Lagrange Origins

8/12/2019 Lagrange Origins

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The Or ig in s o f Eu ler s V ar ia t iona l Ca lcu lu sCRAIG G. FRASER

Communicated by H. Bos

1 I n t r o d u c t i o n

The calculus of variations was established as a distinct branch of analysiswith the publication in 1744 of EULER'SMethodus inveniendi curvas lineas EULERsucceeded in formulating the variational problem in a general way, in identify-ing standard equational forms of solution and in providing a systematic tech-nique to derive them. His work included a classification of the major types ofproblems and was illustrated by a wide range of examples.

EULER'S treatise was the culmination of a line of research that had beguna half-century earlier with the publication of two papers by JAKOB BERNOULLI nthe Acta eruditorum BERNOULLI'S pioneering researches were continued by hisbrother JOHANN and by the English mathematician BROOK TAYLOR. EULER'Sinitial investigations in the 1730s began from a study of the work of these men.

Ironically EULER'S entire approach to the subject would be set aside with theappearance of LA~RAN~E'S method in the 1750s. This method, which veryquickly became standard, involved a variational process that was fundamentallydifferent from EULER'S and that established the subject along new and differentlines. There is a family resemblance shared by the techniques of the BE~NOtJLLIS,TAYLOR and EULER that distinguish them from those employed in the post-Lagrangian stage of the subject.From a larger historical perspective it is clear that the Methodus inveniendicapped and completed the first phase in the development of the calculus ofvariations. The treatise was nevertheless by no means a straightforward productof earlier research. In its basic organization and direction it represented a sub-stantial break with the then established tradition, including EuL~R's own pre-vious work in the subject.

The present paper is concerned to identify some significant characteristics ofthe BERNOULLIS' and TAYLOR'S theory and to document the substantial achieve-ments of EULER'S variational calculus that preceded the Methodus inveniendiThe intent is to illuminate the earlier developments and to explain how and forwhat reasons the subject assumed the form that it did in the treatise of 1744. Inorder to orient the discussion and to provide a basis of reference we begin bylaying out in summary the salient points of the subject as it is developed in theMethodus inveniendi We then turn to an examination of the BERNOULLIS' re-search, with consideration of the ways in which their approach was similar to


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104 C .G . FRASERan d d i f f e r ed f ro m EULER 'S . A s t u d y o f TAVLOR'S s o l u t i o n t o t h e i s o p e r i m e t r i cp ro b l em wi l l r ev ea l i t s s p ec i a l s i g n i f i can ce i n t h e b ack g ro u n d t o h i s i n v es t i g a -t i o n s . F i n a l l y , we d i s cu ss i n s o m e d e t a i l EULER 'S v a r i a t i o n a l p ap e r s o f 1 7 38 an d1 74 1 t h a t w e r e p u b l i s h e d a s m e m o i r s o f th e S t . P e t e r s b u r g A c a d e m y o f S c i en c e .W e c o n c l u d e w i t h s o m e r e f l e c t io n s o n t h e e a r l y h i s t o r y o f t h e c a l c u l u s o fv a r i a t i o n s .

2 . E u l e r ' s T h e o r y o f 1 7 4 42.1 T h e p r o b l e m s c o n s i d e r e d b y E UL ER i n t h e M e t h o d u s i n v e n i e n d i f a l l n a t u ra l l yi n t o t h r e e g r o u p s .

I . P r e s e n t e d i n C h a p t e r 2 , t h e s e p r o b l e m s a r e t h e m o s t b a s i c o n e s , i n w h i c hn o s i d e c o n d i t i o n o r s p e c i a l r e l a t i o n a m o n g t h e v a r i a b l e s i s a s s u m e d .

I I . P r e s e n t e d i n C h a p t e r 3 , t h e s e i n v o l v e c e r t a i n v a r i a t i o n a l i n t e g r a l s i nw h i c h a v a r i a b l e a p p e a r s i n t h e i n t e g r a n d t h a t i s c o n n e c t e d t o t h e o t h e rv a r i a b le s o f t h e p r o b l e m b y m e a n s o f a d i f fe r e n t ia l e q u a t i o n . I n t h e m o d e r ns u b j ec t t h is p ro b l e m is an i n s t an ce o f t h e v e ry g en e ra l p ro b l e m o f LAGR ANGE .

I l I . P r e s e n t e d i n C h a p t e r 5, t h e s e in v o l v e a s id e c o n d i t i o n t h a t is f o r m u l a t e di n t e rm s o f a d e f i n i t e i n t eg ra l. T h e i s o p e r i m e t r i c p r o b l em i s t h e c l a s s ica l r ep re -s en t a t i v e o f t h i s c l a ss o f p ro b l em s .

E UL ER 'S t h e o r y is b a s e d o n h is d e r i v a t i o n o f th e v a r i a t i o n a l e q u a t i o n s f o rt h e s e t h ree c l a ss e s o f p ro b l em s .I . W e a re g i v en a cu rv e i n t h e p l an e j o i n i n g t h e p o i n t s a an d z (F i g u re 1 ) . Th ec u r v e r e p r e s e n t s g e o m e t r i c a l l y t h e a n a l y t i c a l r e l a t i o n b e t w e e n t h e a b s c i s s a x a n dt h e o r d i n a t e y . L e t M , N , O b e t h r e e p o i n t s o f th e i n t e r v a l A Z in f in i t e ly c loset o g e t h e r . W e s e t A M = x, A N = x ' , A O = x , a n d M m = y, N n = y' , O o = y .Th e d i f f e r en t i a l co e f f i c i en t o r d e r i v a t i v e p i s d e f i n ed b y t h e r e l a t i o n d y = p d x .E U LE e p r e s e n t e d t h e r e l a t i o n s

(1)y l y

P - d xy , , _ y 'pr d x

wh i ch g i v e t h e v a l u es o f p a t x an d x ' i n t e rm s o f d x an d t h e d i f f e r en ces o f t h eo r d i n a t e s y , y ' a n d y .

S u p p o s e n o w t h a t Z is s o m e e x p r e s s i o n c o m p o s e d o f x , y a n d p . (E U LE R a l soe m p l o y s t h e s y m b o l Z t o d e n o t e t h e e n d p o i n t o f t h e in t e r v a l A Z ; t h e c o n t e x th o w e v e r a l w a y s m a k e s t h e u s a g e c l e a r . ) T h e d e f i n i t e i n t e g r a l ~ Z d x c o r r e s p o n d -i n g t o t h e i n t e r v a l f r o m A t o Z i s(2) ~ Z d x ( A to M ) + Z d x 4- Z ' d x + . . . ,


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Th e Origins of Euler s Va riational Calculus 105

A . ~ x ~ K L ~ N O P Q R 5

~

7Figure 1.

w h e r e Z , Z ' , . . . a re t h e v a l u es o f Z a t x , y , p ; x , y , p ; . . . . Su p p o s e t h e g i v enc u r v e j o i n i n g a t o z is s u c h t h a t ( 2) is a m a x i m u m o r m i n i m u m . I n c r e a s e t h eo r d i n a t e y b y t h e i n f in i t e ly s m a l l q u a n t i t y nv , o b t a i n i n g i n t h i s w a y a c o m p a r i -s o n c u r v e a m v o z . T h e c h a n g e i n t h e v a l u e o f t h e i n t e g r a l c a l c u l a t e d a l o n g t h eg i ve n a n d c o m p a r i s o n c u r v e s m u s t b y h y p o t h e s i s b e z er o . T h e o n l y p a r t o f t h ei n t eg ra l t h a t i s a f f ec t ed b y v a r y i n g y is Z d x + Z ' d x . E U LE R w r o t e

d Z = M d x + N d y + P d p3 ) d Z ' = M ' d x + N ' d y ' + U d p ' .H e p r o c e e d e d t o i n t e r p r e t t h e d i f f e r e n t i a l s i n ( 3 ) a s t h e i n f i n i t e s i m a l c h a n g e s i nZ , Z ' , x , y , y ' , p , p ' t h a t r e s u l t w h e n y i s i n c r e a s e d b y nv. E v i d e n t l y dx = O,d y = 0 an d d y ' = n v . F r o m ( 1 ) w e s e e t h a t d p a n d dp ' e q u a l n v / d x a n d - n v / d xr e s p e c t i v e l y . H e n c e ( 3 ) b e c o m e s

tz~d Z = P ' ~(4) d xvd Z ' = N n v - P - -d x

T h u s t h e t o t a l c h a n g e i n ~ Z d x ( A t o Z ) e q u a l s ( d Z + d Z ' ) d x= n v . ( P + N ' d x - P ' ) . T h i s e x p r e s s i o n m u s t b e e q u a t e d t o z e r o . W e s e t

P ' - P = d P a n d r e p l a c e N b y N . W e t h e r e fo r e o b t a i n N d x - d P o r(5) N - --dn = O ,d xa s t h e f i n a l e q u a t i o n f o r t h e c u r v e .

I n l a t e r m a t h e m a t i c s ( 5 ) w o u l d b e w r i t t e n a s O f / O y - d ( O f / @ ' ) / d x = 0 a n db e co m e w i d e l y k n o w n a s t h e EULER o r EULER -LAGRANGEe q u a t i o n o f t h e v a r i a -t i o n a l p r o b l e m .I I. I n C h a p t e r 3 EU L ER m o v e d t o a n e w s e t o f e x a m p l e s . H e r e i t is t y p i c a l l yr e q u i r e d t o r e n d e r e x t r e m e a n i n te g r a l o f t h e f o r m y Z d x , i n wh i ch Z i s n o wa fu n c t i o n o f x , y , p = d y / d x a n d a ne w v a r i a b l e / / = y [ Z ] d x , wh ere [Z ] i t s e lf is


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106 C .G . FRASERa fu nc t ion o f x , y , p and ~ [Z ] d x i s to be eva lua ted f rom the in i t ia l absc issa A toA M = x . (Th i s p ro b lem was m o t iv a ted in p a r t b y ex am p les in wh ich Z d ep en d -ed o n x , y , p a n d th e l en g th o f p a th s = ~ o , f i + f d x . ) In c r ease y ' b y nv. A sb efo re th e ch an g es dp , dy ' , dp ' eq u a l n v /d x , n v , - n v / d x respec t ive ly . Le t usca lcu la te the cor responding changes in 17 . We have

1 7 = ~ [ z ] x/ 7 ' = ~ [Z ] d x + [ Z ] d x( 6 ) 1 7 = ~ [ z ] dx [ z ] dx [ z ] dx

1 7 = ~ [ Z ] d x + [ Z ] d x + [ Z ' ] d x + [ Z ] dx , e t c .S u p p o s e d [ Z ] = [ M ] d x + [ N ] d y + [ P ] d p . T h e c h a n g e s i n [ Z ] , [ Z ' ] , [ Z ] , a r e

d . [Z ] d x = nv . d x ( [ f~]x )( c P 3 )(7) d - [ Z ' ] d x = n v . d x I N ' ] - d x )d . [ Z ] d x = O , etc.

Hen ce th e ch an g es in 17, 1 7', 1 7 , 1 7 % . . . eq u a ld . 1 7 = 0

8 ) ( d u 3d . 1 7 = n v . d x I N ' ] d x Jd . 17 = d . 17 ' = d . 17 ~ = e tc .,

wh ere we h av e u sed th e f ac t t h a t d [ P ] = [ P ' ] - [ P ].We n o w ca lcu la t e th e ch an g e in ~ Z d x + Z d x + Z ' d x + etc. tha t resu l t swh en y ' i s i n c r eased b y nv. S u p p o s e d Z = M d x + N d y + P d p + L d l L T h epar t o f the cha nge tha t a r ises f rom the va r ia t ion o f y ' , p and p ' is , as before ,

W h e n y ' is increased by nv al l of the quanti t ies 1 7, 1 7 ', 1 7 , . . . are var ied . Theto ta l ch an g e in ~ Z d x + Z d x + Z ' d x + e tc . due to these var ia t ions i s(10) L d x . d H + U d x . d I I ' + U ' d x . d1 7 + etc .Subst i tu t ing the va lues o f d 1 7, d I I ' , d 1 7 , . . , given by (8) in to (10 ) yields( 1 1 ) n v . d x ( L ' [ P ] ) + n v . d x ( [ N ' ] - ~ x ] ) ( L d x + L ' d x + U ~ d x + e t c . ) .


