ = £T ,
where e= dz 1 will be chosen later on. Since, according to
(2) and (2a) the functions <j)t(f, (') and ■>/^^(^ a) do not both
vanish at (i = aQ, it follows from (50) that we may write
dx , dlj
dr dr
where m is a function of t, which is continuous in the vicinity
of T — 0, and, on account of (oT) and (oS), is representable
in the form ,,, ^ ,,-.,,•-' (u + v) ,
where L f-=0.
T = 0
"Whenever it is possible so to select the sign e that tn is
positive for all sufficiently small negative values of t, we
can construct, according to the theorem II of §33, (/), an
admissible variation of the arc A A' of ©^ for which A./ = 0.
* 3 Subcase A): r odd.^ If we
choose e equal to the sign of n,
)it is positive for all sufficiently
small values of |ti ; see Fig. 41.
Subcase B): r even, m has the same ^ *
sign as nr, no matter how we choose e.
Therefore
1. If «<0, m is positive for nega- \ '
five values of t ; see Fig. 42. ^^^" *"
2. If /i>0, )n is negative for
negative values of r r see Fig.
In subcase A) and subcase Bj)
FIG. 41
s we have
iThis covers the "general" case in which 5 has no singular point at .-l(»-= 1).
2 If we draw a straight line S through the point A' not tangent to (?q, then g
crosses the line £ in case A) ; it lies all on one side of £ in case B) . on the same side
as the arc A A' in case Bj), on the opposite side in case B2). This follows easily
from (57).
§38] Kneser's Theory '203
A,7 = J,(PQ) + J;,{QA') - J.^A') = 0 ,
according to theorem II of §33, d), and therefore the arc
A A' of the extremal @o certainly furnishes no proper^ iiiiiii-
mum, and still less the extremal ©„ (or AB) itself.
But it furnishes not even an improper miiilminti. For"
the envelope % cannot at the same time be itself an extremal,
and therefore the integral J{(^A') can be further diminished
— and consequently A,/ can be made negative — by a suit-
able variation of the arc ^^4'.
The statement that % itself cannot be an extremal can be
proved most conclusively by substituting in the left-hand
side of Euler's differential equation for .r, ij the functions
X = <^ (t, (() , y = ,p(t, a) .
and making use of the characteristic property (•")*.•) of the
envelope.
If we remember the homogeneity properties of F and its
derivatives, and the fact that (f>(f, o), -«/r(/, a) as functions
of t alone satisfy Euler's differential equation, we obtain
after an easy reduction :
F,-~F,. = cT,\4., ,
dr
^y
d
~ dr
F ■ =
^ y
6F,A,
^t
The
argn
iments
of F,,
etc.,
A', y
are
dJ-
' dr '
dp .
dr '
those of (jit, i^t, Fi, A^ are f, a.
Since, according to our assumptions, Fjf/, a) and \{t, a)
iFor the distinction between "proper" and "improper" minimum, com-
pare §3, b).
■^Compare Daeboux, TMorie des Surfaces, Vol. Ill, No. 622, and Zermelo, Dis-
sertation, p 96.
20J: Calculus of Variations [Chap. V
are different from zero for t^fo, a=^aQ, they remain differ-
ent from zero in a certain vicinity of this point. Moreover,
(f>t_ and -yjrf are not both zero. Hence the envelope 5 does
not satisfy Euler's differential equation.^
In subcase B2) the same construction cannot be applied,
and therefore the question cannot be decided by this
method.
Case II : % degenerates into a point. In this case all
the extremals of the set pass through the point A' , and we
can directly apply Corollary II of the theorem on trans-
versals, §33, c).
Accordingly, we have for every
extremal © of the set :
FIG. 44 A .7 = J,. (PA ' ) - J",,^ ( A A ' ) = 0 ,
and therefore the arc A A' of the extremal @o certainly fur-
nishes )io proper minimum.
Summing up the difPerent cases, we may state the
result :
If the end-point B of the extremal AB coincides icith
tlie focal point A' {and a fortiori, therefore, if B lies beyond
^' '■ ^i> /o) //*e arc AB ceases to furnish a minimum, except
in thefolloimng two cases:
1. When the ejivelope % has at A' a cusp of the special
kind defined under subcase B^), the present method fails to
give a decision.^
2. When the envelope degenerates into a point, the arc
A A' furnishes no proper minimum, but it may furnish an
1 Another more geometrical proof can be derived from the fact (see §25, b)) that
only one extremal can be drawn through a given point in a given direction if
i?'j(a;, 2/,a; , 2/) 4=0 for the given point and direction; compare Darboux's proof
(toe. cit.) for the case of the geodesic.
2 Under the restricting assumption that FC.r,,', ?/„', cos y, sin 7) +0 for every 7,
Osgood has shown that the arc A A' actually furnishes a minimum, if the other
suiEcient conditions of §36 are satisfied, Transactions of the American Mathematical
Society, Vol. II (1901), p. 182,
§38] Kneser's Theory 205
improper minimum.' If, however, B lies beyond A\ the
arc AB furnishes not even an improper minimum.^
Thus the necessity of the condition
A(f,a„)4=0 for U<t<U (32a)
is proved for all cases with the one exception just mentioned/
1 The set of geodesies on a spliere which pass through a point affords an example
of this kind.
2 For, from Fj (f,,', n,i) +0 it follows that if a is sufficiently near to «,,, the "dis-
continuous solution" PA'B (see Fig. 44) cannot satisfy the corner condition (24) f)f
§25, c) (compare footnote 2, p. 142), and therefore a variation P ^f N B can b(^ found for
which AJ<0.
3 This agrees with the result derived by Bliss from the second variation (com-
pare §3D) ; the latter method proves the necessity of (32a) also iu the exceptional case.
CHAPTER VI
ISOPERIMETRIC PROBLEMS'
§31). euler's rule
The special example which has given the name to this
class of problems has already been mentioned in §1.
More generally, we nnderstand l)y an isoperimetric prob-
lem one of the following type:
Among all curves joining/' fivo given point f^ 0 and 1 for
which the definite integral
K= C \j(.c, !i,x', u')dt
talces a given value I, to determine the one luhicli minimizes
(or maximizes) anothar definite integral
J= I F(x, y, x', y')dt .
Concerninsf the two functions F and (} we make the same
assumption as in §24, h) concerning F alone. The "admis-
sible curves" are here the totality of ordinary curves which
join the two points 0 and 1 , lie in the domain iR of the fuuc.
tions i'^ and O, and for lohich tlie integral K has the given
indue J . Aside from this one modification, the definition of
a minimum is the same as in the unconditioned problem,
§24, c). We suppose that a solution has been found :
6: x = cl>(t) , y = ^{t) , t.^t^f, ;
and we replace the curve 6 by a neighboring curve
6: X=X-\-$ , y — y ^r, ,
1 This chapter is based chiefly on Weieestkass's Lectures of 1879 aud 1882, and
on chap, iv of Knesee's book.
-Or: joining a given point and a given curve, etc.
206
§39] ISOPERIMETRIC PROBLEMS '2() I
where ^ and ?; are functions of / of class /)' satisfying the
following conditions :
1. They vanish for / = /q and / ^ /i ;
2. In the interval (Vi), they remain in absolute value
below a certain limit p.
3. The integral K taken along CS from ^q t<3 ti has the
same value as if taken along 6 (viz., =1), or, as we write it,
^K = K,,~K,, = 0 ; (1)
a) Admissible varidtions: Our next object is to obtain
an analytic expression for functions |, rj satisfying these con-
ditions, not necessarily the most general expression but one
of sufficient generality for the purpose of deriving necessary
conditions for the minimizing curve.
Such an analytic expression can be obtained, according
to Weierstrass, as follows :
Let j?i, p>, qi-, q> be four arbitrary functions of t of class
D' vanishing at /q ^.nd /j. Then we consider the functions
where e^, e., are constants, and propose so to determine e^ as
a function of e^ that the condition (1) is satisfied for every
sufficiently small value of e^.
For this purpose we notice that the integral Kqi is a func-
tion of e^ , eo which is of class C in the vicinity of e^ ^ 0 , €9 = 0)
and which is equal to Kqi for €i = 0, eo^O. Further, for
Cj^^O, €., = 0 its partial derivative with respect to e^ has the
value fj
Nc = i (G^Ih + Gy q< + G,.p: + G,. g,' ) dt .
Hence if we introduce the assiimjition^ fliaf the curve Q- is
not an extremal for the integral K, the functions p-,, q^ can
ilf G were an extremal for the integral A', the curve 6 (ur at least sufficiently
small segments of it) would in general minimize or maximize the integral K, and it
would therefore be impossible to vary these segments without changing the value of K.
208 Calculus of Variations [Chap, vi
be so chosen that No^O, and the conditions of the theorem
on implicit functions are fulfilled for the equation (1) in the
vicinity of the point e^^O, e^^O. Accordingly, we obtain
a unique solution e^ of the form'
where (ey) denotes, as usual, an infinitesimal. Substituting
this value in f . ?/ we get
(4)
These functions ^, t; have all the required properties for
sufficiently small values of |ei|. The same argumentation
applies to '"partial variations" which vary the curve only
along a subinterval {ft") of (tJi). It is only necessary to
take the functions jJi, 2?2^ Qij <1z equal to zero in the whole
interval (/cA) with the exception of the interior of the sub-
interval {ft").
h) Eulers rule: According" to §25, the total variation
A J^ for the variations (4) may be written
».' t,,
For an extremum it is therefore necessary that
After a definite choice of the functions pii Q-i ^as once been
made the quotient M^/y^ is a certain numerical constant
which we denote by ^ — X :
'Compare p. 35, footnote 2. 2 Compare, in particular, the footnote on p. 122.
§391 ISOPERIMETRIO PROBLEMS 209
We have then the result that the equation
71/, +A.V, = 0 {('))
must be satisfied for all functions 2)i, Qi of class D' which
vanish at /q ^^^^^ h ■ This shows at the same time that the
value of the constant X is independent of the choice of the
functions jjo^ Qz-
If we put H = F + XG, (7)
equation (()) becomes
Hence we infer exactly as in i^25 by the method of §G, that
X (did !i must satisfu ihe differential equations
which are equivalent to the one differential equation
if,,, - H,.„+H,{.v'!j"- y'y') = 0 , (I)
where H^ is defined by :
-"1 — To— — } 7 — To" • UV
y ^ X y X-
We call, again, every curve which satisfies (I) an extremal
for our problem (Kneser).
The above deduction applies t(j so-called "discontinuous
solutions''' as well as to solutions of class C, and shows
that the isoperimetric constant \ has the same constant
value along the different segments of a '^discontinuoiis
solution.''' Moreover we obtain, exactly as in §§*J and 25,
at a corner t^^t?, the ^^corner-condition:'''
1 Compare §9, in particular footnote 3, p. 37.
2 This important remark is clue to A. Mayer, Mathematischt Annalcn, Vol. XIII
(18771, p. 65, footnote; and Weierstkass, Lectures. Even if the minimizing curve
contains unfree points or segments, all those segments of the curve whose variation
is unrestricted (apart from the condition AA: = 0) must satisfy the differential equa-
tion (I) with the same value of the constant A.
210 Calculus of Variations [Chap. VI
H^. = H^
t.j +-II
^.
H..
(10)
All these results may be summarized in the statement
that, so far as the first variation is concerned, our problem
is equivalent to the problem of minimizing the integral
{F-\-XG)dt ,
the curves being subject to no isoperimetric condition.
This simple rule, which is the analogue of a well-known
theorem in the theory of ordinary maxima and minima, is
usually called Enlers rule, according to Euler,' who first
discovered it.
The rule still holds in the case where the point 0 , instead
of being fixed, is movable on a given curve
g: .T- = ^(t) , ^ = 'A(r) .
For, a reasoning similar to that employed in §30, combined
with the remark that for all admissible curves
leads" to the condition ,= tg
H^x'-\- Hyu'
-'''0
= 0. (11)
c) Example XIII : Among all curves of given length joining
tiro given points A and B, to determine the one which, together
with, the chord AB, hounds the viaximnm area.
Taking the straight line joining A aud B for the a'-axis, with
BA for positive direction, we have to maximize the integraP
=i £(■'«'
'' = i ), {■ry-yti)''t
lEuJjER, J/et/iodits inveniendi linens curvas luaximi mininiive proprietute guu-
dentes, 1744; see Stackel's translatiou, p. 101. The first rigorous proof is due to
Weierstrass, Lectures, and Du Bois-Reymond, Mathematische Annalen, Vol. XV
(1879), p. 310. The proof given in the text is due to Weierstrass.
2 For details of the proof we refer to Kneser, Lehrbuch, §33.
•i We substitute this analytical problem for the given geometrical one. without
entering upon a discussion of the question how far the two are really equivalent.
Compare J. I, Nos. 102, 112, and II, Nos. 129-33.
§39| ISOPERIMETRIC PROBLEMS 'ill
while
has a given value, say /, which we siipjDOse greater than the
distance AB.
Since
we get
H = \(xy'-.x--u)+\\ y'+ir ,
Hi=~\- , ^ , (12)
{V x^-\-y -)
and therefore the differential equation (I) becomes
X y — oc y _ 1
(13)
Hence the radius of curvature of the maximizing curve is constant
and has the value jX;, while its direction is determined by the sign
of X.
Again, since H), never vanishes, there can be no corners,' and
therefore the curve must be an arc of a circle of radius |X! . The
center and the radius of the circle are determined by the condi-
tions that the arc shall pass through the two given points and
shall have the given length I . There are two arcs satisfying these
conditions, symmetrioal with respect to the ^--axis.
d) ExAJiPLE XIV : To draw in a vertical plane behreen two
ijiren poi^its a curve of given length such that its center of gravity
■shall be as low as itossible.-
Taking the positive yaxis vertically upward, we have to mini-
mize the integral
J= C '//I y'+y"dt
while at the same time
K= C \ x' + y-'dt
J In
has a given value, say / .
Here
^ = (^ + A)Va- + 7/'-^
1 Compare §25, c) and §28, 6) ; in particular footnote 2, p. 142.
2 Position of equilibrium of a uniform cord suspended at its two extremities.
212 Calculus of Vaeiations [Chap, vi
Using the first of the two diflfereutial equations (8), we obtain at
once a first integral
Vx" + y"
= c .
On account of (10), c must have the same constant vakie all along
the cur\e.
If c = 0 , we obtain ' the solution
X = const. ,
which is possible only if the two given points lie in the same ver-
tical line.
If r=|=0, we obtain as general solution of Euler's equation
two systems of catenaries :
(14)
y -\- \ = zL fi cosh t .
Determination of the constants/ If we suppose .ro<,ri, the
constant /3 must be positive in order that we may have fo< ^i •
Since the curve is to pass through the two given points, the
following equations must be satisfied :
iTo = a + (3t„ , //„ + A = zt /3 cosh A, ,
x^ = a-\- fifi , !ji-\- \ = ± ft cosh /, .
Moreover, the curve must have the given length I ; this furnishes
the further equation
(J (sinh ti — sinh t^) = I .
From these five equations we have to determine the five constants
a, /3, X, /o, ^1-
If we introduce instead of Ai and /i the two quantities^
_ ^i + ^n _ -Ti -}- g-Q — 2a
'^~ 2 ~ '2/3
we derive from the above equations the following :
i2/4-A = 0 is not a solution, since it does not satisfy the second differential equa-
tion (8).
2WEIBESTRA9S, Lectures, 1879.
§iOJ ISOPERIMETRIC PeOBLEAIS 213
Z/i — Z/o = ±2(3 sinh fj. sinh v ,
I ^ 2/3 cosh /A siiih i' .
Hence we s:et
(15)
o^
tanh/x= ±^^L_J^ . (16)
Since we suppose
^^ 1 (^1 — ^'o)" + (i/i — ^o)' >\yi— Uo\ ,
each of the two equations comprised in (16) has a unique solution m.
Further, we obtain from (15) :
and therefore
sinhv_W^-(^,-.,J^^ say = A:. (17)
Since A-> 1 the transcendental equation (17) has one positive root v.
After M and f have been determined, the values of a, i3, X, f„, f,
follow immediately.
Each of the two systems of catenaries (11) contains, therefore,
one catenary satisfying the initial conditions.
§40. THE SECOND NECESSARY CONDITION
We suppose that the general solution' of the diiferential
equation (1) has been found :
x=f{f,a,p,\) , y = g(t,a,^,X) . (18)
It contains, besides the two constants of integration a, /3,
the iso perimetric constant \.
Moreover, we suppose that a particular system of values
of these constants
a = ay, (3 = (3^ ^ A=\o
has been determined" so that the extremal
1 Compare the remarks in §25, a).
2 There are five equations for the determination of the five unknown quantities
214 Calculus of Variations [Chap, vi
u = g{U ttu, p„, A„) ,
passes through the two given points () and 1 ( for / = /q and
/ = /j respectively), and furnishes for the integral A' the
prescribed value / :
We suppose that the functions f, y, ft^ Ut^ ftt^ Utt ^i^d
their first partial derivatives with respect to a, ^, X are con-
tinuous functions of their four arguments in a domain
where ro</oand Ti>ti.
Further, we assume that for the particular extremal Cr,,
/,2 + g?=^() in {T„T,) ,
I'o (20;
ftfjK-f\g,\ 4=0 ,
and that ft(U—fa(lt ^^^fdl^—MJt are linearly independent.'
Finally we retain the assumption introduced in §31) that
®o is not an extremal for the integral K.
a) A lemma on a ccrfdiii fi/jx' of cahnissihlc variations:
In §39 the existence of admissible variations of the form
^ = i{t,c) , 7? = 7?(f, e) (21)
has been established, satisfying the conditions enumerated
on p. 122, footnote 1, and besides the isoperimetric condition
AK = 0
for every suflSciently small value of |e|.
From the latter condition it follows that also
oe
Hence we obtain in particular for e = 0 :
1 Compare § 13, end.
^iO] ISOPERIMETRIC PROBLEMS 215
f ' ( (i.-l> + (^','1 + ^4 i>'+ ^-u<l') dt = 0 , (22,
where
^> = ,^(Y,0, , q=r,,{t,Q) . (23)
If we transform the left-hand side of (22) by integration by
parts, and remember that, as in §25, a),
where
d _ , , rf _
u ^ G,^„- G^.^+G,(yu"-'^"y') ,
6 1 ^-^x'x' ^-^x']i' mm'
\j X y X
we obtain
where
f ' Uwc(f = 0 ,
tv = y p — X q
Since p and q vanish at to and /j , the same is true of ir .
Vice versa, the following leiiima^ holds:
Let to be any function of class D' which satisfies the con-
ditions
w{t,) = (), w(t,) = (), (24)
( ' Uwdt = () ; (25)
then it is always possible to construct an admissible varia-
tion of type (21) for which
a
^yi-x'r})
= IV .
Proof: Since @o is not an extremal for the integral K, it
follows that U^O; it is therefore always possible so to
select a function iVi, of class D\ and vanishing at /o and f^,
that
'Due to Weierstkass; sec Kxesee, Muthauatische Aitnaleu, Vol. LV, p. 100.
216 Calculus of Variations [Chap. VI
Now let
o) = ell- -f- Ci H"| ,
and choose
, y'w — x'(o
^ '-'I ''2 5 / '2l '9*
a; -+ ^ -^ X ^ -\- !/ ^
These functions vanish at /o and /^ for all values of the con-
stants e, ej ; they represent admissible variations if, more-
over, the condition
AA' = 0 (1)
is satisfied.
But by the same process as above, we find :
="= r ' Un-dt = 0 , (26)
^^K
0
e =0
-h-"= ( ' UH\dt^O . (26a)
On account of (26a) we can apply the theorem on implicit
functions to the equation ( 1 ) , and obtain for e^ a unique solu-
tion which, on account of (26), is of the form^
ci=(e)e .
Hence
y'^ — x'-q = w = £W + (e) e ,
which proves our statement.
6) Weiei'strass' s expression for the second variation :
Since Ai^T^O, we may write
AJ=: A J + A„AA' . (27)
Hence if we apply to the increment AF-^\AG Taylor's
formula, we obtain for every admissible variation of type (21)
AJ= f \hJ + H„r] + H,.^'+ H,^.r}')dt
+ 1 r ' {H,J' + • • • + H,.,.r]") dt + (c) e^ ,
'0 .
1 Compare p. 35, footnote 2.
§40] ISOPERIMETRIC PROBLEMS 217
^^^^^^•^ H = F + KG .
The first integral is zero since @o is an extremal.
To the second integral we apply the transformation of
^27, a). We thus obtain the result :
^•^ = gX"(^-(^) + «^''-')"' + W-'- (28)
where H^ and H2 are derived from H in the same manner
as h\ and F^. from F ; see §24, 6) and §27, o). We shall
denote the first term on the right-hand side by ^^-J.
For a minimum if is iherefore necessanj that
X"(^'(^T+^'"'')'"^''^ *29)
and on account of the lemma proved under a) this condition
must be fulfilled for everu function w of class D' icliich sat-
isjies the equations {24) and {25).
c) The second necessarij condition: Since we can con-
struct admissible variations' which vary the arc ©q only
along any given subinterval {ft") of (/q^i), we can apply to
the above integral the reasoning of §11, b). Hence the sec-
ond necessarij condition for a minimum {maximum) is ttiat
H,^0 (^0) (II)
(don<i the arc ©q-
This is tJie ancdogne of Legendre' s condition. Also the
second necessary condition for the isoperimetric problem
coincides, therefore, with the second necessary condition in
the problem to minimize the integral
H{x, y, x', y')dt
without an isoperimetric condition.
1 Compare §39, a).
218 Calculus of Variations [Chap. VI
§41. THE THIRD NECESSARY CONDITION AND THE CONJUGATE
POINT
We assume in the sequel that (II) is satisfied in the
stronger form
H, > 0 along e„ . (II')
It follows, then, by the method of §11, ?>), that (29) is sat-
isfied, provided that the point 1 is sufficiently near to the
point 0.
We have next to determine how near the point 1 must
be taken to the point 0 in order that the inequality (21)) may
remain true. And // is at this point that the equivalence of
the liro prohlems, which ire hare been comparing, ceases.'
In the unconditioned problem the inequality (29) must be
fulfilled for all functions ir of class D' which vanish at /q
and /j ; in the isoperimetric problem only for those which
besides satisfy the equation (25). It is therefore a priori
clear that the condition (29) is certainly fulfilled for the
isoperimetric problem if it is fulfilled for the unconditioned
problem. Hence if we denote by T the upper limit of the
values of ti for which the inequality (29) remains true in
the isoperimetric problem, by T" the corresponding upper
limit for the unconditioned problem, then T is at least equal
to T", but it may be greater, and in general it actually is
greater, as will be seen later.
a) Determination of the conjugate point : The point T
can be determined by a proper modification, due to Weier-
STRASS, of the method for the determination of the conjugate
point in the unconditioned problem:" Since we consider
only those functions w for which
1 This has first been discovered by LundsteOm, '• Distinction des maxima et des
minima dans un problfeme isoperimetrique," Nova acta rty. soc. sr. Upsaliensis, Ser.
3, Vol. Vll (1869) ; compare also A. Mayek, Mathematische Aimalen, Vol. XIII (1878),
p. 54.
2Compare §gl2, 13, 16, 27, b).
§41] ISOPERIMETEIC PROBLEMS 211
X
we may write B'-J in the form
/Li being an arbitrary constant. Transforming the first term
by integration by parts (see §12) and remembering that ir
vanishes at /q a^i^ 'i^ we obtain, if ic' is continuous in (/,)/]),
5V = e- r ' ir I ^ (w) +tiU~]fJf , (30)
^^^^^'" v,(,,) = i/,,,_^^(if,,0 . (31)
To obtain the general integral of the differential equation
* {w) + ;a £7 = 0 (32)
we substitute in the differential equation'
for j^ and y the general integral (18), differentiate with respect
to a, y8, X respectively, and finally put a = aQ, ^^ fSfy, X =^ Xq.
If we denote
^i{t) = gtfa-f,9a
^2(0 =9tf^-ftgp r
0,(t) = (Jtf\-ft9>.)
{t)=0tfa-f,9a\
(0 = 9tf^ — ft9p I {<^ = ^0 , /5 = A , -^ = '^0) ,
the result" is as follows :
Iff means here: i^ + AG.
2 For the computation compare §27, b). In thp differentiation with respect to A
an additional term appears on account of the factor A which occurs explicitly in
F+KG . The immediate result of the differentiation is
2/*(«3W)+(Ga.-|G^) = 0 ;
bu*^ af-^firding to §2j, equation (18),
hence the above result.
«--Jt«-=^'^'
220
Calculus of Variations
[Chap. VI
* (t?, (t)) =0 , * (^,(0) = 0 , ^ {e,{t)) + £7 = 0. (33)
Hence we infer that the function
in which c^ and ('2 are arbitrary constants, is the general
integral of the differential equation (32).
Now if it were })ossible to find values for q, Co, M and a
value /' such that
IV (to) = cA(fo) + ^2^2(^0) + t^OM = 0 ,
IV (t) = c,d,{t') + cAin + H-W) = 0 ,
, r Uivdt = c, r ue,dt-\-c, f U6,dt + fji i U6.idt = 0 ,
»//„ "^^O ^'^'i' ^'u
the second variation could be made equal to zero (and there-
fore presumably At7<0) by choosing iv equal to zero in
(/7i), and equal to this particular integral in {t(jt')-
In order that 8-,/>0 for all admissible functions w, it
is therefore necessary^ that
D{t,t.) =
for
0, (fo)
0. (t)
ue.df
o,{to)
I ue.dt
t^<t^U
o.{t,)
ue.dt
+ 0
(34)
1 Weierstkass, iec<M)-es, 1872. This condition, together with 77,4=0 in {t^^t^). is
al^o sufficient for a permanent sign of &^J (Mayer, Mat/iematischc Aniiaten, Vol. XIII
(1878), p. 53). The proof is based upon tlie following extension of Jacobi's for-
mula (14) of §12 for the unconditioned problem:
^2
(pu -\- qv) ■ir { pu -jr qv) = Hi(p'ii + q'v) — 2q {p'm-\-q'n)
■-^\H•^(p^l + qv){p■
-q'v) - (pm + gn)f/] ,
where u,v,m,n are the functions introduced below, under b), and p and q are two
arbitrary functions. Compare Bolza, "Proof of the Sufficiency of Jacobi's Condi-
tion for a Permanent Sign of the Second Variation in the So-called Isoperimetric
Problems," Transactions of the American Mathematical Society, Vol. Ill (1902), p.
305, and Decennial Publications of the University of Chicago, Vol. IX, p, 21.
§41] ISOPEEIMETRIC PROBLEMS 221
If we denote by /J the root next greater than /„ of the
equation'
D{t,t„) = 0 ,
the above inequality (34) may also be written
The point /J of the extremal ©q is again called the coiijiKidte
of the 2^01 nf /o.
b) The third )wcessc(rij condition : The preceding result
makes it highly probable' that the minimum cannot exist
beyond the conjugate point. And indeed it can be proved^
by a modification of the method employed by Weieestrass
for the analogous purpose in the unconditionetl problem/
that if fo-\ti, the second variation, and therefore also A./,
can be made nejjative.
For the proof it is convenient to throw the determinant
D{t, to) into another form in which its properties can be
more easily discussed.
Let
ti = e.it,) e,(t) - o,(t) ejt) = » (/ , i) ,
V = C\e,(t) + C,0,if) - 0,(f) = r(f, A,) ,
where the constants C^, Co satisfy the equation
These two functions^ satisfy the two differential equations
^ D{t, tff) cauuot vanish identically; see below, under b).
■-Compare remarks in §14, p. 59.
3 The proof has been given by Kxesee, Mdthematische Annalen. Vol. LV (1902),
p. 86. From the statements iu HoRiiAXx's Dissertation (GOttingen. 1887) it appears
that Weieesteass was in possession of essentially ';he same proof, but I have been
unable to ascertain whether he has ever given it in his lectures. I reproduce in the
text Kxeser's proof in a slightly simplified form. In §40 of his Lehrbuch, Kxesee
gives another proof which, however, presupposes that DfKt-^ , fg) +0
* Compare §16, p. 65, footnote 1.
^Xeither m nor r can be identically zero. For since, according to (20), 9j {t) and
0_,(0 are linearly independent and ff, ^^Oin {t„t^). ^iCq) and ^-i^t^^^ are not both zero,
and therefore m^O. it cannot be identicaUy zero since U^Q.