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Th e Origins of Euler 's Va riational Calculus 107R e p l a c e [ L ' ] a n d [ N ' ] b y [ L ] a n d [ N ] a n d s e t L ' d x + L d x + L i~ d x + e tc .e q u a l t o H - ~ L d x , w h e r e H is th e i n t e g r a l o f Z f r o m A t o Z . W i t h t h e s es u b s t i t u t i o n s ( 1 1 ) b e c o m e s

d [P3(12) n v . d x ( H - f L d x ) ( [ N ] - d x ] + n v . d x ( L [ P ] ) ,w h i c h w e m a y r e w r i t e ( u s i n g d ( S L d x ) = L d x ) as

- 2 ;

B y a d d i n g ( 1 3 ) a n d ( 9 ) a n d e q u a t i n g t h e r e s u l t i n g e x p r e s s i o n s t o z e r o w e o b t a i nt h e f i n a l e q u a t i o n f o r t h e p r o b l e md P(14) 0 = N - dxx + [ N ] ( H - ~ L d x ) - d [ P ] ( H d x ~ L d x )

E UL ER c o n t i n u e d i n C h a p t e r 3 t o c o n s i d e r e x a m p l e s t h a t a r e s o m e w h a t m o r ei n v o l v e d , a l t h o u g h t h e b a s i c t e c h n i q u e i s t h e s a m e . H e c o n c l u d e d w i t h a d e -t a i le d a c c o u n t o f tw o e x a m p l e s i n v o l v i n g t h e m o t i o n o f a p a r t i c l e i n a r e s is t in gm e d i u m : t h e f ir s t i s s i m p l y t h e p r o b l e m o f t h e b r a c h i s t o c h r o n e a d a p t e d t o t h ec a s e w h e r e t h e m o t i o n t a k e s p l a c e i n a r e s i s t i n g m e d i u m ; t h e s e c o n d i n v o l v e sd e t e r m i n i n g t h e t r a j e c t o r y o f a p a r t ic l e d e s c e n d in g t h r o u g h t h e m e d i u m b e t w e e nt w o p o i n t s i n s u c h a w a y t h a t i ts t e r m i n a l v e l o c i t y is a m a x i m u m .l i e I n C h a p t e r s 5 a n d 6 E U LE R t u r n e d t o t h e c l a ss ic is o p e r i m e t r i c p r o b l e m t h a tw a s s o p r o m i n e n t i n t h e w o r k o f e a r li e r re s e a rc h e r s. T h e e x t r e m a l i z a t i o n o f~ o Z d x ( A Z = a ) i s c a r r i e d o u t s u b j e c t t o t h e c o n d i t i o n t h a t S o [ Z J d x = c o n s t a n tzn u s t a ls o b e s a t is f i ed in t h e v a r i a t i o n a l p ro ces s . EULER re fe r r ed t o t h i s a sa p r o b l e m o f r e l a ti v e m a x i m a o r m i n i m a , in c o n t r a s t t o t h e a b s o l u t ep r o b l e m s t h a t h a d b e e n i n v e s t i g a t e d i n C h a p t e r s 2 a n d 3 . H e n o t e d t h a ta d i s t in c t v a r i a t i o n a l p r o b l e m w o u l d a r i s e fo r e a c h c h o i c e o f c o n s t a n t i n th es i d e c o n d i t i o n .

U p t o t h i s p l a c e i n t h e t r e a t i se E U L ER h a d e m p l o y e d a v a r i a t i o n a l p r o c e d u r et h a t i n v o l v e d c h a n g i n g a s in g le o r d i n a t e o f t h e e x t r e m a l i z i n g c u r v e , s o t h a ta c o m p a r i s o n a r c w a s o b t a i n e d t h a t d i f f e r ed f r o m t h e a c t u a l o n e a t a s in g lep o i n t . T o h a n d l e t h e i s o p e r i m e t r i c p r o b l e m h e i n t r o d u c e d a n e w v a r i a t i o n a lp r o ce s s. W e n o w v a r y tw o s u c ce s si ve o rd i n a t e s y ' a n d y b y t h e a m o u n t s nva n d oco r e s p e c ti v e ly . ( T h e r e c o g n i t i o n t h a t i t is n e c e s s a r y t o v a r y t w o o r d i n a t e sw h e n t h e r e i s a s i d e - c o n d i t i o n p r e s e n t i n t h e f o r m o f a d e f i n i t e i n t e g r a l o r i g -i n a t e d wi t h JAKOB B ER NOULLI.) H en ce we n o w h av e : ch a n g e i n p i s nv; c h a n g e i np' is (oo0 - nv) /dx; ch a n g e i n p i s - o o / d x ; c h a n g e s i n p ' , p i ~ , . . , a r e z e r o . T h ec h a n g e i n ~ o Z d x b e c o m e s

N - 9 n v . d x + N ' - d x ,


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108 C .G . FRASERw h e r e N ' a n d P ' d e n o t e t h e v a l u e s o f N a n d P a t x , y , p . S i m i l a rl y t h e c h a n g ei n t h e i n t e g r a l t o [ Z ] d x is( 1 6 ) ( [ N ] _ ~ x ] ) . n v . d x + ( [ N , ] d i P ]d x ) . oo9 . d x .T h e ex p re s s i o n i n (15 9 is b y h y p o t h es i s eq u a l t o ze ro . T h e ex p re s s i o n i n (1 6 ) i se q u a l t o z e r o b e c a u s e t o [ Z ] d x i s u n v a r i ed i n t h e v a r i a t i o n a l p ro ces s . L e t u sd e n o t e N d x - d P b y d A a n d [ N ] d x - d [ P - I b y d B . E q u a t i n g ( 1 5 ) a n d ( 1 6 ) t oze ro , w e o b t a i n(17) d A . n v . d x + d A . oc o. d x = 0

d B . n v . d x + d B . o o . d x = 0U s i n g d A = d A + d d A a n d d B = d B + d d B ( 1 7 ) m a y b e w r i t t e n

(18) d A . n v + ( d A + d d A ) . oco = 0d B , n v + ( d B + d d B ) . 0 o9 = 0 .E l i m i n a t i n g n v an d o09 f ro m (1 8 ) w e o b t a i n

(19) d d A d d Bd A d Bw h i c h m a y b e i m m e d i a t e l y i n t e g r a t e d(20) d A = C d B ,w h e r e C i s a n a r b i t r a r y c o n s t a n t .

N o t i n g t h a t d A = N d x - d P a n d d B = [ N - l d x - d [ P ] , w e s ee t h a t t h ee q u a t i o n f o r t h e p r o b l e m i s s i m p l y N - d P / d x + C ( [ N ] - d [ P ] / d x ) = 0 . H en ce(20 ) fo rm u l a t e s w h a t l a t e r b ecam e k n o w n a s E uL E R 'S ru le : t h e ex t r em a l i za t i o no f ~ o Z d x s u b j ec t t o J o [Z ] d x - - c o n s t a n t l e a d s t o t h e s a m e e q u a t i o n a s t h e f r e ep r o b l e m o f e x t r e m a l i z in g t h e i n t e g r a l t o ( Z + C [ Z ] ) d x , w h ere C i s an u n d e t e r -m i n ed co e f f ic i en t o r co n s t a n t m u l t i p l ie r ,

I n P r o p o s i t i o n 5 o f Ch ap t e r 5 EU L ER ex t en d ed t h i s r u l e t o t h e ca s e t r ea t edi n C h a p t e r 3 w h e r e a v a r i a b le o f th e f o r m / 7 = ~ o [ Z ] d x a p p e a r s i n t h ei n t e g r a n d f u n c t i o n Z = Z ( x , y , p , / 7 ) . W e m u s t e x t r e m a l i z e t o Z d x s u b j ec t t o t h ec o n d i t i o n t h a t t o [ Z ] d x i s co n s t an t i n t h e v a r i a t i o n a l p ro ces s . T h e r e l ev an te q u a t i o n f r o m C h a p t e r 3 is (1 4 ) . T h e e q u a t i o n t h a t r e s u lt s in t h e p r e s e n ts i t u a t i o n i s(21) O = N + ( ~ + H - ~ L d x ) [ N ] d ( P + ( ~ + H - ~ L d x ) E P ] )d xwh ere c~ i s an un de r te rm ine d coef f i c i en t. EULER'S us t i f i c a t ion o f the p ro pos i t ion ,a n d t h e c e n t r a l p l a ce t h a t t h i s ty p e o f p r o b l e m o c c u p i e d i n e a rl ie r v a r i a t i o n a lresearch , wi l l be d i scussed in de ta i l i n sec t ion 5 o f th i s pa per .


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Th e Origins of Euler s Va riational Calculus 1092 .2 T w o a n a l y t i c a l f e a t u r e s o f E UL ER S t h e o r y d i s t in g u i s h e d i t f r o m t h e w o r k o fe a r l i e r c o n t i n e n t a l r e s e a r c h e r s : t h e v a r i a t i o n a l p r o b l e m i s f o r m u l a t e d i n t e r m s o fa g e n e r a l i n t e g r a n d o f t h e f o r m Z x , y , p ) ; t h e d e r i v a t i o n i n v o l v es t h e d i f f e r en t i a lo f Z a n d i s e x p l ic i tl y c a r r i e d o u t i n t e r m s o f th e p a r t i a l d e r i v a t i v e s o f t h isf u n c t i o n . O n e c h a r a c t e r i s t i c t h a t s i t u a t e d h i s a p p r o a c h g e n e r a l l y w i t h i n t h ee s t a b l is h e d t r a d i t i o n w a s h i s s y s t e m a t i c d u a l u s e o f t h e d - s y m b o l . T h i s l e t te r isu s e d t o d e n o t e t h e u s u a l c a l c u lu s - d i ff e r e n ti a l o f a v a r i a b l e a s w e ll as t o d e n o t et h e c h a n g e i n a v a r i a b l e t h a t r e s u l t s f r o m t h e v a r i a t i o n a l p r o c e s s . N o w h e r e d i dh e d is t i n g u is h t h e s e tw o q u i t e d if f e r e n t m e a n i n g s o r c o m m e n t o n t h e d u a lch a rac t e r o f t h e s y m b o l . In t h i s r e s p ec t h i s p rac t i ce was s o l i d l y i n l in e w i t hea r l i e r r e s ea rch i n t h e s u b j ec t .

I n s u r v e y i n g t h e Methodus inveniendi a s a w h o l e , C h a p t e r 3 e m e r g e s a s t h ec e n t r e o f t h e t r e a ti s e; t h e n a r r a t i v e r e a c h e s it s h ig h e s t a n d m o s t c o n c e n t r a t e dl e ve l i n t h e t e c h n i c a l e x p o s i t i o n o f p r o b l e m s w i t h a u x i l i a r y d i f fe r e n t ia l e q u a -t io n s . T h e r e m a i n i n g c h a p t e r s a r e m o r e h e t e r o g e n e o u s i n th e i r c h o i c e o f t o p i c sa n d t h e e x p o s i t i o n a s a w h o l e - p a r t i c u l a r l y i n C h a p t e r s 5 a n d 6 - i s l es si n c i s i v e . O n e s h o u l d n o t e e s p e c i a l l y t h e s e c o n d a r y p o s i t i o n t h e h i s t o r i c a l l yp r o m i n e n t i s o p e r im e t r i c p r o b l e m o c c u p ie s i n E UL ER S o r g a n i z a t i o n o f t h esub jec t .

T h e r e a r e s i g n if i c an t p a r t s o f E UL ER S t r e a t is e t h a t h a v e n o t b e e n d i s c u s se dh e r e . T h u s t h e m a n y e x a m p l e s w h i c h h e p r e s e n t s p r o v i d e f u l l a n d d e t a i l e di l l u s t ra t i o n o f t h e t h e o r y . I n a d d i t i o n , t h e m a t e r i a l i n C h a p t e r s 1 a n d 4 is o fc o n s i d e r a b le i n t e re s t f o r a n h i s t o ri c a l u n d e r s t a n d i n g o f t h e c o n c e p t u a l f o u n d a -t i o n s o f e i g h t e e n t h - c e n t u r y a n a ly s is . 1 N e v e r t h e l e s s , t h e p o i n t s t h a t a r e e s s e n ti a lf o r u n d e r s t a n d i n g t h e b a s i c s t r u c t u r e o f h is v a r i a t i o n a l t h e o r y h a v e b e e n s e tf o r th , a n d w e c a n p r o c e e d t o a n e x a m i n a t i o n o f t h e e a r li e r h i s to r y .