222 Calculus of Yae[ations [Chap, vi
^(>,) =0 , vi,(,-) = u (35)
respectively, and both vanish at Iq :
Hence the determinant D{f, /q) reduces, after an easy trans-
formation, to
D{t, Q = mv — VII . (37)
where
m =1 Uiidt , n = \ Ucdt .
From (35) follows :
v<i!{u) - u'^{v) =--Hi{iw'—u'v) = -uU .
Integrating and remembering (30) we get
Hi{uv' — u'v) = — ill . (38)
Again, we obtain by differentiating (37) with respect to / :
D = mv — nu ,
and therefore' 2
Du'-D'u=—r (39)
From the preceding equation it follows that D has af
tQ a. zero' of an odd order, except ivlien m(/o)=^0.
After these preliminaries, we write the second variation
in the form
8V = - £-^• 1^ ' iv' c/f + €^ J" ' IV \y (w) + ixU]df ,
ilf we denote by ?q' the root next greater than t^^ of the equatiou uit) =0, the
relation (39) shows that fy g t[^. For, since u has at t^ a zero only of the first order,
the quotient D/u vanishes for f„ , and therefore
D__C m'^dt
which proves that D 4= 0 for t^j<t< f„.
2Z) cannot vanish identically; otherwise m and therefore also u would vanish
identically, which is incompatible with our assumptions.
§41| ISOPERIMETBIC PROBLEMS 223
where A' is an arbiti'ary positive constant and
Now let u and v denote those particular integrals of the
differential equations
^{n) = () , ^{v) = U
respectively, which satisfy the initial conditions:
then it follows from a general theorem' on differential equa-
tions containing a parameter that
L iu{i) - u (/)) = 0 . L (f (f) - v{i)) = 0 ,
unifornilij in'tli rcsjjccf to ilie intcvcal (/o^i) ^{f ^ ■
Hence, if we put
m = I Utidt , Ti = I Uvdt ,
IJ (t , ti,} = )7ir — Tin .
we have also
Ln{f,U)=D{t,ft>), unifonnly in {f^, fi) .
Now suppose that
tv < U
and that
Then D{i, /q) changes sign at /(,', as has been shown above;
we can therefore choose two quantities t,^ and /^ satisfying
the inequalities
U < h <u<u<u ,.
iPoiNCARE, Mecanique cMeste, Vol. I, p. 38; Picaed, TraitS cf Analyse, Vol. Ill,
p. 137; and E. II A, p. 205. The assumption i7i 4=0 in (t^^t■^) is essential for this con-
clusion.
224: Calculus of Variations [Chap. VI
and so near to /q that D{f, Iq) has opposite signs at f^ and f^.
Now select k so small that also D{f , /q) has opposite signs at
f:i and f^; then D{f, /„) vanishes at least once at a point
Jq between ^3 and f^.
But since D{fQ, /q) is equal to zero, we can determine two
constants c^, C2, not both zero, so that
Ci>7i(A,') +c'2»(C) = 0 .
Now if we choose
ic = cji + r.J- ill {tj^) ,
and give the arbitrary constant /u. the value — Co? then ir sat-
isfies the differential equation
and the conditions (24) and (25).
This function w makes h'-J negative, viz. :
8-J= -ck I u^dt .
It remains to consider the exceptional case^ when ^/ (/q) =^ ^^•
This can only happen when at the same time m(fo)=^0 and
?"(/o') = 0, as follows at once from (39) and (38), if we remem-
ber that i?^i4=0 in (/q^i) and that 11 and u' cannot vanish
simultaneously.
In this case we can make h'-J -^0 \)\ choosing /u,^=() and
w=u in (/oAi) . »• = 0 in (/o'^i) :
and by a slight modification of the method used by Schwaez"
for the proof of the necessity of Jacobi's condition in the
unconditioned problem, it can be shown that S-./ can be
made negative by choosing
iFor this exceptional case, see Bolza, Mathematische Annalen,\o\. LVII (1903),
p. 44.
2 Compare §16, p. 65, footnote 1.
t;41| ISOPERIMETKIC PROBLEMS 225
?(• = /c -f A-.s in {fj',) . ic = ks in (fuf,) .
wliere
(t = i)
"We tlms reach in all cases the result that f/ic ihird iieccs-
sarij roiidiiioii for a iiiiiiiiiiiini is fliaf
Bit, g4z() for U<t<t, , (III)
or
K ^ i, ■
c) Knesers form of the determinant D{f, tn): Let 5(f = f-,o) be
a point on the continuation of the extremal Qa bej'ond the point 0,
taken sufficiently near to 0, or else the point 0 itself. Then it fol-
lows from our assumptions concerning the general solution (18) of
the differential equation (I) that there exists^ a doubly infinite sys-
tem S of extremals passing through the point 5 :
.r = <l>{t,o, h) . l/ = ^{f, a, b) , (40)
and satisfying the following conditions :
1. The extremal (So i^ contained in the system 2, say for
a = cti, , 6 = 6o .
2. The functions
(f>- ^, i^i, ^t, 4>fi' 4'tt
and their first partial derivatives with respect to a and b are con-
tinuous in a domain
n ^f^T, , \a- a„! ^ rf, , \b- b„\ ^ f/, . (41)
where 7\ < t^ < f„ <ti< 1\ and di is a sufficiently small positive
constant.
3. <^2 4_ ^^ -|- 0 in the domain (41) .
4. The value f = /.5, to which corresponds on the extremal (a, b)
ilf .r= /"(#. a. 3. Ai, y = c)[t,a.p.\) represents an extremal passing through the
point 5 (say for t = ^,). the quantities a , (3 . A , f. must satisfy the t\vo equations
S,«g,a,p,A)-<;(fgf,,o„,p„,A(,)=0.
Solving with respect to t-^ and A and remembering (20), we obtain the results stated
in the text.
226 Calculus of Variations
[Chap. VI
the point 5, is a function of a and h . of class C in the vicinity of
From the definition of f-,, according to which.
it follows by differentiation that
<l>t
<f>t
dh
da
86
' + <^„
= 0 , rp,
'du
da
:+^„
= 0
+ «^6
= 0
^t
(42)
5. X is a function of a, b of class C in the vicinity of a„, foo,
and the two derivatives
Xi = A„(ao, 6,,) . ^2 = K{(^Ui, h)
are not both zero, since 0^{f) and ^2(0 '^^^^ *^"o linearly independent
integrals of ^(h) = 0 (compare (33)).
We shall denote by
Y{t,a,b) , G{t,a.b) , K(t,a,b). G^{t,a,b), etc.
the functions of t, a, b into which F, G, H, G^, etc., change on
substituting
x = 4>(t, a,b) , y — xpit, a,b) ,
x'=<i>,{t, a,b) , y'=ij/f(f, a, b) .
The integral K taken along any extremal (a , b) of the system S
from the point 5(^ = ^5) to an arbitrary point ^, is a function of
/, a, b, which we denote by x(^. a, 6) :
(t, a, b)= f G(^ a, b) dt
(43)
Finally we denote by A( f , a , 6) the Jacobian of ^, t/-, x :
d(/,a,6),
T/iew TFeiers/ras6'.s function. D(t, t^,,) differs from the Jacobian
A{t, ao, bo) only by a constant factor :
D{t,t,,)=C^{t,a„b,) . (44)
%in
TSOPEEIMETRIC PROBLEMS
227
Proof: For the partial derivatives of x(t . f(.l>) we obtain the
followiiiii: vahu^s
Ba
Xi = ^ = 'i>i^.r- + ^iG,r ,
Applying the usual integration by parts and remembering that '
0\
%G^' = y'u, o,-l-^G, = -yu,
we get
x„ = f uiii^^cf^,, - <i>,^j at + Fg,, c/.,, + G,. .aJ' - G
The terms outside of the sign of integration reduce to
'9^
9a
on account of (42) .
A similar transformation applies to Xb •
We substitute these values of X(, x«, Xb in ^ ('> a? b) and then
put a = a,j, b = b„, which makes 4 = /50 .
Writing for brevity
^ — ^,^., — ^t^a\
\b = b,,
B = 4), (f>i, — (p, ^,,
a=a(,
b = b,
'0
M
= j UAdt , A^= j
UBdt ,
we obtain for the Jacobian the expression -
A(^ a,,, bo) --MB -NA
(45)
It is now easy to establish the relation (44) ; for if we substi-
tute in one of the differential equations (8) for x, y the functions
4){t, a,b),i'(t, a, b), differentiate with respect to a and then put
a = ao, b — bo, we get
* (A) f A, L^ = 0 ;
similarly :
* (5) 4- A, f/ = 0 .
'Compare equation (18) of §25.
^KxESER, Mathcmatischc Annalen, Vol. LV (1002), p. 93.
228 Calculus of Variations [Chap. VI
Hence if we set
u = X.2A — X^B ,
U and V satisfy the same differential equations as the functions
u,v introduced under h). Moreover, 77 and v vanish for t^^t^,
since, on account of {42),
Hence it follows that
77 = cii (t, f,„) , V = r if, t,„) + c'lt (/, f,,l
where c and e' are constants. Taking^ now D(f, fjo) in the form
corresponding to (37) we obtain immediatel}' the relation (44).
(I) Mayer i< lair of reciprocif [j for isopfrunetric proh-
lems : The problem : To maximize or minimize the integral
J while the integral K remains constant, and the "reciprocal
problem"' : To maximize or minimize K while J remains
constant, lead to ihe same totalifu of extremals.^
For, if we distinguish the quantities referring to the sec-
ond problem by a stroke and make the substitution
1
A =
we have
A = - , (46)
which shows that the differential equations for the two prol>
lems become identical by the substitution \ ^^ 1/A .
Now suppose that in both problems the given end-points
are the same and that, moreover, the values prescribed in
the two problems for the second integral are such that one
and the same extremal @o? fo^ which XQ=t=0, satisfies the
iThis remark had already been made by Eulek; see Stackel, Abhandlungen
aus der i ariationsrechnung, I, p. 102.
§41J ISOPERIMETEIC PROBLEMS 221t
initial conditions for lioth problems. Then iJic ('([iiivalence
of the iiro problems stilJ holds for the second varmtion.
For since
H. = ^ , (47)
Hi has a permanent sign so long as H^ has, and vice versa.
The sign is the same if X is positive, the opposite if X is
negative.
Further, the conjugate to the point 0 is the same in both
problems :
t; = n . (48)
For the system S of extremals through the point 0 is the
same in both problems.
Besides _
U=T ;
lience since the extremal ®o satisfies the differential equation
T+A„f7 = 0 ,
we have, along Qq :
and therefore, according to (45),
A(t, a, ,&„) = - K ^ {f , «o , &o) , (49)
which proves our statement.
This result is due to A. Mayer, and has been called by
him the Uiiv of reciprocifij for isoperimefric problems}
e) Example XIII (see p. 210) : From the expression (12) for Hi
it follows that X must be negative in case of a maximum. Equa-
tion (13) shows, then, that the vector from any point of the curve to
the center must be to the left- of the positive tangent. Of the two
arcs which satisfy the differential equation and the initial condi-
i Mathematische Annalen, Vol. XIII (1878), p. 60; compare also Knesek, Lehr-
buth, pp. 131 and 1-36.
- If, as we always suppose, the positive 2/-axis lies to the left of the positive j--ax'S.
230 Calculus of Vaeiations [Chap. VI
tions only the one above the .r-axis satisfies this condition. Tliis
arc may be represented in the form
X = a„ — \o COS / ) ■ , , ^ ,.~r.^
' . -/„^*</, <A,+ 2^ . (oO)
ij — ^„ — X„ sni / )
Hence we obtain
e^ (t) = - A„ cos / ,
e,{f)= -A„sin^ ,
e,(t)= K .
Again,
FIG. 4.-.
{vx^ + y'^y '
which is equal to — 1/Xo along (^'„, according to (13). This leads
to the following expression for D{t , /„) :
D (/, t^ = 4X" sin o) (sin (a — ta cos w) , (51)
where
Hence we easily infer that the parameter t^, of the conjugate
point is :
n = fo + 27r . (52)
The arc (So satisfies, therefore, the condition
/. < ^o' •
On the other hand, in the problem to maximize the integral
( ' r L {xi/- x'y) + K Vx" + y"^ dt ,
without an isoperimetric condition, the conjugate point t^' is
determined by the equation
0(f, A,)= -A;^ sin (f- 0 = 0 ,
whence '
iThe same result follows from the geometrical interpretation of Jacobi's cri-
terion: The extremals through A are circles of radius A^; their envelope is a circle
about A of radius 2Aq, which is touched by each circle ^ through A at the point dia-
metrically opposite to 4 on g.
§ilj ISOPERIMETEIO PROBLEMS 2;U
HO that, in accordance with the general theory,
t-O ^ 'll
/) Example XIV (see p. 211) : We have here
H, = ^~Jl±L^. ; (53)
hence for a niininuun it is necessary that
7/ + A > 0 .
Of the two sohitions (14) of the differential equation (I) which sat-
isfy the initial conditions, only the one in which the ii^jper sign is
taken in the expression for ^+^, fulfils this condition.
For this solution we obtain
6, (t) = ySo sinh t , $., {f) = /3,(t sinh t - cosh f) , 6.,{f) = /i?„ ,
X ij -X \j 1
Hence follows
{Vx"'-^xry A. cosh- ^ '
Jf,) ' L t-osh i X^'
Jfo ^ L cosh /J,'
f L^^3df = r tanhH'^
and the expression for Z)(f , /„) reduces to '
D {t, U) = f^ (2 cosh {f - /„) - 2 - (/ - A,) sinh (f - A,)) , (54)
or, if we put
f-L
2co
D (t , /,,) = 4ji3i; sinh w (sinh oj — w cosh w) . (54a)
The function sinh cj is positive for every positive w, and the
function
<^ (oj) = sinh CO — (u cosh o)
is negative for every positive to, since </>(w) = 0 and
<fi' (oj) =: — o) sinh OJ .
1 First given by A. Mayer, Mathcmutische Annakn, Vol. XIII (1878), p. 67.
232 Calculus of Variations [Chap. VI
Hence there exists uo co)iJiiyafe point, and the third necessary
coudition is always satisfied.