3. The Paper s o f Ja ko b and Johann Bernoul l i 1701, 1719 )3.1 In 1697 JAKOB BERNOULLt pu bl ish ed in the Acta Erud i torum h i s s o l u t i o n fo rt h e c u r v e o f q u i c k e s t d e s c e n t, th e s o - c a l le d b r a c h i s t o c h r o n e . H e r e h e i n t r o d u c e dt h e t e c h n i q u e o f v a r y i n g t h e c u r v e a t a s in g le p o i n t i n o r d e r t o o b t a i n a c o m -p a r i s o n c u r v e ; t h e c o n d i t i o n t h a t t h e d i ff e r e n c e o f t h e t i m e a l o n g t h e a c t u a l a n dc o m p a r i s o n c u r v e s i s z e r o l e d f o r t h e c a s e w h e r e t h e s p e e d i s p r o p o r t i o n a l t ot h e s q u a r e r o o t o f t h e v e r t i c a l d i s t a n c e t o t h e d i f f e re n t ia l e q u a t i o n o f t h ec y c l o id . A t t h e e n d o f t h e m e m o i r h e r a i s e d t h e i s o p e r i m e t r i c p r o b l e m a sa s u b j e c t o f f u r t h e r i n v e s ti g a t io n , o n e t h a t w a s o f f e r e d a s a m a t h e m a t i c a lc h a l le n g e to r e a d e r s o f t h e j o u r n a l . W i t h t h is a n n o u n c e m e n t t h e p r o b l e mb e c a m e t h e p r i m a r y f o c us o f r e s e a r c h i n v a r i a t i o n a l m a t h e m a t i c s . JA KO B a p p a r -e n t l y b e l i e v e d t h a t e x a m p l e s l i k e t h e b r a c h i s t o c h r o n e i n w h i c h t h e r e i s n o s i d e

1 A discussion of th is material w i l l be presented in the author s forthcom ing s tudy ofthe origins and dev elopm ent of LAGRANGE S analysis.


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110 C .G . FRASERc o n d i t i o n p r e s e n t w e r e r e l a t i v e l y s t r a i g h t f o r w a r d , a n d t h a t t h e i n t e r e s t i n g a n ds u b s t a n t i a l m a t h e m a t i c a l q u e s t i o n w a s p o s e d b y t h e i s o p e r i m e t r i c p r o b l e m . T h ep e r s o n a l t en s i o n b e t w e en t h e b ro t h e r s BERNO UL LI, an d t h e w a y i n w h i ch t h ep r o b l e m w a s r e g a r d e d a s a te s t o f m a t h e m a t i c a l c a p a b i li ty , a l so u n d o u b t e d l yc o n t r i b u t e d t o t h e e m p h a s i s p l a c e d o n i t . C e r t a i n l y t h e f i r s t g e n e r a l t h e o r y w a sd ev e l o p e d a ro u n d i t s s tu d y , an d a l l t h e s i g n i f i can t co n t r i b u t i o n s u n t i l a t le a s tthe 1730s concerned i t s so lu t ion .

JA KO B d ev e l o p ed h i s an a l y s i s i n an i m p o r t a n t m em o i r p u b l i s h ed i n 1 7 0 1 i nt h e Acta Eruditorum T h i s m e m o i r w a s p r e c e d e d i n t h e p r e v i o u s y e a r b ya r e l a t ed p ap e r i n t h e s am e j o u rn a l . T h e s e r e s ea rch es o r i g i n a t e d i n d r a f t st u d i e sc a r r ie d o u t i n 1 6 9 7 , a t t h e s a m e t im e h e w a s c o m p o s i n g t h e p a p e r o n t h eb r a c h i s t o c h r o n e , a n d a r e c o n t a i n e d i n h i s sc ie n ti fi c n o t e b o o k t h e Meditationes

I t i s wel l kn ow n tha t JOHANN BERNOULLI cam e to acce p t h i s b ro th er s ideasfo l l o w i n g t h e l a t t e r s d ea t h i n 1 7 0 5 . I n 1 71 9 h e p u b l i s h ed an a r t ic l e i n t h em em o i r s o f th e Pa r i s A c ad e m y o f Sc i ences in w h i ch h e b as i ca l ly r ew ro t e JA~:OB Sp a p e r , d e r iv i n g i n s o m e w h a t c le a r e r fo r m a n d w i t h a s t r o n g e r g e o m e t r i c a lem p h as i s t h e g en e ra l r e l a t i o n s o f t h e t h e o ry a n d i l l u s t r a t i n g i t w i t h s ev e ra la d d i t i o n a l e x a m p l e s a n d p r o b l e m s . T h e r e i s a t h e o r e t i c a l a n d c o n c e p t u a l u n i t yt o JA KO B S m em o i r o f 1 7 0 1 an d JO HA NN S o f 1 71 9 w h i ch m ak e i t a p p r o p r i a t e t oc o n s i d e r t h e m t o g e t h e r . W e w i l l t h e r e f o r e c o n c e n t r a t e o n a n e x p o s i t i o n o fJA KO B S p ap e r , m en t i o n i n g i n s t an ces w h e re h i s b ro t h e r c l a r if i ed o r s i g n i f ican t l ye l a b o r a t e d u p o n h i s w o r k .3 .2 JA KO B S v a r i a t i o n a l r e s ea rch es w ere p a r t o f h is l a rg e r i n v es t i g a t i o n b y m ean so f t h e ca l cu l u s d u r i n g t h e 1 69 0s o f g eo m e t r i ca l an d m ech an i ca l p ro b l em s . T h ech a rac t e r i s t i c ap p ro a ch d u r i n g t h e p e r i o d i n v o l v ed s e lec t in g o n e o f t h e v a r i ab le so f a p r o b l e m a n d a s s u m i n g t h a t t h e d i f fe r e n ti a l o f t h is v a r i a b le i s c o n s t a n t . O nt h i s b as i s t h e d i f f e r en t ia l s o f th e o t h e r v a r i ab l e s w e re ca l cu l a t ed an d t h e r eq u i -s i t e e q u a t i o n o r f o r m u l a w a s d e r i v e d . T h u s i n t h e f o r m u l a f o r t h e r a d i u s o fc u r v a t u r e o n e o b t a i n e d d i f f e re n t e x p re s si o ns , d e p e n d i n g o n w h e t h e r o n e t o o kt h e ab s c i s sa , t h e o rd i n a t e o r t h e p a t h - l en g t h a s t h e i n d ep en d en t v a r i ab l e . 2

JA KO B S ap p ro a ch t o t h e p ro b l em o f i s o p e r i me t e r s fo l l o w ed a s i mi l a r p a t t e rn .H e a n a l y z e d t h e c u r v e r e la t iv e to a n o r t h o g o n a l c o o r d i n a t e s y s t e m i n w h i c h a na rb i t r a ry p o i n t w as s p ec i fi ed i n t e rms o f t h e ab s c i s s a y , t h e o r d i n a t e x a n d t h el e n g t h o f p a t h z ( F i g u r e 2). ( N o t e t h a t t h e c o n v e n t i o n a d o p t e d h e r e i s th er ev e r se o f EU LE R S , w h e re x i s th e ab s c i s sa an d y t h e o rd i n a t e . ) F ro m t h e v e ryb eg i n n i n g h e d ev e l o p ed h i s i n v es t i g a t i o n a ro u n d t w o ca s e s : w h e re t h e d i f f e r en -t ia l dy i s a s s u med t o b e co n s t an t ; an d w h e re t h e d i f f e r en t i a l dz i s a s s u m e dco n s t an t . ( I n h i s p r e l i mi n a ry s t u d y i n t h e Meditationes h e a l s o co n s i d e red t h ecas e w h e re dx i s co n s t an t . T h i s p o s s i b i l i t y i n t ro d u ced n o t h i n g e s s en t i a l l y n ew ,a n d w a s n o t p u r s u e d i n t h e p u b l i s h e d m e m o i r o f 1 7 0 1 . JOM ANN o n g r o u n d s o fco mp l e t en es s i n c l u d ed t h i s ca s e i n h i s p ap e r . )

2 BOS ]-1974] provides an acc oun t of the ea rly L eibnizian calculus with detailedconsideration of the for m ula for the radius of curvature.


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The Origins of Euler s Va riational Calculus 111

Figure 2.

V

T

A s w e r e m a r k e d e a r li e r , a d i s ti n c t iv e f e a t u r e o f e a r l y v a r i a t i o n a l m a t h e m a t i c sw a s t h e d u a l u s e o f t h e s y m b o l d. I t w a s e m p l o y e d t o i n d i c a t e t h e c u s t o m a r yca l cu lu s -d i f f e ren t i a l o f a v a r i ab l e a s we l l a s t o i n d i c a t e t h e c h an g e i n a v a r i a b l et h a t r e s u l t s f r o m t h e g i v e n v a r i a t i o n a l p r o c e s s . T h u s f o r e x a m p l e w h e n J A K O Bm a d e t h e a b s c i s s a y t h e i n d e p e n d e n t v a r i a b l e t h i s d e c i s i o n h a d i m p l i c a t i o n s f o rb o t h t h e d i f f e r e n t ia l a n d v a r i a t i o n a l p r o ce s s e s. I t m e a n t t h a t t h e u s u a l d i f f e re n -t ia l d y w a s c o n s t a n t , a n d t h a t t h e c a l c u l a t io n s l e a d i n g t o t h e e q u a t i o n o f t h ep r o b l e m w e r e t o b e c a r r i e d o u t r e l a t i v e t o t h i s a s s u m p t i o n . I t a l s o m e a n t t h a tt h e v a r i a t i o n a l p r o c e s s w a s s u c h t h a t t h e a b s c is s a .y w a s n o t v a r i e d i n o b t a i n i n gt h e c o m p a r i s o n c u r v e . U n d e r s t o o d i n t h i s s e n s e i t m e a n t t h a t y w a s e q u a l t oze ro fo r a l l v a lu es o f y .

BERNOUI~LI S ap p r o a ch t o i s o p e r im e t r i c p ro b l e m s was b as e d o n o b t a in in ga c o m p a r i s o n a r c t o a g i v e n c u r v e b y v a r y i n g t w o s u c c e s s i v e p o i n t s o n i t .S u p p o s e a g a i n t h a t w e a r e c o n s i d e r i n g a v a r i a t i o n a l p r o c e s s i n w h i c h t h eab s c i s s a y i s i n d ep en d en t , s o t h a t t h e o rd in a t e x i s t h e v a r i ab l e wh ich i s a l t e r ed .I t is c le a r t h a t i n o r d e r t o p r e s e r v e t h e c o n d i t i o n o f c o n s t a n t l e n g t h i t isn e c e s s a r y t o d i s t u r b t h e c u r v e a t t w o s u c ce s s iv e o r d i n a t e s . T h e i s o p e r i m e t r i cc o n d i t i o n y i e l d s o n e e q u a t i o n i n v o l v i n g t h e t w o v a r i a t i o n a l i n c r e m e n t s . T h eco e f f i c i en t o f eac h i n c re m en t w i ll b e a d i f f e ren t i a l ex p re s s io n i n t h e v a r i ab l e s y , xa n d z . T h e p a r t i c u l a r v a r i a t i o n a l p r o b l e m u n d e r c o n s i d e r a t i o n w i l l f u r n i s ha n o t h e r s u c h e q u a t i o n . U s i n g t h e s e t w o e q u a t i o n s o n e e l i m i n a t e s t h e t w oi n c r e m e n t s t o o b t a i n t h e f in a l d if f e r e n ti a l r e l a t i o n o f t h e p r o b l e m . T h e l a t t e r ,w h i c h m a y b e a n a l y z e d t o y i e l d i n f o r m a t i o n a b o u t t h e n a t u r e o f t h e c u r ve , isr e g a r d e d a s t h e s o l u ti o n o f t h e p r o b l e m .

BERNOUI~LI S i n v es t i g a t i o n was d iv id ed i n to two p a r t s , o n e fo r t h e ca s e wh e rey i s t a k e n a s t h e i n d e p e n d e n t v a r i a b l e a n d o n e f o r t h e c a s e w h e r e z i s t h ei n d e p e n d e n t v a r i a b l e . T w o d i f f e r e n t v a r i a t i o n a l t h e o r i e s i s s u e f o r t h f r o m t h e s ecases .


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112 C .G . FRASER(i) dy i s c o n s t a n t

I n F i g u r e 2 t h e h y p o t h e t i c a l e x t r e m a l i z i n g c u r v e i s ABFGCD, w h e r ey = AH, x = HB a n d z = AB. B, F, G a n d C a r e su c c e s s i v e p o i n t s o n t h e c u r v e .I n o r d e r t o p r e s e r v e t h e c o n d i t i o n o f c o n s t a n t l e n g t h w e v a r y t h e c u r v e a t t w op o i n t s , w h i c h JA K OB t a k e s a s F a n d G . T h e v a l u e s o f t h e o r d i n a t e s K F a n d LGa r e f a n d g re s p e c ti v e ly . T h e s e a r e a l t e r e d b y th e a m o u n t s df a n d dg, g i v i n gr i s e t o t h e d e s i r e d c o m p a r i s o n c u r v e .