The same result is even more easily obtained by usin;»- Kxeser's
method : '
If we let the point 5 coincide" with the point 0 and choose for
the two parameters a . b the quantities
a = t:, . /> = /?,
the system of extremals through the point 0 is represented by the
equations
,r — .t'l, = b[t — a) , y — y„ — b (cosh / — cosh a) , (55)
Hence we obtain
•)(^{t , a , b) — I 1 x'- + ij"' fit = b (sinh t — sinh a) , (56)
and therefore
A {f . a , b) = b' [2 cosh (f -a) -2- (f - a) sinh {t - a)] ,
which for a = ao{=t„),b = b„{ — ^„) reduces to the expression (54)
for D(t,t,).
§42. SUFFICIENT CONDITIONS
The argumentation of §2S applies, with slight modifica-
tions,^ to the present problem, and leads to a fourth necessary
condition for a minimum:
1 Compare Kxesee, Lehrbuch,v. 14;}. -Compare the introductory lines of §41, c).
■These modifications are:
1. The variations ^, tj must now satisfy tlie isoperimetric condition:
in addition to the conditions stated in g2S, a). To obtain sucq variations, let
be arbitrary functions of i of class C satisfying the conditions :
/>j(*o) = 0, <//'^,)=0. p,(f2l=0. Qi(^,)=0,
P-i (h) 13 «2) ~P3 ('2) 1-2 ('2) +0 ' -V, + 0 ,
^V^- having the same signification as in §39, a). Then the functions
f = ejPj + e2P2 + ^3P3 • 1 = ^1914-^292 + ^333
will satisfy all the required conditions if ^1,^2' *3 ^'^^ determined by the equations
which is always possible under the above assumptions concernic.g jj ■ , q..
2. A J has to be replaced by A J+ A^AA'.
^i2\ ISOPERIMETRIC PROBLEMS 238
If we denote by E(.r, // ; j), <] ; p, ?i\\) the function de-
rived from H ^^ F \ XG exactly in the same manner in which
the E-function for the unconditioned problem is dei-ived from
the function F (see ecjuation (-1:8) of ^2S), then the Jhiirlli
ncccA^arjj condiiioii for a iitiniiinniiconsii^ls in the iiicqncilitij'
E (.'•, !j; p.q; P, q\K) ^0 (IV)
which must be fultilled along" the arc ©q for every direction
p. q.
The question arises now whether the four conditions
(I)-(IV) are sufficient for a minimum.
(i) W('ierstrass''s consfruciion: Let
6 : x=^{s) , y = ^{s) , So ^ -s ^ Si , (57)
he any curve of class ( '', different from @q, joining the points
{) and 1, lying in the region' U and satisfying likewise the
isoperimetric condition
K,„ = I ■
for s we take for simplicity the arc of the curve (5.
We propose to express the difference
A t/ := J^,^ — Jdi
in terms of the E-function.
For this purpose we take a point 5 on the continuation of
the arc @o beyond 0, but not on (£, and consider with Kneser*
the doubly infinite system S of extremals through the point 5 :
@: x = <j>{t,a,h) , y = ^{f,a,h) (58)
introduced in §41, c), the arc @q .being given by
x = <f> (t , eta , h) . ^ = <A (^ «o , h,) , fv^t^ti ■
1 Weierstrass, Lectures, 1879.
2In the same sense as in §28, a). sCompare §24, h) and §39.
■* Weierstrass considers instead tha sot of extreiiials through 0. Compare p.
3-10, footnote 1.
234
Calculus of Variations
[Chap. VI
We shall say that for the curve (S Weicrstvass's cnusfnicfion
/.s- posnihlc^ if the point 5 can be so chosen that the follow-
ing conditions are fulfilled :
A) Through every point 2 of the curve (E there passes a
uniquely defined extremal Go of the system 2 :
C5, : .c = cf>(t, a, . K) , if^ip it , a, , 6.) , (59)
lying wholly in the regic^n U and such that the integral K
taken alon^: ©•> from 5 to 2 has the same value as when taken
from 5 to 0 along @o and then from 0
to 2 alonof 6 :
-K52 — K-^i) + K^
02 >
(60)
FIG. 46
and when 2 coincides with 0 or 1 , the
extremal ©o coincides with ©o*
This means analytically : There
exists a system of three single-valued functions
such that
t = t (s) , a = a (s) ,
h{s)
(61)
<f>{t{s),a{8),h{s))=4>{s) ,
il^{t{s),a{s),h{s)) = 'ip{><) ,
where %(/, a, b) has the same signification as in equation
(43), and
X(s) = r G (^(.s), ^(.s-), ^'(.s'), r{^)) ^'« •
Moreover :
t (so) = A, , a (so) = «o , b (So) = &o ,
f (si) = ti , a (si) = ao , 6 (si) = h .
(62)
B) The three functions /(s), «(s), 6(s) are of class 6'' m
(SflSi).
1 Compare Knesek, Lehrbuch, p. 133.
§42] ISOPERIMETEIO PROBLEMS 235
C) If ^2 be any value of s of the inteTvnl (-Vi) and we
denote : '
f., = f (.s-.,) . a.2 = a {s.>) , b2 = b (.Sj) , t:,-, = t-, (a, , ho) ,
then the functions
4>, ^, 4>i, 4'i^ 4>ii, "Ptt
and their tirst partial derivatives with respect to (i and b are
continuous in the domain
(l? being a sufficiently small positive quantity, and moreover
the function" A.(rt, b) is continuous at (oo, Im).
These conditions admit of the followiniy sfeometrical
interpretation : *
We adjoin to the two equations (58) the equation
z = x{f, ((, b). (58a)
Interpreting then x, y, z as rectangular co-ordinates in
space, the equations (58) and (58a) represent a curve in
space, ©', whose projection upon the x, ?/-plane is the ex-
tremal (S, and whose ^-co-ordinate indicates at every point /
the value of the integral K taken along ® from the })oint 5
to the point /.
We thus obtain, corresponding to the system 2 , a doubly
infinite system S' of curves in space, all passing through the
point 5 :
a:- = a-3 , U = y, , z = 0 .
The particular curve ©J adjoined to the curve Q^) passes,
besides, through the two points 0' and 1':
0' : x = Xo , y = y^ , z = z^, = Km ,
1/ : X = Xi , y — yi , z = Zi = K-,,,-^1 .
In like manner we adjoin to the curve (i a curve in space,
iFor the notation soe §41, c). 2Conipare §41, c).
SWeieksteass, Lectures, 1879; compare also Knesee, Lehrl.nch, p. 140.
236 Calculus of Variations [Chap. VI
6'. by combining with the two equations (57) the third equa-
tion
z = x(:^) + K,. . (57a)
The curve 6' passes likewise through the points 0' and 1'.
The above assumptions A) and B) may then be couched
in geometrical language as follows :
Through every point 2' of the curve G' there passes a
uniquely defined curve of the set S' ; it changes continu-
ously as the point 2' describes the curve &' from 0' to 1' and
coincides with Qq when 2' coincides with 0' or 1'.
Under the assumption that Weierstrass's construction
is possible for the curve (i, we consider as in §20, b) and
§28, (1) the integral J taken from 5 to an arbitrary point
2(3 = 82) of 6 along the uniquely defined extremal ©_>, and
from 2 to 1 along (S, and denote its value regarded as a
function of ,so by 8(82) :
Then as in §20, />)
AJ=-[s(s,)->S(.So)] .
The integral K taken along the same path has the constant
value 1 + Kr^Q : _
since KQi = KQ2-\- K2i = l and K-^2=^Ko2^ ^50- Hence it
follows that we may write
im = ('1^ + ,, 'L^A +('H^+ ,,. ^) . (03)
ds2 \ ds.2 dso / \ d,§2 ds.2 )
Proceeding now as in §28, d) and remembering that the
extremal ©2 satisfies the differential equations
^^~dt^^'^-^ ' ^"-Jt^^^=^ '
where
we obtain the result
§421 ISOPERIMETRIC PROBLEMS 237
dS(.%) ^,- - - - ,x ■.
.^ " = - E(^-,, ^2.: i>2, g^; P2, q2\h) ,
the direction-cosines po, q> and ];2 5 ^2 referring to the curves
©2 and 6 respectively.
The result can again easily be extended to curves G hav-
ing a finite number of corners.
Thus we finally reach the result' that wlienever IVcicr-
sfrass^s construction is jwssible for thr cm- re 6. JVcici--
strass's theorem also Jiohls :
A J = I E (^"2 , //,, ; p, , q., ; 2h , q-2 1 h) ds^ . (64)
h) Hence we infer that AJ^^O whenever
E {J-2 , ^2 ; Ih , ^2 ; P2, ^2 1 ^2) ^ 0 throughout (s^> Sj) .
If, moreover, the E-f unction vanishes only icJten p>^^Pii
^i — 'li^ ''"'t/ if besides
A [to, a.2, bo) =t= 0 along (S ,
A./ cannot be zero, and therefore
AJ> 0 .
Proof:' If we differentiate equations (Gl) with respect to
6', we obtain
^ df , da , db
9f -j- + (p., -r ~r ^1, T = (f> ,
(is as (Ifi
. dt , da , db -,
"^f ;7 + '^■' 77 +'/''' ,^ = "A ,
(IS (Is (Is
df , da db
'^^ds'^^"ds^^"ds=^ ■
Xow if p2=^Pi, ^i^^'li, we have at the point 2 :
and therefore, since*
• Weierstrass, Lectures, 1879; compare Knesee, Lehrbuch, ii. 134.
-Due to Kneser, Lehrbuch, p. 134. 3 Compare §41, c).
238 Calculus of Vaeiations [Chap, vi
x'=G , Xt=G >
also'
x'= l^'Xt '
on account of the homogeneity of (jf.
Substituting these values in the above equations, we see
that either
A (^2 , a-i , b,) = 0 ,
or else
da , db „
— — u , — — — u .
ds ds
Hence if __
A(/., a^, ?>.^) 4: 0 along- 6,
a? and b-^ must be constant along (S, and, on account of (02),
their constant values must be
a (s) — a,, , b (s) = % ,
that is: 6 is identical with the extremal %, which is in con-
tradiction to our assumption that 6 shall not coincide with
©0. Hence the statement is proved.
c) In many examples the above theorem is sufficient to
establish the existence of an extremum.
Example XIII (see p. 229) : The system 2 is the totalifij of
circles through the 2^omt 5 :
a- — Xj = 6 (cos f — cos a) ,
y — yr> ^ b (sm f — sm a) ,
the parameters being a = tr,, b= — X .
The ordinate z erected at the point t of the circle (a , b) is the
length of the arc of this circle from the point b(t = a) to the point t :
z = \b{t-a)\ . (66)
The system S' of curves in space is therefore a sj^stem of helices.
Through every point {x,y,z) for which
z > V{x-jc^ + {y-yr,f > 0 , (67)
iThis means geometrically: If c?2 touches 5, then also (J.,' touches G'.
2 The result remains true if A (fg. ('21 ''2) =0 at a finite number of points.
§42] ISOPERIMETRIC PROBLEMS 239
thrre passes one and but one curve of the system 2 foi" which
a<f<a + 27r, h>0 . (68)
Moreover the inverse functions /, a, b of a-, y, z thus defined are
regular^ in the vicinit}- of every point (.rg, y^, z<^ satisfying the ine-
quality (fiT), and take, at the points (.ro, //n, 2;,,) and (.r, , //,. Zi) the
values /„, t/„, 60 and fi, Oo, 6u respectiveh.-
Now we join the two points 0 and 1 by an ordinary curve 6,
whose length has the given value / and which does not pass
through 5.
Then for every point 2 of (S the sum of the lengths of the arc
50 of the circle Q,» and of the arc 02 of (I is greater than — never
equal to — the distance between the two points 5 and 2, which in
its turn is greater than zero, since (5 does not pass through 5, /. e.,
the condition
1 Proof : Ou setting
f+a t—a
the equations for the determination of t , <i , b become
X — .(5 = — 26 sin y sin w ,
2/ -2/5= 26 cos 7 sin u>, (69)
z = 26w .
Hence if we put
and suppose
/ '• i
V (a; -a-j)' -;-((/- (/-I =u.
0<co<7r, we get It = 26 sin u),
and therefore we obtain for the determination of w and y tlie equations:
sin w • / (V rr.n\
= r, y-ij^-t(.c-Xr^)-ue'. ((0)
where r = u z . Since, according to equation (67) , 0 < c < 1 , the transcendental equa-
tion for 10 has one and but one solution in the interval : 0 < w < n- .
Moreover if 0<i'2<l be any particular value of r, this solution o> is regular in
the vicinity of V = r.j, since the derivative of the function sinio'w is +0 for 0<(o<7r.
Similarly the equation for y has a unique solution in the interval 05y<2ir,
which is a regular function of .c . y in the vicinity of every point (.x^ , V2) different from
(■':,■ y-J-
The values of w and 7 being found, the quantities t,a, h are obtained immedi-
ately. They satisfy the inequalities (68) and are regular functions of .r, y , z in the
domain (67).
2 For, of the two arcs of circles of the system 2 which pass through the point (.c , y)
and have the given length z, the one is described in the positive sense (so that the
center is to the left) if we start from the point .5, the other in the negative sense.
For the former the inequalities (681 are fulfilled, for the latter, they are not.
On the other hand the arcs 50 and 51 of e,, are, according to §41, e), described in
the positive sense, and are therefore contained in the above system of uniquely de-
fined solutions.
240 Calculus of Variations [Chap. vi
-2> y {.r,-.r,f +(>/,- u,f>0
is fulfilled.'
Hence it follows that Weierstrass's construction is possible
for the curve 6 .
Fiu'ther we find easily that
E {x2, 1)2 ; ih, Qi ; Th> <h ! K) = 'V^fl — cos a,,) . (71)
where a.2 is the angle between the positive tangents to the two
curves @2 and (i at the point 2 .
X2 is negative in (soSi) (since it is equal to — bo)^ and 02 cannot
vanish identically in (.s'„.s,) .
For, according to (51),
A ( fo , a.i, b-i) = 4 A.2 sin W2(sin cd., — Wjcos io.^) ,
and therefore
A (fj , «., , 62) =(= 0 in (So6'i) ,
since 0 < w.2 < tt .
Hence it follows that
AJ<0,
and thus we reach the result that the arc of circle @„ furnishes a
greater value for the area J than any other ordinary euvve of the
same length u-hich can be drau-n betu-een the two points 0 and I.