A t t h e b e g i n n i n g o f t h e p a p e r JA KO B e s t a b l is h e d i n h is T h e o r e m 4 a r e l a t i o nw h i c h e x p r e s s e s th e f a c t t h a t t h e v a r i a t i o n o f t h e p a t h - l e n g t h~ox/ 1 + dx/dy)2dy is z e ro . In P r o b l e m 1 h e a p p l i e d t h e t h e o r e m t o m a x i m i z eo r m i n i m i z e t h e a r e a u n d e r t h e c u r v e A P R S Q V ( F i g u r e 2 ). T h e o r d i n a t eF = H P is s o m e g i v e n fu n c t i o n o f t h e a b s c i s s a AH = x. T h e d e r i v a t i v e o f t h isf u n c t i o n w i t h r e s p e c t t o x i s d e n o t e d b y h . T h e e q u a t i o n JA K OB o b t a i n e d a ss o l u t i o n i s(22) hdzddd x - 3hddxddz - dhdzdd x = 0 ,w h e r e w e h a v e s u b s t i t u t e d = f o r t h e s y m b o l ~ w h i c h JA K OB u s e d t od e n o t e e q u a l i t y , a

D e t a i l e d d e s c r i p t i o n s o f JA K O B'S s o l u t i o n t o P r o b l e m 1 a r e a v a i l a b l e i nl i t e r a tu r e . 4 I t is s u ff ic i e n t h e r e t o n o t e t h a t t h e b a s i c i d e a i s s i m p l y t h e o n eg e n e r a l i z e d b y EU L ER i n h i s d e r i v a t i o n o f E uL ER 'S r u l e i n t h e o p e n i n g p a r t o fC h a p t e r 5 o f t h e Methodus inveniendi. T h e c o n d i t i o n ~o x / 1 + dx/dy)~dY = c o n -s t a n t l e a d s t o t h e e q u a t i o n

dx + dg = 0 .

T h e v a r i a t i o n a l i n t e g r a l i s ~ o Z d y w h e r e Z = Z x) a n d dZ = Ndx. S e t t i n g t h ev a r i a t i o n o f t h is i n t e g ra l e q u a l t o z e r o w e o b t a i n t h e e q u a t i o n(24) i df + n +d n) d g= 0 .E l i m i n a t i n g df a n d dg f r o m ( 2 3 ) a n d ( 2 4 ) g i v e s

dN \dz251 - 2 d x\ d z

B e c a u s e dy i s c o n s t a n t , w e h a v e b y d i f f e r e n t i a t i n g dz 2 = dy 2 + dx 2 t h e r e l a t i o ndxd2x = dzd2z. U s e o f t h e l a tt e r m a k e s ( 25 ) a f te r s o m e r e d u c t i o n s b e c o m e

3 BERNOULLI had used the symb ol = in the Meditationes, but in his publ ishedwritings p referred to us e D ES CA RTES S 30 .4 See [GoLDSTINE 1980, 50 -55 ] and [ F E I G E N B A U M 1985 , 53-56] .


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Th e Origins of Euler s Va riational Calculus 113(26) N d 3 x d z - 3 N d 2 x d 2 z - d N d 2 x d z = 0 .N o t i n g t h a t c ? Z / # x = N = h , we see tha t th i s i s s im ply eq ua t io n (22), JAKOB Ss t a t e d s o l u t i o n t o P r o b l e m 1. 5I n P r o b l e m 2 JA KO B p r o c e e d e d t o m a x i m i z e o r m i n im i z e t h e a r e a u n d e ra c u r v e w h o s e o r d i n a t e is a f u n c t i o n o f t h e p a t h - l e n g t h z = A B . A g a i n t h ec o n d i t i o n t h a t S o x / 1 + ( d x / d y ) 2 d y = c o n s t a n t p r o v i d e d b y m e a n s o f T h e o r e m4 o n e o f t h e r e l a t i o n s n e e d e d i n t h e s o l u t io n . T h e o t h e r w a s o b t a i n e d b y s e t t in gt h e v a r i a t i o n o f g i v e n in t e g r a l e q u a l t o z e r o . E l i m i n a t i n g t h e v a r i a t i o n a l i n -c r e m e n t s f r o m t h e s e e q u a t i o n s JA KO B e v e n t u a l l y a r r i v e d a t t h e f i n a l e q u a t i o n(27) h d x (d z ) 2 d d d x = 2 h d z Z d d x 2 + h d x 2 d d x 2 + d h d x d z 2 d d x ,w h e r e h is n o w t h e d e r i v a t i v e o f t h e i n t e g r a n d f u n c t i o n w i t h r e s p e c t t o z .

T h e u l t i m a t e d e v e l o p m e n t o f th i s p a r t o f JA KO B S i n v e s t i g a t i o n w o u l d b each i ev ed b y EULER i n h is p a p e r o f 1 74 1. I t s h o u l d b e n o t ed t h a t i t is n o tp o s s i b le t o i n t e r p r e t JA K O ffS d e r i v a t i o n o f (2 7) i n t e r m s o f t h e v a r i a t i o n a l t h e o r yp r e s e n t e d in C h a p t e r 5 o f t h e M e t h o d u s i n v e n i e n d i . W e d i s cu s s t h i s p o i n t i ng rea t e r d e t a i l i n s ec t i o n 5.(ii) d z i s c o n s t a n t

I n J A KO ffS s e c o n d m e t h o d t h e p a t h - l e n g t h is t a k e n a s t h e i n d e p e n d e n tv a r ia b l e. I n F i g u r e 2 w e n o w h a v e B F = F G = G Z ; i n a d d i t i o n t h e p o i n t s F a n dG a r e v a r i e d i n s u c h a w a y t h a t t h e d i s t a n c e s B F , F G a n d G Z r e m a i n u n a l t e r e d .T h e s i t u a t i o n i s d e p i c t e d i n s o m e w h a t g r e a t e r d e t a i l in t h e d i a g r a m ( F i g u r e 3 )p ro v i d ed i n JOHANN S p ap e r , i n wh i ch B a b c e i s t h e ac t u a l cu rv e , B a g i e t h ec o m p a r i s o n c u r v e , a n d w h e r e a b = b c = c e a n d a b = a g = 9 i = i e . I n T h e o r e m5 JA K O ~ o b t a i n e d a n a n a l y t i c a l r e l a t i o n t h a t e x p r e s s e s t h e c o n d i t i o n t h a t t h el e n g t h o f t h e a r c B C ( a e i n JO HA NN S a c c o u n t ) is u n c h a n g e d i n t h e v a r i a t i o n a lp r o c e s s . H e t h e n a p p l i e d t h i s t h e o r e m i n P r o b l e m 3 , t h e h a n g i n g - c h a i n p r o b l e m .G i v e n a f le x i bl e h e a v y l in e o f d e f i n it e l e n g t h s u s p e n d e d b e t w e e n t w o p o i n t s i t isn e c e s s a r y t o d e t e r m i n e t h e s h a p e i t w i l l a s s u m e i n s t a t i c e q u i l i b r i u m i n o r d e rt h a t i t s c en t r e o f g ra v i t y b e a t t h e l o w es t p o i n t p o s s i b le . J OHANN i n t u rne x t e n d e d t h e m e t h o d t o g i ve a n a l t e r n a ti v e s o l u t i o n t o h is b r o t h e r s P r o b l e m 2.H e a l s o a p p l i e d i t t o s o l v e t h e b r a c h i s t o c h r o n e p r o b l e m s u p p l e m e n t e d w i t h a ni s o p e r i m e t r i c c o n d i t i o n . T h e r e w a s t h e r e f o r e a d e f in i t e c l u s t e r o f e x a m p l e s i nw h i c h t h e m e t h o d w a s e m p l o y e d .

Th i s p a r t o f t h e B ERNOULLIS h eo ry w as t ak en u p b y n e i t h e r TAYLOR n o r t h eEU LER ; t o m y k n o w l e d g e i t d i d n o t b e c o m e a c o m m o n p a r t o f l a t e r v a r i a t io n a lr e s e a r c h . I t is t h e r e f o r e r a t h e r a n o m a l o u s w i t h i n t h e h i s t o r y o f t h e s u b je c t . Ino r d e r t o u n d e r s t a n d t h e b a s i c i d e a i t w i l l b e u s e f u l t o c o n s i d e r t h e m e t h o d f r o m

5 In ob tain ing (26) fro m (25 ) we use the following relations, derivable in a straight-forward way by calculation: d ( d x / d z ) = ( d 2 x d y 2 / d z 3 ) , and d a ( d x / d z ) = ( d 3 x d y 2 d z 2-- 3(d2x)2 dy 2 dx)/dz5


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114 C .G . FRASERB N p R S

...iD e I qE ~ ..i c

F e ~

Figure 3.

a m o r e f o r m a l m a t h e m a t i c a l v i e w p o in t . C o n s i d e r a n y p r o b l e m i n w h i ch t h ei s o p e r im e t r i c c o n d i t i o n ~ o x / 1 + ( d x / d y ) 2 d y = co n s t a n t i s p re s en t . B ecau s e t h et o t a l p a t h - l e n g t h i s u n c h a n g e d i n t h e v a r i a t i o n a l p r o c e s s , w e h a v e t h e f r e e d o mt o i n t e r p r e t t h e p r o b l e m i n s u c h a w a y t h a t z = ~ , , / 1 + ( d x / d y ) 2 d y b e c o m e s t h ei n d e p e n d e n t v a ri a b l e . C o n s i d e r t h e e x a m p l e o f t h e h a n g i n g c h a i n . L e t p = p ( y )b e t h e f u n c t i o n t h a t g i v e s t h e w e i g h t o f t h e c h a i n f r o m A t o B F i g u r e 2). L e tp = d p /d y . I t i s n e c e s s a r y t o m a x i m i z e t h e i n t e g r a l ~ o X p ( y ) x / 1 + (d x /d y )~ d ys u b j e c t t o t h e s id e c o n d i t i o n ~ o x / 1 + ( d x / d y ) a d y = co n s t an t . W e s e t z = ~x / 1 + ( d x / d y ) Z d y a n d m a k e z t h e i n d e p e n d e n t v a r ia b le . L e t q = q ( z ) b e t h ew e i g h t o f t h e c h a i n f r o m A t o B . L e t h = d q / d z . T h e v a r i a t i o n a l p r o b l e m n o wb e c o m e s o n e o f m a x i m i z i n g ~0 x h d z ( l = t o t a l l en g t h o f s t ri n g ), wh e re y is n o wa d e p e n d e n t v a r i a b l e a n d x is g iv e n in t e r m s o f y a n d z b y m e a n s o f t h ed i f f e r e n t i a l e q u a t i o n d y 2 + d x 2 = d z 2 .

T h e r e a r e t w o w a y s o f p r o c e e d i n g t o t h e s o l u t i o n o f t hi s p r o b le m . W e c o u l dd o a s JA KO B a n d JO HA NN d i d a n d v a r y t w o p o i n t s o f th e e x t r e m a l i z i n g c u r v e i ns u c h a w a y t h a t e a c h e l e m e n t o f t h e p a t h - l e n g t h i s u n a l t e r e d . W e t h e n o b t a i no n e e q u a t i o n o f c o n d i t i o n , w h i c h JA K OB p r e s e n t e d i n h is T h e o r e m 5. T h ec o n d i t i o n t h a t t h e v a r i a t i o n o f ~ o x h d z i s z e r o l e a d s t o a n o t h e r e q u a t i o n . B ye l i m i n a t i n g t h e v a r i a t i o n s o f t h e o r d i n a t e s f r o m t h e s e t w o e q u a t i o n s w e a r r iv ea t t h e f i n a l d i f f e r en t i a l r e l a t i o n o f t h e p ro b l em . JAKOB o b t a i n ed t h e l a t t e r i n t h ef o r m2 8 ) d q d y a d d d x + 3 d q d x d d x 2 = d y 2 d d q d d x .