The same reasoning, slightly modified,- leads to the theorem
1 If we had taken, instead of the system of extremals through 5 . the system
through 0, the above inequality would be true only with certain exceptions which
would require a special discussion. Compare p. 233, footnote 4.
2The curve 0 is now closed; accordingly the points 0 and 1 coincide. If we let
also the point 5 coincide with 0 and consider two points 3 and 4 of i for which
•'*o'^'^:i*^'^4'^'''i • ^^^ obtain by the same reasoning as above
S{s^)-S{s.,
Now let .s, and s^ approach s,, and s, respectively, then we get
-I A.^ ( 1 — cos aj) ds.2 ,
J,ii being the area of a circle of the given perimeter I. Hence
The previous method is not applicable when the curve iT begins at the point 0
with a segment of a straight line, because then the inequality (67) is not satisfied for
th(^ point 3. In this case, take the point 3 beyond the end-point 6 of this rectilinoiir
segment and let 3 approach 6. Then ^(s^) approaches again Jq, with the same result
as before.
0
^■i'2\ TsOPERIMETRIC PROBLEMS "J4l
that among all closed cnrrcs of givoi Ipugth the circle i)icludes
the ma.riinum area.
Example XIV (see p. 231) : Any admissible curve 6 l)eiug
given, we choose the point 5 so that for every point 2 of 6
X2 > .Tj .
Then through every point 2' of the space curve (§,' one and ])ut
one curve of the system ' 2 ' :
X — x-^= h(t — a) ,
?/ — ^5 = 5 (cosh t — cosh a) , (72)
z =: b (sinh / — sinh a) ,
can be di'awn for which ^iq 47
t>a , 6 > 0 .
This follows from the determination of constants given in
§39, d). At the same time it is easily seen, in the same manner as
in the preceding example, that all the conditions for Weier-
strass's construction are fulfilled.
Further we find
E (0-2 , z/2 ; p2, q-i-, p2,q2\ K) = (2/2 + \') d — fos a.,) . ( 73 >
where a., has the same signification as in (71). But. according to
§41,/),
ij., -\- A., = 5., cosh to > 0 ,
since 62 >0, and a^ cannot vanish identically along 6 since
A(f,, a,, 62)4=0
alousr (5 • Heuce we infer that
^01 > '-^ui , i- c-,
the catenary @,, lias its center of gravity lower than, any other
ordinary curve of equal length which can be drawn between the
two points 0 and 1.
d) ^^FiehV about the (irc Qq : Returning now to tlie
general case, we meet with a peculiar difficulty which has
1 Compare equations (55) and (56).
242 Calculus of Variations [Chap. VI
no analogue in the unconditioned problem. Sn})pose that
for the are @q, which we assume to be free from multiple
points, the conditions
i/i>o (in
and
^<^u' (III';
are fulfilled.
Does it follow, then, that the arc ©q can be surrounde('
by a neighborhood (p) such that for every admissible curve
6 which lies wholly in this neighborhood, Weierstrass's
construction is possible '?
In the unconditioned problem and under the analogous
assumptions, this question could be answered in the affirma-
tive;' fo)' flic isopcfiuicfric pj'ohJrm fhc (jii<'>ffi<)ii lias )iof
yet been ((HSircrcd.
Only the following milder statement can be })roved :
If conditions (II') and (III') are fulfilled, a neighbor-
hood^ (p') of the space curve @o' adjoined to the arc ©^ can
be assigned such that JVeiei'strctss^s constriictioyi is jMfssihIc
for every adiuissible ciirre S ivJiose corresponding space
ciirrc lies icliolly in f/ie neighborhood {p' ) of ©J.
The proof proceeds by the following steps :
1. If conditions (II') and (III') are fulfilled, we can take
the point 5 so near to 0 that for the system of extremals
through the point 5 not only the conditions enumerated in
§41, c) are satisfied, but, besides, the following:^
A (t, «,,, b,) ^ 0 for U^t^f, . (74)
1 Compare §28, d) and §.34.
-We understand by the neighborhood (p) of the arc e,,' the portion of space
swept out by a sphere of radius p' whose center describes the arc i',/.
3 For the proof remember (44), and notice that the condition for a permanent
sign of S J may also be written
D{ti.t)^0 for t^,^t<f^,
(compare §41, a)). The statement follows then by a slight modification of the
analogous proof given by C. Jordan. Cours cV Analyse, Vol. Ill, No. 393.
§42] ISOPEEIMETKIC PROBLEMS '243
S
12. Bv an extension of the method of §34 we can now
prove the existence of a "ticld"' §>;[. about the arc ©' :
If 53 j;. denotes the domain
A, — e ^ / ^ f , + e , \a — ao\^k , \b — b„\^k ,
and §»/,. the image of 'B/^- in the .r, ij, ^-space detined by the
transformation
X = cf>{f, a, b) , ij = \lj(t, a,b) , 2: = x(^«,^>),
then the two positive quantities A: and e can be taken so
small that the correspondence between S;^. and ^l is a one-
to-one correspondence, and that at the same time
A(/, a, 6)4-0 (75)
in S,..
The sintrle-valued functions i, <i,h of ,r, /y, z thus de-
fined are of class C in ^^., and a neighborhood (p ) of the
arc G"o' can be inscribed in ^/..
It follows now easil}" that for every admissible curve 6
irliosc (idjoiiicd .^jxtcc curi'c lies ichollij in the ''fichV ^l,
Weierstr ass's construction is possible.
(') SiiJlficiciif conditions for <i soni-sh'ong niiiiiiiiiun :
Suppose iK)W that in addition to the conditions (II') and
(III') the inequality
E(.*', /y; p,q; P,q\K)>0 (IV)
holds along the arc ©^ for every direction /), q except
I>=p,q=q.
Then it follows from continuity considerations that we
can take /i"-so small that
E {J--2 , y-i ; i>2 , (h ; Ih , q^ \ K) > 0
along every admissible curve (S satisfying the above addi-
tional condition, except at the points where P2=P2^ ^2='Q2)
at which E vanishes.
244 Calculus of Vaeiations [Chap. VI
From Weierstrass's theorem and the ineqilality (75)
it follows now that for every such curve (S
AJ> 0 .
Hence, if we modifv our orimnal definition of a minimum
and say : "The arc @o furnishes a semi-stroiir/ minimum for
the integral J if there exists a neighborhood {p') of the
adjoined arc Go such that AJ^O for every admissible curve
(£ whose adjoined space curve 6' lies wholly in this neigh-
borhood (p')/' we can enunciate the
Theorem y The ejctrenial Qq {which ice suppose free from
)iu(1fij)le points) furnishes a semi-stroiKj ininimum for the
infegrol J ivith tlie isoperiynetric condifion K^^l, if ihe
eondiiions (//'), (///'), {IV') are fulfiUed.
It must, however, be admitted that the restriction which
we impose in the "semi-strong" minimum upon the varia-
tions of the arc CS",,, is rather artificial and alters completely
the character of the original problem."
1 Weieesteass, Lectures. 1X82; compare Knesee, Lehrlntch. %%'\C-) and 38.
Mayer's hiir of reciprocity extends to the sntHcient conditions for a semi-stronp
extremum, since, in the notation of J;+li''i, 1^ = 1 AE. Coniparo Kneser, Lehr-
buch, § 36.
2 As a matter of fact the i^recodiug theorem does not contain a solution of the
isoperimetric problem originally proposed, but a solution of the following problem,
which is usually (but unjustly I considered as e(iuivalent to the isoperimetric prob-
lem, viz. :
Among all curves in space which pass through the two points
x = .i\,, // = 2/,j, 2 = 0 and j- = x^. y = y^. z = l
and satisfy the diflerential equation
— = fT(.<-, y.x,y),
to determine the one which maximizes or minimizes the integral
J= I i-'i.f, ;v, ■>-', y') ^ft .
CHAPTER VII
HILBERT'S EXISTENCE THEOREM
§43. INTRODUCTORY REMARKS
If a function f{oc) is defined for an interval {ah), it has
in this interval a lower (upper) limit, finite or infinite, which
may or may not be reached. If, however, the function is
continuous in [oh), then the lower (upper) limit is always
finite and is always reached at some point of the inter-
val : the function has a minimum (maximum) in the interval.
Similarly, if the integral
J= \ F{.r,y,x',y')dt
is defined for a certain manifoldness M of curves, we can, in
general, not say <(, priori whether the values of the integral
have a minimum or maximum. But the question arises
whether it is not perhaps possible to impose such restric-
tions either upon the function F or upon the manifoldness
M, (or upon both), that the existence of an extremum can be
ascertained a 'priori.
In a communication to the "Deutsche Mathematiker-
Vereinigung" {Jahresherichte, Vol, VIII (1899), p. 18-1),
HiLBERT has answered this question in the affirmative. He
makes the following general statement :
"Eine jede Aufgabe der Variationsrechnung besitzt eine
Losung, sobald hinsichtlich der Natur der gegebenen Grenz-
bedingungen geeignete Annahmen erfullt sind und notigen-
falls der Begriff der Losung eine sinngemasse Erweiterung
erfahrt," and illustrates the gist of his method by the ex-
ample of the shortest line upon, a surface and by Dirichlet's
245
246 Oalculus op Vaeiations [Chap. Vll
problem. In a subsequent course of lectures (Grottingen.
summer, 1000) lie gave the details of his method for the
shortest line on a surface, and some indications' concerning
its extension to the problem (^f minimizing the integral
J= { ' F{ji-, y. !j')il,r .
• -A)
We propose to apply, in this last chapter, Hilbert" s
method to the problem of minimizing the integral ■
J'= I F{x, y, x', y')dt ,
with fixed end-points, under the following assumptions,
where S denotes, as before, a region of the .r, //-plane, and
Uq a finite closed region contained in the interior of S :
A) The function F{x, y, x', y') is of class C" and sat-
isfies the homogeneiiy coiKlitioii
F{a-, !,, k.r', ky') = kF{x, y, x', y') , k\> 0
throughout the domain
SI: ix,y) ill a , .r'^ + /y'^4=0 .
B) The function F(x, y, cos 7, sin 7) is positive through-
out the domain
S„: (oc, y) in IS„ , 0 ^ y ^ 27r .
C) Tlic function Fi{x, //, cos 7. sin j) is positi re ihrowgh-
out the domain QIq-
iln his thesis, Eine neuc Methodc inder Variationsrechnung (GOttingen, 1901),
§§5-14, Noble has discussed the details of the iiroof for this case. But his con-
clusions do uot possess the degree of rigor which is indispensable in an investiga-
tion of this kind. In particular, the reasoning in §§9, 10 and 1.3 is open to serious
objections.
2For the special case where F is of the form /(.i-, y)\ x"- — y"-, Lebesgue has
given a rigorous existence proof by an elegant modification of Hilbert's method
in a recent paper, "Integrale, longueur, aire," AniuiU di Matematica (.3), Vol. VII
(1902), pp. .312-359. Lebesgce applies Hilbert' s method also to the more difficult
case of a double integral of the form
i J V EG-F^ dudv .
%ii] Hilbert's Existence Theorem 247
D) The region ISo is coiirex (i. c, the straight line join-
ing any two points of Uo li*?s entirely in the region ISq) and
contains the two given points whieh we denote' with HiL-
BERT by A^ and A^.
Under these assumptions we propose to prove
1. That for every rcdijiaWc curve £ in the region ISq the
gmeralized infcgral J* (according to Weierstrass's defini-
tion) Itas (I. deicrminate finite inhie.
2. That there always exists, in the region ISo, at least one
rectifiable curve 2q, joining the two given points A'^^ and A\
which furnishes for the generalized integral ^7* an ahsolnfc
)iiiiti)in()H trifli respect to the totalitij of all rectijiabh' citrres
tcJiich can be di-oini hi ^S^ofrotu A^ to A^.
3. That this minimizing curve i'o is either a single arc of
an e.rtre))i(d of class C\ or else is made up of a finite
number or of a numerable infinitude of such arcs separated
Jl
by points or segments of the boundary of th^^ region Ho.
§44. theorems concerning the generalized integral Jf
In §31 we have considered Weiers trass's extension of
the meaning of the definite intesrral
r''
J= F(,v, y,y, y')dt
to curves havin^: no tangfent.
Another definition of the generalized inteo^ral has been
given by Hilbert^ in his lectures. This definition, while
' The advautage of this notation will appear in §45.
2Hilbert's own deflnitiou is as follows (see Noble, loc. cit., p. IS). Let 11 , be a
partition of the arc AB of a continuous curve into segments. Consider the totality
of all analytic curves which can be drawn from AtoB and which have at least one
point in common with each of the segments. Let J, denote the lower limit of the
values of the integral J taken along these curves. Next, let n._, be a new partition
derived from 11, by subdivision, Jo the corresponding lower limit, and so on. Then
HiLBERT defines the upper limit of the quantities: Jj , J2, J3, ■ • • , J„ . • • • if it be
finite, as the value of the definite integral J taken along the arc .-1 B .
248 Calculus of Variations [Chap. VII
leading to the same value for the generalized integral as
Weierstr ass's definition, is better adapted to our present
])urpose, especially in the simplified form which has been
given to it by Osgood.'
n) Hilhcrf -Osgood'' s (Icjiin'tioii of ilw (jcncralizcd inie-
(irol : We shall use the following notation: P' and P"
l)eing any two points of the region U^, we denote by
iH(P'P") the totality of all ordinary curves which can be
drawn in the region ISq from P' to P", and by i[P'P") the
lower limit of the values which the integral
J= fF{x, y, x', !/')(lf
takes along the various curves of M{P'P").
This loiver Jim it is cdwaijs positive. For, according to
A) and B), the function F{jc, y . cos 7, sin 7) has a positive
minimum value 7// in the closed domain SIq- Hence, if 6 be
any curve of M,{P'P"), we obtain, by taking the arc as
independent variable on the curve 6,
0<m\P'P"\^v,l^J^{P'P") , (1)
where I denotes the length of the curve 6 and |P'P"| the
distance between the two points P', P". Hence it follows
that
0<m|P'P"l^/(P'P") . (2)
After these preliminaries, let
S: x = <l>{t) , y = tp{t) , to^t^t,
be a continuous curve lying wholly in the region So- If
the functions <^(/), '^{t) are not differentiable, the integral J
taken along S has no meaning. In order to give it a mean-
ing also in this case, we consider any partition H of the
interval (/q/i)
'Osgood, Transactions of the American Mathematical Society, Vol. II (1901), p.
294, footnote.
§44] Hilbert's Existence Theorem 240
11: /„ < T, < r. • • • < T„_, < f, ,
and denote by
the corresponding points of the cnrve ^.
Then we form the sum
» — 1
The upper limit of the vahies of S^ for all possible parti-
tions n we define as the value of the integral J taken
along the curve S from A to B, and we denote it by
J**{AB), or simply J**.
It is easily seen that ^S'n uiay also be detined' as the lower
limit of the values of the integral J taken along all ordinary
curves which can be drawn in Sq from ^ to J5 and which
pass in succession through the points P^, P-,, • • •, Pn-i-
Hence it follows that it is always possible to select a
sequence \^^,\ of ordinary curves joining A and B, lying in
Sq, and such that
LJ,=Jf* .
v=oo
The above definition of the generalized integral is a
direct g^eneralization of Peaxo's" definition of the length of
a curve. For, in the particular case
the sum >S^n reduces to the length of the rectilinear polygon
with the vertices A, P^, P2, • • • , P^ -\^ B.
We must next investiojate under what conditions the gen-
eralized integral t/f * is finite, and show that for ordinary
1 This is the form which Osgood gives to Hilbert's definition; see the refer-
ence on p. 248, footnote ] .
2Pean'o, Appliciizioiii geometriche del Calcolo Infinitesinuile, p. 161.
2e50 Calculus of Vaeiations [Chap, vil
curves the generalized integral is identical with thr ordi-
nary definite integral.
h) Coiidifioiis for flic Jiiiitcitc.^s of fitc (jciicrdlizcd infv-
(jrol: The function F{.r, //, cos 7, sin 7) has a iinite maxi-
mum value M in the domain (Uq. Hence it follows that for
every curve 6 of M{P'P")
i(F'P")^JAP'P")^Ml , (2a)
/ denoting again the length of the curve (5. AVe may choose
for the curve 6 the straight line P' P" , since, according to
assumption D), the line P' P" lies wholly in the region 21,,.
Then we obtain the further inequality
i{P'P")^M\P'P"\ . ()})
From (2) and (3) follows at once
11-1 n — 1
ni 2^ I P.. P.,^, , ^ ^'ii ^ ^1/ 2) I P.. P.+i I • (4)
But the upper limit of the sum
II— 1
is, according to Peano's definition, the length of the eurve
i^ Hence we obtain the
Lemma: In order that f/ie gciieroh'zcd iiitcijnd Jf* nioij
he finite, it is necesscu-ij ond sujjicivnt that the curiae S shall
have a finite length (in Peano's sense).
We confine ourselves, therefore, in the sequel to continu-
ous curves S having a finite length ("rectifiable curves" in
Jordan's terminology).' From (4) it follows further that
m\AB\^ J ** (A B) ^ ML , (5)
where L denotes the length of the curve S.
U. I, No. 110.
§14] Hilbert's Existence Theorem 251
c) Properties of ilie generalized integral : From the two
characteristic properties of the lower limit it follows readily
that for any three points P, P', P" of iS,, the inequality
holds :
i{PP') + i {P'P") ^i {P P") . (6)
Hence it follows that if TTj denotes a partition derived from
n by subdivision of the intervals of 11 . then
Hence we easily infer that we get the same upper limit ./f *
for the values of S^ if we confine ourselves to those parti-
tions n for which
Tv-f-l — T,, < 8 ,
(v = 0 , 1 , 2 , • • • , ;/ — 1 ; T„ = f, , T„ = /i) ,
h being an arbitrary positive quantity.
Following now step by step the same reasoning which
Jordan uses in his discussion of the length of a curve, we
can easily establish the following properties of the general-
ized integral, always under the assumption that the curve "i
is rectifiable :
1. The generalized integral Jf^ {AE) is at the same time
the limit which the sum >S'ii approaches as all the differences
T^^.1 — T^ approach zero.'
Combining this result with the inequality (4) we obtain
the new inequality
wi^ Jp(AB) . (7)
2. If P be a point on the curve ii between A and P,
dividing the arc S into the two arcs S^ and So. then also the
integrals J**(J.P) and Jf*{PB) are finite, and"
J** (A B) = J** {A P) + J** (P B) . (8)
3. The generalized integral Jf^{AP) is a continuous^
1 Compare J. I, No. 107. 2Comparo .J. I. No. 108. ^ Apply (S) and (.-)).
252 Calculus or Variations [Chap, vil
function of the parameter / of the point P and increases
continuallv as P describes the arc AB from ^ to jB.
(/) Comparison tcifh W\'ierstrass''s definition of the gen-
eralized integral : If P' and P" are two points of Uq whose
distance from each other is less than the quantity Pq defined
at the end of §28, e), P' and P" can be joined by an extremal
@ of class C which furnishes for the integral J a smaller value
than any other ordinary curve which can be drawn in the
region Sq from P' to P". If the extremal (S itself lies
entirely in the region SIq, the value which it furnishes for
the integral J" is equal to i{P'P"); if @ lies partly outside
of iSq. this value is equal to or less than i{P' P").
Now consider any partition 11 for which
T^+l — T^, < 8 , (l/ = 0 , 1 , • • • , Ji — 1) ,
S being chosen so small that |P'P"| </Oo for any two points
P', P" of S whose parameters /', /" satisfy the inequality
\t' — r'|<S. Then we can inscribe in the curve 2 a 2^olij-
gon of minimizing cxtreinah with the vertices
As in §31, (/), let U^i denote the value of the integral J
taken along this polygon of extremals.
If the curve 8 lies entirely in the interior of Eo- ^ can be
taken so small that the polygon lies in the region iSy, and
therefore
Un=Su .
Hence Jf* may in this case also be defined as the limit
of Uu.
If 8 has points in common with the boundary of S^, Un
may be less than S^.
Nevertheless, also in this case the limit of Un for
LAt^O is Jf*.
In order to prove this statement we ccuisider, along with
§i5] Hilbekt's Existence Theorem 253
the two sums Su and Uu, the sum V^ defined in ■i^^^l, fj,
/. e., the value of the integral J taken along the rectilinear
polygon AP^P-i- • • Pn-iB. Since the region Eo is convc.r,
this polygon lies entirely in iSq, find therefore we have the
double inequality
Uu ^ Sn ^ Fn . (9)
From the first part of this inequality it follows that Uu has
a finite upper limit ^-/f*. This upper limit is at the same
time the limit which Uu approaches for Z.At = 0, as can be
inferred' from the fact (proved in §31, e)\ that if 11' be a
partition derived from IT by subdivision, then Uu'^ Uw
Hence it follows, according to §31, c) and (/), that Vu ap-
proaches the same limit as Uu ', therefore we obtain, on
account of (9), and remembering the equations (77) and (SO)
of §31 :
Jf* = J* , (10)
i. e., we have the result that Hilhert-Osgood's definition
leads for the generalized integral to the same value as
Weierstrass'' s definition.
Hence it follows, according to §31, 6), that /or an, "ordi-
nary'" curve the generalized integral coincides loith the
ordinary definite integral.
§45. hilbert's construction
We are now prepared to apply Hilbert's method to the
integral" ^7*.
Accordingly we consider the totality of all rectifi'able
curves S which can be drawn in the region iSo from the
point A'^ to the point A^. The corresponding values of the
integral Jf have a positive* lower limit. We propose to
1 Compare J. I, No. 107.
20n account of (10) we may use the symbol Jt iustcad of J^*.
s According to (o).
254 Calculus of Variations [Chap, vil
prove that under the assumptions A)-D) enumerated in
§43, ihere exists at least one rectijiahle curve So drairii in
^from A^ to A^ fo7- whicJi the integral J* adualhj reaches
its lower limit.
a) Construction of the point A-' : We consider the totality
of ordinary curves M(A^A^) which can be drawn in the region
®o from A^ to A^, and denote the lower limit i{^A^A^) of the
corresponding values of the integral J by K'
i {A" A) = K .
We can then select' an infinite sequence of curves
(Si , 1^2 ' ■ ■ ■ » ^vi ■ ■ ■ >
belonging to M{A^A^) such that the corresponding sequence
of values of the integral </, which we denote by
approaches K as limit :
L J, = K .
v=.aa
On the curve G^ there exists" one and but one point Al such that
'^^v V"-'^' ^'J = 2^^ ■
These points Al are infinite in number;^ they lie in the finite*
1 Compare Jordan's definition of "point limite,' loc. rit.. No. 20, and an analo-
gous remark in E. II A, p. 14.
2 Since F is positive along 0„ the integral J taken along the curve (>„ from A' to
a variable iJointP, increases continually as P describes the curve i!^ from ^ to ^ ;
hence it passes through every value between 0 and J^ once and but once.
if They need not all be distinct; the conclusion holds even if there are only a
finite number of distinct points among them. For in this case an infinitude of the
points A^ must coincide with at least one of the distinct points; this point has then
I' \
the properties of the point A^ .
*The existence of the accumulation point A'' can also be proved tinthout making
use of the finiteness of ffi^ . From (1) it follows that
2m
Hence if we select G> J^Cv = 1 , 2, 3, • - •), which is always possible since L J^, is
finite, the points A^, lie in the interior of the circle (A , G '2m), and therefore have
an accumulation point.
§45] Hilbert"s Existence Theorem 255
closed region Uq ; hence there must exist at least one
})oint A^ in So such that every vicinity of A^ contains an
infinitude of the points Al. Moreover, we can select a sub-
sequence \Q„ } of the sequence \(l^l such that
LAi =Ai .
h) Hilberfs Jcuima conccrniufi the point A'-: We con-
sider next the totality of curves
Then the fundamental lemma holds that the lower limit of
the corresponding values of the integral »/ is ^K:
i ( A"A^) = I i (A"A') = hK . (11)
Proof: We denote by 6^ the curve made up of the arc
A^Al oi the curve 6^ and of the straight line ^4^, ^'; the
latter lies entirely in So since So is convex.
According to (2a) the integral J taken along the straight
line Al A^ is at most equal to M' A\ _A^\. Therefore
Z.J,. {A'Al)^^K
k=:f: 'A-
since
LlJ,^ =^K and L \ A' Ai 1=0.
Hence it follows from the characteristic properties of the
lower limit that
i{A'A'^)^^K .
In the same way we prove that
i{AiA')^\K .
But, on the other hand, according to (t5) :
i {A'Ai) + i (AiA') ^ i {A" A') .
The three inequalities are compatible only if separately:
/ (A^Ai) = \K and / {AK\') =\K .
256 Calculus of Variations [Chap, vil
c) TJic points A'i'"' : Repeating the process of section a)
with the points A^ and A^ we obtain a new point, A*, lying in
the region iSo and having the characteristic property that
In like manner we derive from the two points A'^ and A^ a
point, A^, satisfying the relation
i {A^Ai) = i (AU') = \ i {AiA') = \K .
By an indefinite repetition of this process we obtain an
infinite set of points
]^^1' !
g = 0, 1, 2, -..,2"
'-0,1,2,..
all lying in the region Eo and having the characteristic
property that y jl <i±l\
i (A2»A 2« j = i„ iiT . (12)
More generally
i{A'-A-' ) = {T'--r')K , (13)
it '=^ "— ^
where n' , n" are integers, q , 7" odd integers, and
0^t'<t"^1 .
For, reducing r' and r" to the same denominator
/_^ ,,_q + r
'' 2" ' '" 2" '
we obtain, according to (6) and (12),
/ yAA^") ^ ^ i (a2« A 2-)=^^K,
A2« A 2" / ^ 'S^ i (^A2" A 2« j = ^^ ff ,
2M_
(a 2" A'j ^ T] i VA2"A 2» j = -| i:
H=g-*-/-
%io\ Hilbert's Existence Theorem 257
/ ( A"A2'7 + i (^A2» A 2" ) + / \A 2» A') ^ iv .
But on flu* other hand, we have, on account of ((>).
K = i [a^A') ^ i {a"A^) + '■ [a^'A^') + t (a^^ A') .
The two inequalities are compatible only if in each of the
above three formulae the equality sign holds, which ])roves
(13).
From (2) and (V-\) follows the important inequality
lA-A-'|^(r"-r')-- , (14)
where A^ A"^' \ denotes as^ain the distance between the two
points A''., A''".
Let us now denote by
Jc{r) , !j(t)
the rectangular co-ordinates of the point A^, r being one of
the fractions q/2^'' considered above. Then
\x(r')-.r(r")'^\A'-A'''\ , | ^ (V) - // (r") | ^ | A^ A-" | ,
and therefore on account of (14)
|^(r')-a^(T")|^(T"-r')- , j
V (15)
(?) The remaining points of Hilberfs curve: The mean-
ing of the two functions £c(/), !j{t), which so far have been
defined only for values of t of the set
, _) q{ g = 0, 1, 2. •••,2»-l ,
^~) 2^ ( ' >/=0,l,2,..-,
can now be extended to all values of t in the interval
O^/^l
as follows:
From the inequalities (15) we infer by means of the gen-
258 Calculus of Variations [Chap. Vll
eral criterion' for the existence of a limit, that if the inde-
pendent variable / approaches in ihc set S any particular
value t = a of the interval (01), then the functions .r(/), //(/)
approach determinate finite limits. In symbols, the limits''
L X (f) and L ij (f)
exist and are finite.
Moreover, if a itself belongs to the set *S', then
Lx{t) = x{a) , Ly{t) = y{a) . (16)
t\s t\s
t=a t=ii
Tf (I does not heloiuf to the set S, ivc define, according to
Hilbert, the fimctions x(f) (did i/{t) for t ^^a h/j the eqiici-
tioiis {10).
The two functions .r(/), y{t) thus defined for the whole
interval (01) are continuous and "o/ limited variation.''''^
For, the two inequalities (15), which have been proved for
values t' <iT" of ttie set S, can easily be shown to hold for
any two values f <t" of the interval (01), by considering
two sequences \tI\ and \tI' \ belonging to the set S and
such that
Lrl=f' , L tI' = t" .
From the inequalities (15) thus extended, it follows at once
that the two functions x{t), //(/) are continuous and "of
limited variation."'