U s i n g t h e r e l a t i o n d 2 y d y = - d 2 x d x o b t a i n e d b y d i f f e r e n t i a t i n g


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Th e Origins of Euler 's Va riational Calculus 115@2 dx 2= dz 2 w i t h dz co n s t an t ) h e i n t eg ra t ed (2 8 )(29) dd x: dq dy 3 = _ 1 : adz 2 ,w h e r e a is a c o n s t a n t o f in t e g r a t i o n . 6

T h e s o l v i n g p r o c e d u r e j u s t o u t l i n e d is in it s g e n e r a l f o r m s i m i l a r t o t h ee a r l i e r o n e i n w h i c h t h e a b s c i s s a w a s t a k e n a s t h e i n d e p e n d e n t v a r i a b l e . I t i sh o w e v e r n e c e s s a r y t o n o t e t h a t i n t h e p r e s e n t c a s e t h e r e i s a p l a u s i b le a l t e r n a -t i v e a p p r o a c h t o t h e s o l u t i o n . I t w o u l d b e p o s s i b l e t o u s e a v a r i a t i o n a l p r o c e s si n w h i c h o n l y o n e v a l u e o f t h e a b s c i s sa is a l te r e d , a n d i n w h i c h e a c h o f t h es u c ce s s iv e v a l u e s o f t h e o r d i n a t e is c h a n g e d . I n t h is c a s e o n e w o u l d o b t a i na s i t u a t i o n l ik e t h e o n e d e p i c t e d i n F i g u r e 4, i n w h i c h t h e p o i n t s c , e a n d s o o na r e d i s p l a c e d a l o n g t h e i r r e s p e c t iv e o r d i n a t e s . A l t h o u g h E U LE R n e v e r p u r s u e dthe BERNOULLIS's e c o n d m e t h o d , h e d i d d e v e l o p a f o r m a l g e n e r a l v e r si o n o f s u c ha v a r i a t i o n a l p r o c e s s i n C h a p t e r 3 o f t h e Methodus inveniendi. U n d e r s t o o d f r o mt h e v i ew p o i n t o f h is t h eo ry t h e BBRNOUI~I~m' m e t h o d am o u n t s t o a p ro ce d u re f o rr e d u c i n g a p r o b l e m i n w h i c h a n i s o p e r i m e t r i c c o n d i t i o n i s p r e s e n t t o o n e i nw h i c h t h e r e is n o i n t e g r a l s i d e c o n d i t i o n b u t i n w h i c h t h e v a r i a b l e s o f t h ep r o b l e m a r e c o n n e c t e d b y m e a n s o f a d i f f er e n ti a l e q u a t i o n . I n t e r m s o f th eo r g a n i z a t i o n o f t h e Methodus inveniendi i t w o u l d i n v o l v e th e r e d u c t i o n o f t h ei s o p e r i m e t r i c p r o b l e m , t r e a t e d i n C h a p t e r 5 , t o t h e c l a ss o f p r o b l e m s i n v e s t -i g a t e d i n C h a p t e r 3 .I n t h e E u l e r i a n a p p r o a c h j u s t o u t l i n e d t h e v a r i a t i o n a l integral f o r t h eh a n g i n g c h a i n i s o f t h e f o r m ~ xhdz w h e re x = ~ o x / 1 - ( d y / d z ~ dz. T h e a p p r o -p r i a t e d i a g r a m h e r e is F i g u r e 4 , w h e r e t h e d e p e n d e n t v a r i a b l e y is v a r i e d o n l y a tt h e s i n g l e p o i n t b a n d w h e r e t h e a u x i l i a r y v a r i a b l e x i s c h a n g e d a t b a t e a c hs u c c es s iv e p o i n t o f th e c u r v e . T h e r e l e v a n t e q u a t i o n o f s o l u t i o n i s g iv e n b y ( 14 ),i n w h i c h w e r e p l a c e x b y z a n d / 7 b y x :(301 h d z = 0 .I n t e g r a t i n g t h i s e q u a t i o n y i e l d s

i dx3 1 ) hdz- dywh ere c i s a co n s t an t . W e n o w co m b i n e (3 0 ) an d (3 1 ) , s e t h = dq/dz, a n d u s e t h ef a c t t h a t dydZy dxdZx = O, o b t a i n i n g

d2x c(32) dqdy 3 - dz 2 ,

6 JAKOB [1991, 232] h ad obta ined this equ ation in the Meditationes for the casewhere q = z . I t should be noted that in h is notebook he considers the hanging chain inhis first example; the result that wo uld later be prese nted as Pro blem 1 is given in thefifth example.


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116N P

C. G. FRASERR S T U

Figure 4 .

ab = bc = ce = ef= fg= . . .ab= ag =gi = i j =jk = kl= . . .

l ~ C

a n e q u a t i o n t h e s a m e a s B E R N OU L L I'S (2 9). 7I t m u s t b e e m p h a s i z e d t h a t t h e p r e c e d i n g a n a l y s is is n o t a n a c c e p t a b l e

i n t e r p r e t a t i o n o f t h e B E R NO UL LIS ' t h e o r y , w h i c h w a s b a s e d o n a d i f f e re n t v a r i a -t i o n a l p r o c e s s t h a n t h e o n e i n v o l v e d i n th e d e r i v a t i o n o f e q u a t i o n (32 ). F o re x a m p l e , i n t h e E u l e r i a n p r o c e d u r e , (3 2) is o b t a i n e d d i r e c t ly a n d n o t a s a r e s u lto f e l i m i n a t io n o f v a r i a t io n s f r o m t w o s e p a r a t e e q u a t io n s . T h e r e is n o e v i d e n c et h a t t h e a l t e r n a t iv e a n a l y s i s w h i c h w e h a v e p r o p o s e d e v e r o c c u r r e d a s a p o s s i -b i li ty t o t h e B ER NO UL LIS . F u r t h e r m o r e , E U L E R d i d n o t h i m s e l f c a r r y o u t t h e

7 I t shou l d pe r ha ps be n o t ed t h a t one cou l d a l so em pl oy such a n a l t e r na t ivevar ia t io na l process ( su i tably mo di f ied) in the BER NOULLI ' f i rs t met ho d in w hich dyis t aken a s cons t an t. C ons i de r t he so l u t i on t o JA K O B's P r o b l em 1 . W e le t z - - z ( y ) bet he dependen t va r i ab l e , x becom es an ax i l i a r y va r i ab l e g i ven i n t he f o r mx = SYox/ dz/dy) 2 - l dy. W e a p p l y e q u a t i o n ( 1 4 ) f r o m C h a p t e r 3 o f t h e Methodusinveniendi. I n th is e q u a t i o n x b e c o m e s y, y b e c o m e s z a n d / / b e c o m e s x s o t h a t w e ha v e .

w he r e h is t he de r i va ti ve o f t he i n t eg r a nd f unc t i on w i t h r e spec t t o y . W e can r eexpre s st h i s equa t i on i n t he f o r m

The r e su l t i s i den t i ca l w i t h t he so l u t i on t ha t w ou l d r e su l t f r om an app l i ca t i on o fEU LER'S r u l e t o t he exam pl e t r ea ted a s an i sope r im e t r i c p r ob l em .

The r e is how eve r an i m por t a n t d if fe rence be t w een t h is ca se and t he one i nvo l ved i nt he BERN O U LLIS second m e t hod . I n t he l a t t e r t he w ho l e m ode o f f o r m u l a t i ng t heques t i on na t u r a l l y sugges ts an a l t e r na ti ve t r ea t m en t a l ong t he p r i nci p le s o f Ch ap t e r 3 o ft he Methodus inveniendi.


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Th e Origins of Euler 's Va riational Calculus I 17d e r i v a t i o n o f (3 2); o u r a c c o u n t i n d i c a t e s o n l y h o w s u c h a d e r i v a t i o n w o u l dp r o c e e d u s in g t h e t h e o r y o f C h a p t e r 3 o f t h e M e t h o d u s i n v e n i e n d i f o l l o w i n g t h er e d u c t i o n s u g g e s t e d b y t h e B ER NO UL LIS' s e c o n d m e t h o d , s T h e l a t t e r c a n p e r h a p sb e s t b e s e e n as s o m e t h i n g o f a c u r i o s i t y w i t h i n t h e h i s t o r y o f m a t h e m a t i c s .

D es p i t e t h e p e r i p h e r a l ch a ra c t e r o f t h i s p a r t o f t h e B ER NOULL~S' w o rk s o m eo f t h e i s su e s c o n c e r n i n g t h e n a t u r e o f t h e i s o p e r i m e t r i c v a r i a t i o n a l p r o c e s sw h i c h h a v e a r i s e n i n o u r d i s c u s s i o n w i l l c o m e u p a g a i n - - i n a d i f f e r e n t h i s t o r i -c a l a n d m a t h e m a t i c a l s e t t i n g - - w h e n w e c o n s i d e r E V LE R'S p a p e r s o f 1 738 a n d1741.

4 . T ay lor s Re search (1715)4 .1 A l t h o u g h J OHANN fa i t h fu l l y p re s e n t ed J AKOB'S s o l u t i o n s tO t h e t h reei s o p e r im e t r i c p r o b l e m s h e a ls o i m p a r t e d h is o w n p a r t i c u l a r d i r e c t i o n t o t h es u b je c t . H i s a p p r o a c h w a s m o r e g e o m e t r i c , r e f e r r i n g a t e a c h s t a g e o f e a c hd e r i v a t i o n t o t h e v a r i o u s r e l a t i o n s t h a t s u b s i s t e d a m o n g t h e p a r t s o f t h e c u rv e .In h i s a t t e m p t t o c l a r i fy h i s b ro t h e r ' s an a l y s i s h e i d en t i f i ed ex p l ic i t g en e ra le q u a t i o n s a n d p r o p e r t i e s w h i c h w e r e i n t e n d e d t o f o r m u l a t e t h e u n d e r l y i n gp r i n c i p l e s o f t h e s u b j ec t .

JO HA NN 'S p a p e r w a s m o t i v a t e d i n l a rg e p a t h b y t h e a p p e a r a n c e i n 1 71 5 o ft h e En g l i s h m a t h em a t i c i an B R OOK TAYLOR'S M e t h o d u s i n c r e me n t o r u m. I n P r o -p o s i t i o n 1 7 o f t h a t t r e a t is e TA YLO R h a d p r e s e n t e d a s o l u t i o n t o t h e i s o p e r i m e t r i cp r o b l e m t h a t i n c o r p o r a t e d t h e r e s u l ts c o n t a i n e d i n P r o b l e m s 1 a n d 2 o f JA KOB'S1 70 1 p a p e r . T AY LO R d i d n o t m e n t i o n JA K OB , a l t h o u g h h e w o u l d l a t e r a c k n o w l -e d g e h is p a p e r a s t h e g e n e r a l s o u r c e o f h is o w n s o l u t io n . JO HA NN t o o k e x c e p t i o np u b l i c l y t o w h a t h e s a w a s a f a i l u r e t o c r e d i t h i s b r o t h e r ' s w o r k . I n h i s p a p e r o f1 71 9 h e w i s h e d t o e s t a b l i s h t h e p r e c e d e n c e a n d m a t h e m a t i c a l s i gn i f ic a n c e o fJAKOB'S ideas.

FEIGENBAUM [1 9 8 5 , 5 3 - -6 3 ] h a s m ad e a d e t a i l ed co m p a ra t i v e s t u d y o fTAYLOR'S P r o p o s i t i o n 1 7 an d JAKOB 'S P r o b l em s 1 an d 2 i n w h i ch s h e s h o w s t h e

s I t should be note d that qui te apart from the quest ion of the underly ing variat ionalprocess, we h ave in the BERNOULLIS' second m etho d a systematic form al proc edu re forsolving a certa in class of isoperimetric problems. Applied to the hanging-chain pro ble m ,it leads in a straightforward way to the equation of solution. Nevertheless neither EULERno r ( to my knowledge) m ode rn researchers mak e use of th is approach. EULER presentedhis solution o f the hanging-chain pro blem in w73 of Chapter 5 of the Methodusinveniendi wh ere i t is treate d as a conv entio nal isoperimetric pro blem and solved in theusual way b y an applica tion o f EULER'S rule .I t should be noted more general ly that the reduct ive approach in quest ion herewo uld lead for EULER's the ory to instances where th e integra nd in the expression for theauxi l iary variable / / i s g iven by , f l - p 2 . N o such ins tances are listed in Sect ion V ofCARATHI~ODORY'S [1952, lvi-lxii] Vo llst/indiges Ve rzeichnis de r B eispiele Eule rs in de rVariat ionsrechnung .


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118 C .G . FRASERp r e c i s e p o i n t s o f s i m i l a r it y a n d d i f fe r e n c e i n t h e w o r k o f t h e t w o a u t h o r s .A s i m i l a r s t u d y o f TAYLOR an d JOHANN'S p a p e r o f 1 71 9 w o u l d b e o f s o m ei n t e re s t . 9 S i n c e o u r p r i m a r y c o n c e r n i s in t h e l a t e r d e v e l o p m e n t o f t h e s u b j e c twe s h a l l o n l y d es c r i b e TAYLOR 'S s o l u t i o n i n e n o u g h d e t a i l t o a l l o w a c o m -p a r i s o n wi t h EULER 'S wo rk .4 .2 I n P r o p o s i t i o n 1 7 i t i s n e c e s s a r y t o d e t e r m i n e t h e c u r v e A B C o f g i v e nl e n g t h w h i c h m a x i m i z e s o r m i n i m i z e s t h e a r e a u n d e r a c u r v e a b c w h e r e t h eo r d i n a t e E b o f a b c i s c o m p o s e d i n a n y g i v e n w a y o f t h e a b s c i ss a D E = z t h eo r d i n a t e E B = x a n d t h e p a t h - l e n g t h A B = v (F i g u re 5 ) . Th e a rea i s r eg a rd ed a sb e i n g d e s c r i b e d b y t h e s e g m e n t E b o f t h e l i n e E B . I n i t s g e n e r a l o u t l i n eTAYLOR'S so lu t i on wa s m od el le d a f te r JA~ZOB BERNOULLfS. H e a l lo w ed tw os u c c e s s i v e o r d i n a t e s t o v a r y . T h e i s o p e r i m e t r i c c o n d i t i o n f u r n i s h e d o n e e q u a -t i o n , a n d t h e c o n d i t i o n t h a t t h e a r e a u n d e r a b c i s a n e x t r e m u m f u r n i s h e da n o t h e r e q u a t i o n . B y e l im i n a t i ng t h e t w o v a r i a t i o n a l i n c re m e n t s f r o m t h es ee q u a t i o n s h e o b t a i n e d t h e f i na l r e l a t i o n o f t h e p r o b l e m e x p r e s s e d in t e r m s o fN e w t o n i a n f i u x io n s o f t h e v a r i a b le s .