1 Compare E. II A, p. i:i.
2 The notatiou accordiQK to E. H. Moore, Transactions of the American Mathe-
matical Society, Vol. I (1900), p, 500.
:i Compare .J. I, No. 67. Let f(t) be finite in the interval (t^yt^), and let
n: t^,<r,<r, • - • <r^^_,<t,
be a partition of this interval. If then the upper limit of the sum
n--l
^ = 11
for all possible partitions n is finite,/(<) is said to be "of limited variation."
§46] Hilbert's Existence Theorem 259
Hence the curve S,, defined by the two equations
So: .r = x(f) , !J = y{t) , O^t^l (17)
is continuous and has a finite length/ /. c, it is a rcrlijidhle
rnrre. As / increases from 0 to 1 the point (.r, y) describes
the curve Sq from the point A^ to the point A^. Moreover,
the curve Sq lies eniirely in fhc region ^q, since Sq is clo.^ed.
^-16. properties of hilbert's curve
It remains now to prove that the curve Sq actually mini-
mizes the integral J* and has the further properties stated
in §43.
a) Minimizing property of HiJherfs curve: The funda-
mental equation (13) which has been proved for values
t', t" of the set S only, can easily be extended" to any two
values /'< /" of the interval (01) :
i{A'A'")^(f"-t')K . (13a)
But from (13a) it follows immediately that tJic ijciierdlized
iutegral
I Compare J. I, Nos 105, 110.
-' For the proof, we introduce the same two sequences j t,, | , | t;, {■ as above.
Then we have, on account of (61,
/ (a' a"'') + i (a'^^'a'"'' ) + i iA^'" a'" ) 5 M a''a'") .
Passing to the limit 1"= =o we obtain, on account of the continuity of tlie functions
x(t),y{t).
L a' a""'' =0 . L\ a""'' a'"\ =0 ,
and therefore, on account of i3),
Moreover
L i (a''a''") = 0 , Li (a'"'' a'") = 0
Li(A'''A''n = (.t'~t)K ,
on account of (13). Thus we obtain
i {a'' a' )^ (f-'-f^K .
And by the method employed in proving U3) wc finally show that the inequality sign
is impossible.
260 Calculus of Vakiations [Chap. Vll
tdken aloitfi Hilbert's cur re I'o 's finite (ind ihat its value
is cqiKil to i(A^A^).
For let n be any partition of the interval (01) :
11: T„ = 0 < T, < To • •• < T„_, < 1 = T„ .
Then we obtain, according to (13a),
H-l
.S'n = V '■ (A^^'A^"^') =K = i {A' A') .
Hence also the upper limit of the values of aS„ is equal to
K, that is Jt^{A°A' ) = / ( AW ) . ( 18)
From the definition of the symbol i{A^A^) as lower limit
it follows now that if 6 be any ordinary curve drawn in So
from A^ to A^, then
J*(A»A')^J,(A»A') .
Moreover, if ^ be any rectifiable curve drawn in Eo
from A^ to A^, and e any preassigned positive quantity, we
can always find, according to §44, a), an ordinary curve 6
of MiA^A'^) such that
I Jf{A'A') - J,{A"A') I < e .
Hence it follows that
J*{A'A')^Jf{A"A') . (19)
This proves the theorem enunciated at the beginning of this
section :
If the conditions A)-D) enumerated in %43 are fulfilled,
then tJiere cdways exists at least one red if able curve join-
ing the two points A^ and A^ and lying entirely in tlie region
iSo, whicli furnishes for the integral
J = j F{x,y, x', y')dt ,
generalized, an absolute minimum luith respect to the totality
of rectifiable curves which can he drawn in %^from A^ to A^.
%i'o] Hilbert's Existence Theorem 261
b) Analytic characfei' of Hilbcrfs cin-rc: Let T' denote
the totality of those values of / in the interval (01) which
furnish points of the curve Sq in the interior of the region
Uo, T" the totality of those which furnish points of 2q on
the boundary of IS,,. From the continuity of ^q it follows
that every point' /' of T' is an inner' point of T' . Hence
an interval (a/S) contained in (01) and containing /' in its
interior can be determined such that all points in the inte-
rior of (a/3) belong to J", whereas the end-points belong to
T" except when they coincide with the points 0 or 1. The
set T' consists, therefore, of a finite or infinite number of
such intervals (a/3) which do not overlap. According to a
theorem of Cantor's,^ the totality of these intervals is
numerable, so that we may denote them by
The curve 2q consists, therefore, either of a finite number or
of a numerable infinitude of interior arcs separated by points
of the boundary of So
We are going to prove, according to Hilbert, that each
interior arc of Sq is an arc of an extremal of class* C".
For let P(f) be a point of Hilbert's curve Sg in the
interior of the region iR,,- Then according to §28, e) a
circle (P, a) can be constructed' about P such that any two
points P', P" in the interior of the circle can be joined by
an extremal G of class C" which lies entirely in the region
Uo and which furnishes a smaller value for the integral J
than any other ordinary curve which can be drawn in Uq
from P' to P".
1 Except the end-points of the interval (01) in case they should belong to T'.
2 Compare J. I, No. 22. 3 Mathematische Annalen, Vol. XX, p. 118.
*From our assumption C) it follows according to §6, c) that every arc of an
extremal of class C which lies in ?So, is ipso facto also of class C".
5 Let d be a positive quantity, taken so small that the circle (P, e) lies in the
interior of S,,, and let Pq be defined for the region 51^ as in §28, e). Then choose for
<T the smaller of the two quantities d/Z and Py 3.
262
Calculus of Variations
[Chap. VII
FIG. 48
On account of the continuity of the functions -rit), y{f)
there exists a vicinity (/ — S, / + 8) of / such that the arc of
the curve Sq corresponding to the interval' (/ — S, / + S) lies
wholly in the interior of the circle (P, a). Let Pi(fi) and
Pgf/g) be two points of this arc (/i< f^).
\ and denote by ©2 the minimizing ex-
p\ __a, tremal joining P^ and P3.
We propose to prove fin if f/ic arc
P1P3 of Hilbert's curve Sq ^•'^ identi-
cdl icith the extremal Qo-
Consider any point Poit-i) of the arc
P1P3 of £0 a»tl denote by ©3, d^ the
minimizing extremals joining Pj, Po and Po, P3 respectively.
Then it follows from the minimizing properties of the
extremals ®i, ®2, ©3 and from (13a) that
J^^iP.P,) = i{P,P,) = {U -fOK ,
J,^{P,P,) = i{P,P,) = if,-QK ,
J^JP.P,) = i{P,P,) = {t,-t,)K ■
hence, adding:
J^^{P,P:d = J^,{P,P,} + J,^,{P,P,) .
The extremal 60 furnishes therefore the same value for
the integral J as the curve made up of the two arcs ii^ and
©1. But this is in contradiction to the minimizing prop-
erty of ©2 ^^idess the compound curve 63, ©| coincides with
©o. Therefore the point P2 must be a point of (£"0 ; moreover
J,^ {P,P,) = i {P,P,) = (f, - f,) K .
Conversely, every point of the extremal do belongs at the
same time to the arc P1P3 of Sq- For, let P4 be any point
of @2 between P^ and P3, and let
u = J,..^{P^P,) .
Then
iQr (0, S), or (1 - 8, 1) in case P coincides with the point A or A ,
^iG\ Hilbert's Existence Theorem 263
0 < u < J.,,(P,P,) = (f, -t,)K .
Hence if we define ti l)y the relation
i( = (/, - /,) K ,
/^ lies between /^ and f-^ and is therefore the parameter of
some point P4 of Sq between i\ and Pg. The point 1\ be-
longs therefore also to ©2 ^^^ we have
JrjP, P,) = (f, -t,)K = J^JP, P,) .
Hence it follows that P4 must coincide with P4 since F is
positive along ©o-
Prom the relation between f^ and the quantity // (which
may be taken as the parameter on ©o), it follows, moreover,
that the points are ordered on both arcs in the same manner,
which completes the proof that the arc P1P3 of 2^y is iden-
tical with the extremal ©2-
Hence it follows that Hilbert's curve 2q is of class C"
and satisfies Euler's differential equation in the vicinity
of every interior point P, and therefore every interior arc of
Sq is indeed an arc of an extremal of class C" .
From the assumption B) that F is always positive it fol-
lows finally that Hilbert's curve 2^) can have tio iimUiple
points.
ADDENDA
P. 58, 1. 5: lu order to justify the terms "next greater," "next
smaller," it must be shown that an integral u of a homog'eneous
linear differential equation of the second order
d'u , cln , ,,
can have only a finite uuniber of zeros in an interval (ab) in
which 2^ iiiid q are continuous.
Proof: According' to the existence theorem (compare footnote
1, p, 50), u is of class C" in (ab). Suppose ti had an infinitude of
zeros in (ab); then there would exist in (ab) at least one accumu-
lation point (comjjare footnote 1, p. 178) for these zeros. Now
either it (c) =t= 0 ; then a viciuit}- of c can be assigned in which
i({x)=^0. Or else u{c) = 0; then ii (c) =^ 0 (compare footnote 3,
p. 58), and
n{c+h)=h{u'{c) + (h)) ;
hence a vicinity of c can be assigned in which c is the only zero of
u{x). In both cases we reach therefore a contradiction with the
assumption that c is an accumulation-point.
The same lemma has to be used, p. 108, 1. 6 up; p. 135. 1. 13;
p. 200, 1. 4; p. 221, 1. 1.
P. 59, 1. 11. Simpler as follows:
Choose X2 so that Xi < a?2 < Xo and at the same time X2 < Xi (the
quantity introduced on p. 55). Then A(j?, X2) and A(.r, x<,) are two
linearly independent integrals of (9). Applying Sturm's theorem
to these two functions we ol^tain the result that
A (x , Xo) =t= 0 in {Xf, , Xi) .
P. 62, 1. 6. Simpler proof: <Py{x, 7o) and A(x, Xo) are integrals
of Jacobi's differential equation; both vanish for x = Xo without
being identically zero. Hence they can differ only by a constant
factor. Compare footnote 2, p. 58, and footnote 1, p. 137.
P. 81, 1. 18. From what has been proved in the first paragraph
of p. 81, it follows that ^& is indeed a region in the specific sense
of §2, a).
265
200 Calculus of Variations
P. 83, 1. 13. Add:
d) The Field-Integral for the set of extremaU through the
point A.
Let P(a"2, y-i) be any point in the field ^^ formed by the set of
extremals through the point A{Xf,, >/-,), and let 7.' = '/'(•A, y-i) be the
value of 7 for the unique extremal of the field which passes through
the point P. Then the integral J taken along this extremal
@2'- y = <f>(x, y,)
from the point A to the point P is a single-valued finiction of
X2, J/2 which we denote by J{X2, t/2). Its value is
JioPo, y2)= I Fix,(i> (x , y,) , <^,. ( X . y_)) dx ,
where it is understood that 72 is replaced by its expression f (X2, yz)
in terms of X2 and yo ■
The partial derivatives of J{x2, yi) with respect to ^'2 and yz
have the following values:
'^^'l'"''^^' = P(^2 , Z/2 , Ih) - lhF,j. {x. , ?/2 , Ih) ,
" (15a)
O J^H'K-J^lf y2f I>2) )
where jfj^ denotes the slope of the extremal Q,2 at the point P.
For
^-^^^-^ = F{X2, y2, P2) + ^^£^' (i^.</>v + F„<t>.y)dx ,
9^/2 9(/2./^-
If we transform the integral as iu §20, c), and make use of (12) we
obtain (15a).
In many respects it would have been preferable first to prove
the formulae (15a) and to make use of them in the demonstration of
Weierstrass's theorem.
Compare the analogous formulae (44) in §37, and the still more
general formula? (14) in §34.
P. 142, 11. 4 and 5. After e insert: -\-2mTr where m is an integer.
Addenda litw
F. 151, 1. 14. Add: This result is due to Erdmann; compare
Journal fur Mafheinaiik, Vol. LXXXIT (1877), p. 29.
P. 152, 1. 8. Weierstrass himself gives the couditiou iu the
follovviug slightly different form :
+
Let §2 and h denote the numerical values of the angles which
the directions p2, ^2 and^)2? </? respectively make with the diiection
p2 -, (J2, SO measured that 5^ and S2 are < tt . Then
- + + + _ _
sin 8. ; sin 82 = E (.r.^ , y.^; P2,q2; P2, ^2) :
E (j"2 , i/2 ; Th , ^2 ; ih , q-i) • (♦'-ia)
This form of the condition follows immediately from (64). For on
account of (48) equation (64) may be written
lhl^\' {^2 > 2/2 > Ih , qd + q-iF",. (.ra , y.^ , fh , q^ —
PiK' ("^2 , 2/2 , 2h , q^) + q2i^\ (^2 , 2/2 , p-2 , q^) .
But _ + _ ^
p2 = I [sin S2i>2 + sin 82^>2]
. ■*■ - ■'- +
q2 = I [sin 82 q2 + sin 82 (/j] >
where Z is a factor of proportionality. Substituting these values in
the last equation, we obtain (64a).
F. 169, 1. 7, and p. 175, 1. 15. Instead of "region" read "domain.'
Compare 5:; 2, a).
P. 169, 1. 8. Instead of: "of the set," read: "to the set."
F. 172, 1. 13. Add reference to Kneseb, Lehrbuch, p. 48.
F. 178, 1. 18. After " abgeschlos.sen " add the reference: E. I,
p. 195.
F. 180, 1. 18. x\dd: Hence it follows that S'^ is a region in the
specific sense of ^2, a).
F. 182, 1. 7, and p. 185, 11. 4 and 6. The image of a region by a
transformation of the kind here considered is again a region.
Hence SI, (S^, ^Ic are indeed regions.
F. 200, 1. 7. Add: /,; is therefore identical with the quantity des-
ignated on p. 155 by t'u' . The use of the notation /o in the present
discussion is justified by the fact that in Kneser's theory the con-
jugate point appears as a special case of the focal point correspond-
ing to the case when the transversal X degenerates into the point A.
F. 246, 1. 1. HiLBERT has published the details of his proof of
268 Calculus of Variations
Diiichlet's principle in the Festschrift zur Feier des 150-jdhrigen
Bestehens der konigl. Gesellschaft der Wissenschaften zii Gottin-
gen 1901, and in the Mathematische Annalen, Vol. LIX (1904), p.
161.