TAY LO R i n t r o d u c e d h is v a r i a t i o n a l p r o c e s s i n h i s p r e l i m i n a r y L e m m a 4 . I nF i g u r e 6 th e p o i n t s E a n d F o f t h e e x t r em a l i z i n g c u r v e a r e a s s u m e d t o p a r t a k eo f a n u p w a r d a n d d o w n w a r d m o t i o n . T AY LO R s et A E = d E F = c F G = f a n dB E = a I F = b H G = c . H e u s e d f l u x i o n a l d o t n o t a t i o n t o d e n o t e c h a n g e s t h a tr e s u l t f r o m t h e g i v e n v a r i a t i o n a l p r o c e s s . H e d e v e l o p e d h i s a n a l y s i s i n t e r m s o ft h e f l u x i o n a l v a r i a t i o n s d an d d. I t i s n o t d i f fi cu l t t o s ee t h a t ci i s t h e f t u x i o n a lv a r i a t i o n o f t h e o r d i n a t e a t E ; d is m i n u s t h e f l u x i o n a l v a r i a t i o n o f t h e o r d i n a t ea t F . Le t y = 2 / ~ w h e r e 2 a n d i a r e t h e u s u a l N e w t o n i a n f l u x io n s o f x a n d v.U s i n g t h e i s o p e r i m e t r i c c o n d i t i o n h e a r r i v e d a t t h e e q u a t i o n

33) ~ - ~ + y

I n P r o p o s i t i o n 1 7 h e c a l c u l a t e d t h e v a r i a t i o n i n t h e a r e a u n d e r t h e c u r v ea b c ( F i g u r e 5). T h e p a r t o f t h is a r e a u n d e r c o n s i d e r a t i o n i s(3 4) ~ P + ~ P ' + ~ P ,w h e r e P , P ' a n d P d e n o t e t h e v a l ue s o f t h e o r d i n a t e c o r r e s p o n d i n g t o t h ep o i n t s A , E a n d F ( F i g u r e 6 ) r e s p e c t i v e l y o f th e e x t r e m a l i z i n g c u r v e . ( N o t e t h a tt h e p o i n t A i n F i g u r e 6 c o r r e s p o n d s t o t h e p o i n t B i n F i g u r e 5 .) C l e a r l y t h ef l u x i o n a l v a r i a t i o n j 6 i s e q u a l t o z e r o . H e n c e t h e t o t a l v a r i a t i o n i n t h e a r e a i s

9 In i ts larg er scope and strong er geo me tric emphasis JOHANN'S app roa ch wasdistinguished fro m TAYLOR'S. Nevertheless, h e had seen the Englishman's solu tion w henhe wrote his paper, and the question arises as to i ts possible influence. CertainlyJOHANN'S first fundam ental eq uatio n is simply a statem ent in differential form o f theresul t (equat ion (3 3) above) obtained by TAYLOR in Lem ma 4.


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The Or ig ins o f Eule r s Var ia t iona l Ca lcu lus 119

3

; E - a vFigure 5.

Figure 6.

(35 ) ~ /6 ' + ~ /6 .B e c a u s e t h e a r e a i s a m a x i m u m o r a m i n i m u m , ( 3 5 ) i s e q u a l t o z e r o :(36) /6, + 15, = 0 .

T AY LO R p r o c e e d e d t o e x p r e s s t h e f l u x i o n a l v a r i a t i o n o f P i n t e r m s o f t h ev a r i a t i o n s 3, 2 a n d ~ a n d t h e p a r t i a l d e r i v a t i v e s Q , R a n d S o f P w i t h r e s p e c t t oz , x ad v:(37) /6= QYc R2 S~ .U s i n g t h is f o r m u l a h e o b t a i n e d t h e fo l l o w i n g v a l u e s f o r t h e c h a n g e s i n P a n dr :

/5, = Rd + Sa~( 3 8 ) / 6 = - R ' ~ - S rB y e x p r e s s i n g t h e v a r i a t i o n s d a n d / i n t e r m s o f ~i a n d ~ h e n o w r e d u c e d (3 6) t o

( 3 9 )R + S - d

a R,+S,C


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120 C .G . FRASERN o t e t h a t a/d = y a n d c / f = y = y + 2)~ + y;. Su bs t i tu t in g t hes e valu es in to (39)an d eq u a t i n g (3 9 ) an d (3 3 ) , h e o b t a i n ed

p R + S y4 0 ) p + ) ) - R ' + S ' ( y + 2p + y) 'H e s e t R ' = R + l~ a n d S ' = S + S . E n t e r i n g t h e s e s u b s t i t u t i o n s i n t o ( 4 0 ) h ea r r i v e d a t t h e f i n a l e q u a t i o n o f t h e e x t r e m a l i z i n g c u rv e :(41) Rp - R y + ~yp + 2 S 9~ - S y y = 0 .

In co ro l l a r i e s TAYLOR co n s i d e red t h e t w o ca s e s i n w h i ch v i s ab s en t an d i nw h i c h x i s a b s e n t f r o m t h e e x p r e s s i o n f o r P . H e w e n t o n t o s u p p o s e t h a t t h ea b s c i s s a z i s a l s o a b s e n t , a n a s s u m p t i o n t h a t l e a d s f o r t h e c a s e s i n q u e s t i o nd i r ec t l y t o J AKOB B ER NOULLI'S P r o b l em s 1 an d 2 . (A l t h o u g h h e d i d n ' t m en t i o nB ER NOULLI o r r e f e r t o t h e eq u a t i o n s h e h a d o b t a i n ed . ) FEm ENBAUM [1 9 8 5 ,6 1 - -6 2 ] d e s c r i b e s t h e r e s u l t s TAYLOR rea ch e d i n t h e s e c o ro l l a r i e s . W e co n c l u d eo u r a c c o u n t b y i n d i c a t in g h o w o n e w o u l d o b t a i n JA K OffS e q u a t i o n (27 ) f r o mTAYLOR'S (41 ). W e a re co n s i d e r i n g t h e ca s e w h e re P i s a fu n c t i o n o f v a l o n e s ot h a t R = 0 . (4 1 ) b eco m es(42) SY~9 + 2S~9~9 - S y y =- 0 .I n t e r m s o f t h e n o t a t i o n u s e d b y JA K O B B ER NO U LL I w e h a v e h = S a n dd x / d z = y a n d ( 4 2 ) b e c o m e s

d h d x d ( d x ) ( d ( d x ' ] ~ ] 2 d x d 2 [ d x \(43) d y d z d y _ + 2 h \ d y \ d z J ] h d z ~ y a ~ d z ) = 0B e c a u s e d y i s c o n s t a n t w e h a v e dzd2z = dxd2x . U s i n g t h i s r e l a t i o n a n d p e r f o r m -i n g t h e n ec es s a r y d i f f e r en t i a t i o n s i n (43 ), o n e o b t a i n s JAKOB 'S eq u a t i o n (27 ). l ~4 ,3 T AY LO R f o r m u l a t e d h is s o l u t i o n t o P r o p o s i t i o n 1 7 i n t h e i d i o m o f th ef l u x io n a l ca l cu l us a n d e m p l o y e d k i n e m a t i c i m a g e r y i n i n t r o d u c i n g t h e r e q u is i tei n f i n i t e s i m a l p r o c e s s e s . N e v e r t h e l e s s , h e s h o w e d a s t r o n g e r s e n s e t h a n d i dJOHANN BERNOULLI fo r the un de r ly ing an a ly t ica l ch ar ac te r o f JAKOB'S inves t iga -t i on . J OH AN N e m p h a s i z e d a n d d e v e l o p e d f u r t h e r t h e g e o m e t r i c a s p ec t s o f t h es u b je c t ; h is d e r i v a t i o n o f t h e v a r i a t i o n a l e q u a t i o n s w a s c a r r i e d o u t i n d e t a i l e dr e f e r e n c e t o t h e i n f i n it e s im a l g e o m e t r i c a l e l e m e n t s o f a d i a g r a m . H e i n t r o d u c e de x p l a n a t o r y p r i n c i p l e s a n d c o m m e n t a r y t h a t w e r e i n t e n d e d t o c l a r i f y t h e n a t u r eo f h is b r o t h e r ' s m e t h o d .

10 TAYLOR ende d his discussion with Co rollar y 5. Altho ugh not directly relev ant tothe qu est ion of the d evelopm ent of the isoperimetr ic theory th is resul t is o f some in terestfrom the view point of the larger histo ry of the calculus of variations. It form ulates w hatlater bec am e know n as the inverse problem: given an e qua tion o f the form (41), i t isnecessary to find a qua ntity P such that the pro blem o f maximizing or minimizing thearea under the curve with general ordinate P gives rise as solution to (41).


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Th e Origins of Euler 's Va riational Calculus 121TA YLO R'S e x p o s i t i o n b y c o n t r a s t w a s a m o d e l o f c o n c is i o n . H e p r o c e e d e d

d i r e c t l y t o a s in g le e q u a t i o n t h a t c o m b i n e d t h e s e p a r a t e r e s u l t s c o n t a i n e d i nJA KO B'S f i rs t t w o p r o b l e m s . I n h is s o l u t i o n h e i n t r o d u c e d f u n d a m e n t a l m o d i f i c a -t i o n s i n t o t h e l a t t e r ' s t h e o r y . H e s h i f t e d t h e o v e r a l l m a t h e m a t i c a l a p p r o a c ha w a y f r o m t h e q u e s t i o n o f t h e s p e c i f ic a t i o n o f t h e p r o g r e s s i o n o f t h e v a r i a b l e s( i n w h i c h a g i v e n s o l u t i o n w a s u n d e r s t o o d t o i n v o l v e a s u i t a b l e d e t e r m i n a t i o no f t h e i n d e p e n d e n t v a r i a b l e ) t o a s t u d y o f t h e s t r u c t u r e o f th e i n t e g r a n df u n c t i o n . T h e l a t t e r w a s e x p l i c it l y t r e a t e d a s a f u n c t i o n o f s e v e r a l v a r i a b l e s i nh is e q u a t i o n (3 7). B y w o r k i n g w i t h t h e f l u x i o n o f t h is f u n c t i o n a n d c a r r y i n g o u tt h e d e r i v a t i o n i n t e r m s o f i ts p a r t i a l d e r i v a t i v e s (g i ve n b y ( 3 7 )) h e e s t a b l i s h e dn e w a n a l y t i c a l c r it e r i a f o r th e o r g a n i z a t i o n a n d i n v e s t i g a t i o n o f v a r i a t i o n a lp r o b l e m s .

5. Euler s Papers (17 38 , 1741)5 .1 EULLR 'S ea r l i e s t in t e r e s t i n v a r i a t i o n a l m a t h em a t i c s o r i g i n a t ed i n h i s s t u d yo f t h e p r o b l e m o f t h e c u r v e o f q u i c k e s t d e s c e n t i n a r e s is t in g m e d i u m , H ep r o p o s e d t h i s p r o b l e m i n 1 72 6 i n w h a t a p p a r e n t l y w a s h i s v e r y f ir s t p u b l i c a t i o ni n s c i e n c e . I n 1 7 4 0 h e p u b l i s h e d a s o l u t i o n i n w h i c h h e c o r r e c t e d a n d e x t e n d e de a r l i e r w o r k c a r r i e d o u t b y JA KO B H ERM AN N. M o r e g e n e r a l l y h e w a s d u r i n g t h e1 73 0s i n t e r e s t e d i n t h e m e c h a n i c a l q u e s t i o n o f m o t i o n i n a r e s i s ti n g m e d i u m ,a t o p i c t h a t w a s t r e a t e d e x t e n s i v e ly i n h is Mechanica ana ytica of 1736.