P. 246, 1. 2. I had at my disposal a set of notes of this course
for which I am indebted to Professor J. I. Hutchinson.
P. 247, 1. 17. After "numerable" add the reference: E. I, A,
p. 186.
P. 253, 1. 17. After "result" add: due to Osgood; see the ref-
erence on p. 248, footnote.
INDEX
[The numbers refer to the pages, the subscripts to the footnotes.]
Absolute maximum, minimum, 10.
Accumulation-point, of a set of points.
Admissible cuetes, 9, 11, 101, 121, 206.
Amplitude, of a vector, 9.
Bliss's condition, for the case of two
variable eud-poiuts, 113.
Boundary conditions: along segment
of boundary. 43, 149; at points of tran-
sition, 42, iSO, 267; when minimizing
curve has one point in common with
boundary, 152, 267.
Boundary, of set of points, 5.
Beachistocheone, 126, 13."), 146; determi-
nation of constants, 128., ; case of one
variable end-point, 106i.
Catenoid (see Surface of revolution of
minimum area).
CiECLE, notation for, 9.
Class C,C,C/' D,D'..: functions
of, 7; curves of, 8, 116; curves of class
K, 161.
Closed: region, 5; set of points, 178, 267.
Conjugate points, 60; for the case of
parameter-representation, 135; for iso-
perimetric problems, 221 ; geometrical
interpretation, 63, 1.37 ; case where the
two end-points are conjugate, 65,, 204.
Connected set of points, 5.
Continuous functions: definitions and
theorems on: existence of maximum
and minimum, 1.34, ^02 ; sign, 21,; uni-
form continuity, 80o; continuity of com-
pound functions,2l3; integrability, I'l.^.
Continuum, 5.
Convex eegion, 247.
Co-oedinates: agreement concerning
positive direction of axes, S.
Coenee: defined, 8, 117; corner-condi-
tions, 38, 126, 210.
Ceitical point, 109,.
Cueves : (a) representable in form
«=/(«), 8; of class C,C',.. D, 8; (h)
in parameter- representation, 115.,: of
class C, C". 116; ordinary, 117 ; regular,
117; rectiiiable, 116,; of class K, 161;
Jordan curves, 180.
Curvilinear co-ordinates: ingeneral,
181 ; Kneser's, 184.
Definite integrals : theorems on : in-
tegrable functions, 12.,, 89j ; first mean-
value theorem, 'H^; connection with
indefinite integral. 89., : integration by
parts, 20, ; differentiation with respect
to a parameter, I63.
Derivatives: notation, 6, 7; progres-
sive and regressive, 7, ; reversion of the
order of differentiation in partial de-
rivatives of higher order, I83.
Differential equ.vtions: existence
theorem, 284; dependence of the gen-
eral integral upon the constants of in-
tegration, 543; upon parameters, 71..,
223,.
Discontinuous solutions, 36, 125, 209.
Distance: between two ijoints, nota-
tion, 9.
Domain, 5.
End-points, variable (.see Variable end-
points).
Envelope: of a set of plane curves in
general, 624, 1.374 ; of a set of extremals.
62; theorem on the envelope of a set of
geodesies, 166; extension of this tin -
orem to extremals, 174; case when the
envelope has cusps. 201 ; case when the
envelope degenerates into a point, 204.
Equilibeium, of cord suspended at its
two extremities, 211, 231, 241.
Equivalent peoblems, 183, 197, 228.
Erdmann's coenee condition, 38.
Euler's (differential) equation, 22;
Du Bois-Reymond's proof of, 23; Hil-
bert's proof of, 24 ; Weierstrass's form
of, 123; assumptions concerning its
general integral, .54, 130; cases of re-
duction of order, 26,, 29.
Euler's isoperimeteic rule, 2in.
Evolute, of plane curve, 1743.
Existence theorem: for a minimum
" ini Kleinen," 146; for a minimum " ini
Grossen," 245; for differential equa-
tions, 2X; in particular for linear dif-
ferential equations, .50.
Extraordinary vanishing of the E-
function, 142.
Extremal: defined, 27, 123, 209; cdm-
structiou of extremal through given
point in given direction, 28, 124; set of
extremals through given point, 60; sft
of extrenials cut transversely by a
given curve, HI ; construction of ex-
tremal through two points, sufficiently
near to each other, 146; problems with
given extremals, 30.
Extremum: defined 10 (compare Mini-
mum, Maximum 1.
Field: defined, 79; theorem concerning
existence of, 79; applied to set of ex-
tremals througli .4,82; improper, S3., ;
for case of parameter-representation,
!44, 176; for isoperimetric problems,
241; field-integral, "266.
First necessary condition (seeEuler s
differential ecjuation).
First variation : defined, 17; vanishing
of the, 18; transformation by integra-
tion by parts, 20, "22; for case of
variable eud-points, 102, ; for case of
269
270
Calculus of Variations
parameter-representation, 122, 123; for
isoperimetric problems, 209.
Focal point : of a transverse curve on
an extremal: defined, 109; equation
for its determination, according to
Bliss, 10^, 155 ; according to Kneser, 200 ;
geometrical interpretation. 111, 156:
case where end-point B coincides with
focal point, 201.
Fourth necessary condition (see
under Weierstrass).
Free variation, points of, 41.
Function Ef.j-, ?/: p, p): defined_34, 75;
relation between E (.<•, y; p, p) and
Fify-, 76; geometrical interpretation
of this relation, 77.
Function Ei (.c, 2/ ; p, p ), 76.
Function E(.r, ;/; p,q; /*, ?): defined,
i:-W; homogeneity properties, UO; rela-
tion between E-fuuction and F,. Ill;
ordinary and extraordinary vanishing,
142; Kneser's geometrical interpreta-
tion, 195.
Function Ei (.r, y; ]>, q; P, g ), 145.
Function F^, 121.
Function F^, 1.32.
Fundamental lemma, of the Calculus
of Variations, 20.
Generalized integral. 157, 248 (com-
pare Integral taken along a curve).
Geodesic curvature, 129.
Geodesic distance, 176.
Geodesic parallel co-okdinates, 164.
Geodesics, 128, 146, 155; Gauss's theo-
rems on, 164, 165; theorem on the en-
velope of a set of, 166.
Hilbert's: construction, 253: existence
theorem, 245; invariant integral, 92, 195.
Homogeneity condition, 119; conse-
(luences of, 120.
Implicit functions, theorem on, 35™.
Improper: field, 83.; maximum, mini-
mum, 11.
In a domain, use of the word explained,
.5,6.
Infinitesimal, 6.
Inner point, 5.
Integeability condition, 29.
Integrable functions, theorems on,
125. 89„.
Integr.al, taken along a curve, defini-
tion and notation. 8; for case of par-
ameter-representation, 117; condition
for invariance under parameter-repre-
sentation, 119; extension to curves
without a tangent, (a) Weierstrass's,
157, (6) Hilbert-Osgood's, 248.
Integration, by parts, 20, 20i.
Interval, defined, 5.
Invariance, of E and Fi, 183.
IsoPERiMETEic constant, 209; Mayer's
theorem for case of discontinuous solu-
tions, 209i.
IsopERiMETRic PROBLEMS : in general.
206-44; special, 4, 210, 229, 238; with
variable end-points, 1132.
.Iacobian, 572.
Jacobi's condition. 67 : proofs of its
necessity, 65,, 66; Weierstrass's form
of, 135; Kneser's form of, 136; for case
of one variable end-point, 109, 155, 200;
for isoperimetric problems, 225, 226.
Jacobi's: criterion, 60,135; differential
equation, 49, 133; theorem concerning
the integration of Jacobi's ditterential
equation, 54, 1.35; transformation of
the second variation, 51.
Jordan curve, 180.
Kneser's: theory, 164-205; curvilinear
co-ordinates. 184 ; sufficient conditions,
187; theorem on transversals, 172.
Lagrange's differential equation,
•>■>
Legendre's condition, 47; Weier-
strass's form of, 133; for isoperimetric
problems, 217; Legendre's differential
equation, 46.
Length of a curve: Jordan's defini-
tion, 157, ; Peano's definition, 2492.
Limit: definition and notation, 1^; uni-
form convergence to a, 19i ; criterion
for the existence of, 258,.
Limited variation, functions of, 2583.
Limit : lower and upper. 83, lOj ; attained
by continuous function, 134, 8O2.
Limit-point (see Accumulation-point),
Lindelof's construction, 64.
Linear differential equations of
the second order : existence theo-
rem, hOi ; .Abel's theorem, 582 ; Sturm's
theorem, SSj.
Lower limit, 83, 10,.
Maximum (see Minimum).
Mayer's law of reciprocity for isoperi-
metric problems. 229, 244,.
Mean-value theorem, first, for definite
integrals. 24^.
Minimum : of a continuous function, 184,
8O0 ; of a definite integral, absolute and
relative, 10; proper and improper, 11:
weak and strong, 69, 70; for case of
parameter-representation, 121; semi-
strong in case of isoperimetric prob-
lems, 244; existence of a minimum "im
Kleinen," 146; Hilbert's a-priori exis-
tence proof of a minimum "im Gros-
sen," 245-63.
Neighborhood of a curve, 10; neigh-
borhood(p) of a curve, 1.3, 121.
Neighboring curve, 14,.
Numerable set of points, 261, 268.
One-sided variations (see also Boun-
dary conditions) : analytic expression
for, 42, 148; necessary conditions for a
minimum with respect to, 42, 149; suf-
ficient conditions, 42.
Open region, 5.
Ordinary curves, defined, 117.
Index
271
Oedinaky vanishing of the E- func-
tion, 142, 206.
Osgood's theorem concerning a char-
acteristic property of a strong mini-
mum, 190.
Parameter eepeesentation, curves in,
115.
Parametee-teansfoemation, 116.
Paetial derivatives (see Derivatives).
Partial variation, of a curve, .37.
Point-by-point variation, of a curve,
41.
Point of a set, 124.
Positively homogeneous, 119.
Progressive derivative, 7,.
Proper minimum, 11.
Rectifiable curves, II60, 250i, 251i,
25I2, 2.')l3 (compare Length).
Region: defined, 5; open, .5; closed, 5.
Regressive derivative, 7,.
Regular curves, 117; functions, 2I2;
problems, 2',i, 40, 97, 125.
Relative maximum or minimum, 10, IO4.
Second necessary condition (see Le-
gendre's condition).
Second variation, 44-67; Lesendre's
transformation of, 46; Jacobi's trans-
formation of, 51 ; for case of variable
end-points, 102, ; Weierstrass's trans-
formation of, for case of parameter-
representation, 131 ; for case of variable
end-points in parameter-representa-
tion, 102, 155; for isoperimetric prob-
lems, 216-25.
Semi-steong exteemum, 244; sufficient
conditions for, 244.
Set of points: definition, 10,; inner
point of, 5 ; boundary point of, 5 ; accu-
mulation points of, 178i ; closed, 178,
267; numerable, 261, 268; upper and
lower limits of one-dimensional set, 83,
IO2; connected, 5; continuum, 5.
Sign of square roots, agreement con-
cerning, 2,.
Slope eesteictions, 101].
Solid of eevolution, of minimum re-
sistance, 73 1, 1423.
Steong exteemum: defined, 70; sufiS-
cient conditions for (see Sufficient con-
ditions).
Strong variation, 72.
Sturm's theorem, on homogeneous lin-
ear differential equations of the second
order, 58^.
Substitution symbol, 5, 6.
Sufficiency proof, for geodesies, 165.
Sufficient conditions for weak mini-
mum, 70.
Sufficient conditions foe strong
minimum: when x independent vari-
able, in terms of E- function, 95; in
terms of Fy ;/■, 96; for one-sided varia-
tions, 42, ; in case of one movable end-
point, 109; in case of two movable
end-points, llSo.
Sufficient conditions for strong
minimum : for case of paranif^ter-repre-
sentatiou, Weierstrass's, 14.3-46; exten-
sion to curves without a tangent,
Weierstrass's proof, 161, Osgood's
proof, 192; Kneser's sufficient condi-
tions for case of one movable end-
point, 1X7; for isoperimetric problems,
Weierstrass's, 237, 243.
Surface of revolution of jnNiMUM
AREA, 1, 27, 48, 64, 97, 153.
Taylor's theorem, ll,.
Third necessary condition (see Ja-
cobi's condition).
Third variation, 59,.
Total differential, 253.
Total variation, 14.
Transverse; curve transverse to an ex-
tremal, 106; condition of transversality.
36, 106; in parameter-representation,
155 ; for isoperimetric problems, 210.
Transversal: to set of extremals, 168;
degenerate, 169; Kneser's theorem on
transversals, 172,,
Unfree variation, points of, 41.
Uniform continuity, 80,.
Uniform convergence, to a limit, 19,.
Upper limit, 83, 10,,.
Variable end-points: general expres-
sion of first variation for case of, 102, ;
of second variation, 102, ; one end-point
fixed, the other movable on given curve,
treated (a) by the method of differen-
tial calculus, 102-113, (b) by Kneser's
method, 164-205 (for details see Trans-
versality, Focal ijoint. Sufficient con-
ditions) ; case when both end-points
movable on given curves, 113.
Variation: of a curve, 14,; total, 14;
definition for first, second, etc., 16:
special variation of type e>), 15; of type
«o(x, e).18; for case of parameter-repre-
sentation, 122, 122, ; weak and strong, 72.
Varied curve, 14,.
Vicinity (5) of a point, 5.
Weak exteemum: defined, 69; sufficient
condition for, 70.
Weak variations, 72.
Weieesteass's: construction, 84, 144,
234; corner-condition, 126; E-function,
35, 1.38; form of Euler's equation, 123, of
Legendre's condition, 133, of Jacobi's
criterion, 135; fourth necessary condi-
tion, 75, 1.38, 233; lemma on a special
class of variations, 33, 1.39; transforma-
tion of second variation, 131.
Weierstrass's sufficient conditions,
95, 96, 143; extension to curves without
a tangent, 161; for isoperimetric prob-
lems, 237, 243.
Weierstrass's theorem (expression of
A,/ in terms of the E-function), 89, 144;
Hilbert's proof of, 91 ; for case of vari-
able end-points, 189, 194, 195; for iso-
perimetric problems, 237.
Wronskian determinant, 57,.
Zermelo's theorem, on the envelope of
a set of extremals, 174.
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