A l t h o u g h t h es e e a r ly r e se a r c h es i n v o l v e d q u e st i o ns o f m a x i m a a n d m i n i m at h e y s t o o d a p a r t f r o m t h e m a i n l i ne o f v a r i a t i o n a l w o r k t h a t h a d b e g u n w i t hJAKOB BERNOULLI'S pa pe rs o f 1697 an d 1701 . EULER was inv es t ig a t in g a spec i f i ce x a m p l e u s i n g m e t h o d s t h a t w e r e s u i t e d t o t h e p r o b l e m a t h a n d b u t w h i c h w e r en o t o t h e r w i s e g e n e r a l i z a b l e . S o m e t i m e i n t h e m i d d l e 1 7 3 0 s h e b e c a m e i n t e r e s t e di n t h e t h e o ry d ev e l o p ed b y t h e B ER NOULLIS an d TAYLOR. H e co n t r i b u t ed t wop a p e r s o n t h e s u b j e c t t o t h e S t. P e t e r s b u r g A c a d e m y o f S c i en c e . A l t h o u g h t h e yw e r e p u b l i s h e d i n 1 7 3 8 a n d 1 7 4 l t h e r e s e a r c h p r e s e n t e d i n t h e m s e e m s t o h a v eb een co m p l e t ed s ev e ra l y ea r s ea r l ie r . 11

A t t h e b e g i n n i n g o f t h e p a p e r o f 1 73 8 E U L E R p r o p o s e d a c l a s s i fi c a ti o n o fv a r i a t i o n a l p r o b l e m s a c c o r d i n g t o t h e n u m b e r o f s id e c o n d i t i o n s t h a t a r ep r e s e n t . I f t h e r e a r e n o s u c h c o n d i t i o n s w e h a v e th e f r ee o r a b s o l u t e p r o b l e m ,e x a m p l e s o f w h i ch a r e t h e b r a c h i s t o c h r o n e o r t h e s u r f ac e o f r e v o l u t i o n o fm i n i m u m r e s i s t a n c e . N e x t w e h a v e p r o b l e m s i n w h i c h a s i d e c o n d i t i o n i n t h ef o r m o f a d e f i n i te i n t e g r a l is p r e se n t . T h e t r a d i t i o n a l i s o p e r i m e t r i c p r o b l e m is

i1 In a le t ter to MAUPERTUIS dated M arch 16, 1 746 EULER stated that he hadwri t ten the Methodus inveniendi(excluding the two appendices) while he was still in St.Petersburg. Since he left for Berlin on June 19, 1741 this would mean that the work wascom pleted by the spring of 1741. Th e pap er of 1741 mu st therefore have been com pletedsom etim e earlier. See CARATHI~ODORY's [1952, xi, w3] discussion.


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122 C .G . FRASERt h e s t a n d a r d h e r e , a n d E U LE R m e n t i o n e d i n t hi s c o n n e c t i o n t h e n a m e s o f JAKO Ban d JOHANN BERNOULU, TAYLOR an d HERMANN 12 F u r t h e r c la ss es o f p r o b l e m sw i l l b e o b t a i n e d b y a d d i n g a s e c o n d , t h i r d , a n d s o o n , c o n d i t i o n .

EULER ran i n t o d i f fi cu l ti e s i n h i s i n v es t i g a t i o n o f t h e f r ee v a r i a t i o n a l p r o b l emi n t h e c a s e w h e r e t h e v a r i a b l e s i n t h e i n t e g r a n d a r e c o n n e c t e d b y m e a n s o fa d i f f e r e n t i a l e q u a t i o n . T h e p r o t o t y p e f o r t h i s p r o b l e m o c c u r s w h e n t h e i n t e g -r a n d f u n c t i o n c o n t a i n s t h e p a t h - l e n g t h s = ~ o ~ / 1 + p Z d x p = d y / d x ) . A n i m -p o r t a n t e x a m p l e c o n c e r n s th e m o t i o n o f a p a r t ic l e i n a r e s is t in g m e d i u m ,a s u b j e c t t h a t w a s o f p a r t i c u l a r i n t e r e s t t o h i m .

T o u n d e r s t a n d t h e p o i n t a t i s s u e h e r e c o n s i d e r t h e v a r i a t i o n a l i n t e g r a lS o Z x , y , p, s ) d x , w h e r e i t i s a s s u m e d t h a t n o i s o p e r i m e t r i c c o n d i t i o n i s p r e s e n t .T h e c o r r e c t g e n e r a l e q u a t i o n o f s o l u t io n i s t h e o n e d e r i v e d in C h a p t e r 3 o f t h eM e t h o d u s i n v e n i e n d i , g i v en a s eq u a t i o n 1 4) ab o v e :

d P14) 0 = N - dxx + [ N ] H - j L d x ) - d x ~d [ P ] H ~

In t h e c a s e a t h an d 17 = s , IN ] = 0 , [P ] = d y/d s, H - j L d x = j a L d x and 14)b e c o m e s44) 0 = N - ~ x + L ~ s s - L d x d ~ s = 0 .

I n w o f h is p a p e r o f 1 74 1 E U LE R o b t a i n e d t h e s o l u t i o n45) d P d yO=N- La-

i n w h i c h t h e t e r m i n v o l v i n g t h e i n t e g r a l ~ x L d X i s miss ing.T h e e r r o r c o m m i t t e d b y E U L E R is a s i g n i fi c a n t o n e a n d v i ti a t e s a s u b s t a n t i a l

p a r t o f t h e t h e o r y t h a t h e d e v e l o p e d in t h e t w o p a p e rs . I t w o u l d a ri se w h e n e v e rt h e i n t e g r a n d c o n t a i n s a v a r i a b l e H w h i c h i s c o n n e c t e d t o t h e o t h e r v a r i a b l e s o ft h e p r o b l e m b y a r e l a t io n o f th e f o r m / 7 = t o [ Z ] d x . A n e x p l a n a t i o n o f i tso r i g i n i s r e a d i l y a v a i l a b l e . C o n s i d e r e x a m p l e s i n w h i c h t h e i n t e g r a n d c o n t a i n st h e p a t h - l e n g t h s b u t w h i c h a r e n o t s u b j e c t t o a n y i n t e g r a l s i d e c o n d i t i o n . T h ep r o b l e m s a r e t r e a t e d b y E U LE R b y a p r o c e s s i n v o l v i n g t h e v a r i a t i o n o f a s i n gl eo r d i n a t e . T r a d i t i o n a l l y w h e n s u c h p r o b l e m s h a d b e e n c o n s i d e r e d a n i s o p e r i m e -t r i c c o n d i t i o n h a d a l w a y s b e e n a s s u m e d a n d t h e r e q u i s i t e p r o c e s s i n v o l v e d t h ev a r i a t i o n o f t w o o r d i n a te s . N o w i t t u r n s o u t f o r p r o b l e m s i n w h i c h t h e v a r i -a b le s a r e r e l a t e d b y a d i f f er e n t ia l e q u a t i o n t h a t t h e c a l c u l a t i o n o f t h e v a r i a t i o np r o c e e d s d i f fe r e n tl y , d e p e n d i n g o n w h e t h e r o r n o t a n i s o p e r i m e t r i c c o n d i t i o n i sa s s u m e d . C o n s i d e r t h e e x a m p l e p r e s e n t e d i n t h e p r e c e d i n g p a r a g r a p h . W h e n w ec a l c u la t e th e v a r i a t i o n a c c o r d i n g t o t h e t r a d i t io n a l p r o c e d u r e o f v a r y i n g t h e

12 In intro duc ing his analysis of the isoperimetric prob lem in w of his pap er o f1741 EULER referred onc e aga in to this gro up of researchers.


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Th e Origins of Euler's Va riational Calculus 123o r d i n a te s y ' a n d y b y t h e a m o u n t s n v an d o~o we o b t a i n a s t h e co e f f i c i en t o f nvt h e t e r m(46) N ' d x - d P + L ' q d x - L d q d x ,w h e r e q = d y / d s , L ' a n d N ' a r e t h e v a lu e s o f L a n d N a t x ' a n d L is t h e v a l u eo f L a t x . I m p l i c i t i n t h e c a l c u l a t i o n o f (4 6) is t h e a s s u m p t i o n t h a t~ o x / 1 + p2 d x = c o n s t a n t . ( T h e d e r i v a t i o n o f t h is r e s u l t i s p r e s e n t e d i n d e t a i lb e l o w i n t h e a n a l y s i s l e a d i n g t o e q u a t i o n ( 5 6 ) . N o t e t h a t a r e s u l t e q u i v a l e n t t oa spec ia l case o f (46) (w here P = 0 ) is c on ta i ne d in TAYLOR'S eq ua t io n (39).)W h e n w e p e r f o r m t h e s a m e c a l c u l a t i o n i n t h e c a s e w h e r e t h e r e i s n o i s o p e r i m e -t r ic c o n d i t i o n w e o b t a i n a s t h e c o e f f i c ie n t o f n v(47) N ' d x - d P + L ' q d x - d q ( L ' d x + L ' d x + U ~ d x + . . . )H e r e t h e v a r i a t i o n o f t h e s i ng l e o r d i n a t e y ' l e a d s f o r t h e v a r i a b l e s t o a c h a n g ei n i ts v a l u e a t e ac h o f t h e s u cces s iv e v a l u es x ' , x , . . . ; t h u s th e r e a r e c h a n g e s i n~o Z d x o v e r t h e e n t i r e i n t e r v a l fi 'o m x t o a , n o t j u s t i n a n e i g h b o u r h o o d o f x .

T h e c r u c i a l f a c t h e r e is t h a t i n t h e i s o p e r i m e t r i c p r o b l e m b e c a u s e o f t h ec o n d i t i o n I 0 x / 1 + p 2 d x = c o n s t a n t t h e v a r i a t i o n o f t h e v a l u e o f ~ o Q d x wil l bel i m i t e d to a n e i g h b o u r h o o d o f x . A l l e a r l ie r w o r k - m o s t p r o m i n e n t l y JA KO BBERNOULLI'S P ro b l em 2 an d TAYLOR'S P ro po s i t io n 17 - in w hich s was em -p l o y e d a s a v a r i a b l e c o n c e r n e d s u c h a p r o b l e m . I n h is d e r i v a t i o n o f (4 5) E UL ~Rw a s c o n s i d e r i n g t h e f re e p r o b l e m a n d w a s u s i n g a s i n g l e - o r d i n a te v a r i a t i o n a lp r o c e s s . N e v e r t h e l e s s h e c o n t i n u e d t o a s s u m e t h a t ( 4 6 ) p r o v i d e s t h e c o r r e c te x p r e s s i o n f o r t h e v a r i a t i o n . N e g l e c t i n g t h e h i g h e r - o r d e r t e r m a n d s e t t i n gN ' = N , L ' = L , w e h a v e e x p r e s s i o n t h a t a p p e a r s i n ( 4 5 ) .

I t s h o u l d b e n o t e d t h a t f o r v a r i a t i o n a l i n t e g r a l s o f t h e f o r m ~ o Z ( x , y , p ) d x ,i n wh i ch t h e re a r e n o a u x i l i a ry v a r i ab l e s s u ch a s s i n Z , t h e co e f f i c i en t o f n v ist h e s a m e e x p r e s s i o n i n b o t h t h e o n e - o r d i n a t e a n d t w o - o r d i n a t e v a r i a t i o n a lp r o c e ss e s . T h e r e is (i t w o u l d s e e m ) n o o b v i o u s r e a s o n w h y E U L ~R s h o u l d h a v eb e e n a w a r e t h a t t h i s s i t u a t i o n c h a n g e s w h e n s o c c u r s i n Z . I t w a s t h e n a m a j o rr e v e l a t i o n w h e n h e r e a l i z e d th a t i t s p r e s e n c e le a d s t o a n i n t e g r a l t e r m i n t h ev a r i a t i o n a l e q u a t io n . S o m e t i m e b e t w e e n t h e c o m p l e t i o n o f t h e m a i n b o d y o f t h es e c o n d p a p e r a n d i t s p u b l i c a t i o n i n 1 7 4 1 h e a r r i v e d a t t h i s u n d e r s t a n d i n g . A ne x p l a n a t i o n a n d d e r i v a t i o n o f th e c o r r e c t e q u a t i o n is s k e t c h e d b y h i m i n t h el a s t f o u r p a r a g r a p h s o f t h e p a p e r . H e p r e s e n t e d t h e r e in o u t l i n e f o r m t h ed e r i v a t i o n t h a t is d e v e l o p e d in m o r e d e t a il in C h a p t e r 3 o f t h e M e t h o d u sinven iendi , r e p r o d u c e d b y u s a b o v e i n s e c t i o n 2.1 i n t h e a n a l y s is l e a d i n g t oe q u a t i o n ( 1 4 ) . I t s h o u l d b e n o t e d t h a t t h i s m a t e r i a l i s h i g h l y i n c o n g r u o u s i nre la t io n to the m ai n c on te n t of h is p ap er of 1741. CARATHI~ODORY'S [-1952, XXX]c o n j e c t u r e t h a t t h e s e p a r a g r a p h s w e r e i n s e r t e d i n t h e p a p e r a t a l a t e r d a t e j u s tp r i o r t o i t s p u b l i c a t i o n , a t a t i m e w h e n w o r k o n t h e M e t h o d u s i n v e n i e n d i w a sa l r e a d y w e l l u n d e r w a y , s e e m s e x t r e m e l y p l a u si b l e.

R e c o g n i t io n o f t h e e r r o r t h a t h e h a d c o m m i t t e d s e em s t o h a v e g a l v a n i z e dE UL ER a n d l e d t o a n i n t e n s e b u r s t o f r e s e a r c h a c t i v it y in v a r i a t i o n a l m a t h e -m a t i c s . In t h e M e t h o d u s i n v e n i e n d i h e r e je c t e d t h e t r a d i t i o n a l o r g a n i z a t i o n o f t h e


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12 4 C. G , FRASERs u b j ec t , c o n c e n t r a t i n g i n C h a p t e r s 2 a n d 3 o n t h e d e t a i le d d e v e l o p m e n t o f t h ef re e p r o b l e m a n d a s si g n in g t he h i s to r i ca l ly p r o m i n e n t i s o p e r i m e t r ic p r o b l e m t oa s e c o n d a r y p o s i t i o n a t t h e e n d o f t h e t r ea t is e .

5 .2 A l t h o u g h E U LE R'S p a p e r s o f 17 38 a n d 1 74 1 c o n t a i n a n i m p o r t a n t e r r o r ,t h e r e i s a l s o m u c h o f i n t e r e s t i n t h e m ; i n s e v e r a l r e s p e c t s t h e a n a l y s i s iss u b s t a n t i a l l y s u p e r i o r t o t h e c o r r e s p o n d i n g t r e a t m e n t i n t h e M e t h o d u si n v e n i e n d i . I n t h e c a se o f t h e t r a d i t io n a l i s o p e r i m e t r i c p r o b l e m h e a r r i v e d a ta c o n s i s t e n t t h e o r y , o n e t h a t r e p r e s e n t e d t h e e l e g a n t d e v e l o p m e n t o f t h eB ER N OU LU S' a n d T AY L OR 'S t h e o r y . I n d e e d i f o n e i s i n t e r e s t e d i n t h e c o m p l e t em a t h e m a t i c a l e x p l i c a t i o n o f JA K OB 'S P r o b l e m s 1 a n d 2 it w il l b e f o u n d i nE U LE R'S p a p e r o f 17 41 a n d n o t i n h i s m o r e f a m o u s M e t h o d u s i n v e n i e n d i . W et u r n n o w t o a n e x a m i n a t i o n o f t h is s u b je c t. 13

E UL ER 'S a p p r o a c h w a s b a s e d o n t h e f u r t h e r d e v e l o p m e n t o f t h e i d e a t h a tu n d e r l i e s t h e i s o p e r i m e t r i c r u l e ( E U LE R 'S r u l e ) , o u t l i n e d s u b s e q u e n t l y b y h i mi n C h a p t e r 5 o f t h e M e t h o d u s i n v e n i e n d i a n d d e s c r i b e d b y u s a b o v e i n s e c -t i o n 2.1. ( I n w h a t f o l lo w s w e a d h e r e t o t h e n o t a t i o n e m p l o y e d i n t h e p a p e r s o f1 73 8 a n d 1 74 1, w h i c h d i f fe r s s o m e w h a t f r o m t h a t o f t h e M e t h o d u s i n v e n i e n d i . )A s s u m e w e w i s h t o e x t r e m a l i z e t o Q d x s u b j e c t t o a s id e c o n d i t i o n g i v e n i n t h ef o r m o f a d e f in i t e i n te g r a l . L e t o a b c d ( F i g u r e 7 ) b e t h e p r o p o s e d e x t r e m a l i z i n gc u r v e . W e h a v e O A = x , O B = x ', O C = x , O D = x , . . . . A a = y , B b = y ' ,C c = y ' , D d --- y ' , . . . . W e v a r y t h e t w o s u c c e s s i v e o r d i n a t e s y ' a n d y b y t h ea m o u n t s b f i a n d - c 7 , o b t a i n i n g i n t hi s w a y a c o m p a r i s o n c u r v e o a f lT d . T h ec a l c u l a t i o n o f t h e c h a n g e i n th e i n t e g r a l ~ o Q d x a l o n g t h e t w o c u r v e s l e a d s t o a ne q u a t i o n o f t h e f o r m(48) P . b f i - ( P + d P ) ' c 7 = 0 .

13 In his influential hist oric al e ssay CARATHI~ODORY1-1952] fails in m y view to givea jus t es t im at ion of the theo ry c onta ined in EULER'S pap ers of 1738 and 1741. In essencehe sees the pas sage f rom the ea r li e r papers to the t rea ti s e o f 1744 as one f rom e r ror tot ruth. In reference to the 1738 m em oir he writes (p . xxix): So erhie l t er die auBerorden-t l ich kompliz ier ten Formeln, die in zwei Tabel ten auf p , 28 und 32 verzeichnet s ind.Schon die Um st/ indl ichkei t dieser Form eln h~i tte ihn s tutz ig m ach en sollen. Th e pap erof 1741 is dismissed in a s imilar sum m ary fashion: Au ch die zwei te Arb ei t E56 yo n 1736is t mi t so lchen unha l tba ren Resu l ta ten gesp ick t .O ne po int tha t CARATHt~ODORY m ake s (referr ing to the las t pa r t of the m em oir o f1738) is tha t EULER'S analys is is no t eas i ly ada pta ble to the case where the re is mo rethan one s ide condi t ion presen t . However i t i s no t c lea r why th i s should be rega rded asa pa r t i cu la r w eakness o f the theory tha t i s s e t fo r th in the p apers o f 1738 and 1741. Thefact is that when auxi l iary variables (such as the path-length) occur in the varia t ionalintegrand i t becomes diff icul t to handle mult iple s ide condi t ions . No progress on thisques t ion i s made in Chapte rs 5 o r 6 o f the M e th o d u s i n v e n i e n d i . What is rea l ly needed isa general me thod of multipliers, and this was bey ond the scope of the theo ry in i tspre-Lagrangian phase . (See a lso notes 17 and 18.)


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Th e Origins of Euler s Variational Calculus 125

D

2vI c

Figure 7.

S i mi l a r l y t h e ch an g e i n t h e in t eg ra l t h a t ap p ea r s i n t h e s id e co n d i t i o n l ead s t o~ h e e q u a t i o n(49) R . b f i - ( R + d R ) . c 7 = 0 .B y e l i m i n a t i n g b f i a n d c 7 a n d s o l v i n g w e o b t a i n P + C R = 0 a s t h e e q u a t i o n o ft h e p r o b l e m .

I n t h e t w o p a p e r s t h e r e i s c o n s i d e r a b l e i n t e r n a l d e v e l o p m e n t a n d r e f i n e m e n to f th e an a l y s i s . I n t h e f i rs t EU L ER s u m m ar i ze d h i s v a r i o u s r e s u l ts i n t h e fo rm o ft ab le s , m a t e r i a l t h a t w as s y n t h es i zed i n th e l a t e r w o rk i n a s in g l e fo rm u l a t i o nan d d e r i v a t i o n . I t is o f s o m e i n t e r e s t t o fo l l o w h i s e f fo r t s a s a s t u d y i n t h ef o r m a t i o n a n d e v o l u t i o n o f a m a t h e m a t i c a l t h e o r y . In t h e c a se w h e r e th ei n t e g r a n d Q is o f th e f o r m Q = Q ( x , y , p ) ( p = d y / d x ) t h e r e s u l t i n g v a l u e fo r P i sp r ec i s e ly th e ex p re s s i o n t h a t ap p e a r s i n t h e can o n i ca l E E L ER eq u a t i o n fo r th ef r ee v a r i a t i o n a l p ro b l em. O n e i s ab l e t o o b s e rv e i n t h e t w o p ap e r s h i s g r ad u a lp r o g r e s s i n i d e n t if y i n g t h e s t a n d a r d g e n e r a l fo r m s a n d e q u a t i o n s o f t h e s u b je c t.

W h e n t h e i n t e g r a n d Q a ls o c o n t a i n s t h e p a t h - l e n g t h , th e d e r i v a t i o n o f t h eex p re s s i o n fo r P i s r a t h e r mo re co mp l i ca t ed . T h e t ab l e s i n t h e p ap e r o f 1 7 3 8co n t a i n v a r i o u s p a r t i a l r e s u l ts t h a t h e w as ab l e t o o b t a i n . I n w 1 6 7 o f h i sp ap e r o f 1 7 4 1 h e s u cces s fu l ly h a n d l ed t h e s u b j ec t i n a s i n g le g en e ra l d e r i v a t i o n .F o r s o m e r e a s o n ( w h i c h w e d i s c u s s b e l o w ) h e c h o s e n o t t o i n c l u d e t h i s m a t e r i a li n C h a p t e r 5 o f t h e M e t h o d u s i n v e n i e n d i .

A g a i n w e a r e d e a l i n g w i t h t h e c u r v e o a b c d d ep i c t ed i n F i g u re 7 , w h e reO A = x , A a = y a n d o a = s . W e h a v e O B = x ' , O C = x , O D = x ' , . . . .A a = y , B b = y ' , C c = y , D d = y ' , . . . , o b = s ' , o c = s , o d = s ' , . . . . W e a r eg i v en t h e i n t eg ra l ~ o Q d x w h ere (2 i s co mp o s ed o f x , y , p = d y / d x a n d


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25/39

Th e Origins of Euler s V ariationa l Calculus 127S u m m i n g t h e s e e x p r e s s i o n s a n d e q u a t i n g t h e r e s u l t t o z e r o , w e o b t a i n(56) ( - d V + L ' q d x + M ' d x - L ' d x d q ) b f l = ( - d V ' + L ' q ' d x + M d x ) c 7 .F o l l o w i n g t h e s t a n d a r d p r o c e d u r e w e s e t A = ( - - d V + L ' q d x + M ' d x- L ' d x d q ) , B = ( - d V ' + L ' q ' d x + M ' d x ) a n d c a l c u l a t e d P / P = ( B - A ) / A :

d Pm(57) p - d d V + L ' d x d q + d x d ( L ' q + M ' )- d V + d x ( L ' q + M ' )T h i s e q u a t i o n m a y b e i m m e d i a t e l y i n t e g r a t e d t o y i e l d

5 8 )L dx dq

P = e ~ - a v + a x (L q + U ) ( - - d V + d x ( L q + M ) ) .T h e e x p r e s s i o n f o r P i n ( 5 8 ) i s t h e g e n e r a l o n e t h a t o b t a i n s f o r a n a r b i t r a r y

v a r i a t i o n a l i n t e g r a n d o f t h e f o r m Q ( x , y , p , s ). E q u a t i o n ( 5 8 ) r e p l a c e s s e v e r a l o ft h e m a n y s p e c ia l r e s u lt s E U LE R h a d d e r i v e d i n h is p a p e r o f 1 73 8. H e p r o c e e d e di n w 3 2 t o e x t e n d (5 8) t o t h e c a s e w h e r e t h e i n t e g r a n d Q c o n t a i n s ina d d i t i o n t o x , y , p a n d s th e h i g h e r - o r d e r d e r i v a t i v e r d e f i n e d b y dr = pd x . 14 I tw a s c l e a r h o w t h e r e s u l t c o u l d b e f u r t h e r e x t e n d e d t o i n c l u d e a r b i t r a r y h i g h -e r - o r d e r d e r i v a t i v e s .5 . 3 I t s h o u l d b e n o t e d t h a t ( 5 8 ) a n d t h e p r o c e s s b y w h i c h i t i s d e r i v e d m u s t b ei n t e r p r e t e d d i f fe r e n tl y , d e p e n d i n g o n w h e t h e r o r n o t t h e v a r i a b l e s i s p r e s e n t i n(2. I f s is n o t p r e s e n t t h e n L = 0 a n d P r e d u c e s t o t h e s t a n d a r d f o r m( - d V + M d x ) . H e r e i t i s n o t n e c e s s a r y t h a t t h e d i s t a n c e s o a b c d a n d oa f l Td b ee q u al . T h e . e x p r e s s io n f o r P m a y b e u s e d i n c o n j u n c t i o n w i th a n y o t h e r g i ve ni n t e g r a l s i d e c o n d i t i o n o f t h e f o r m ~ o W ( x , y , p ) d x - - c o n s t a n t . E U LE R p r o v i d e dj u s t s u c h a n e x a m p l e t o i l l u s t r a t e ( 5 8 ) . H e c o n s i d e r e d c u r v e s A M (F i g u re 8 ) fo rw h i c h t h e a r e a A Q M p o s s e s s e s s o m e g i v e n c o m m o n v a l u e . A m o n g t h i s c l a s s i tis n e c e s s a r y t o f in d t h e o n e t h a t e x p e r i e n c e s t h e l e a s t r e s i s ta n c e a s i t m o v e st h r

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