Calculus o f Variation s 5 for al l tes t function s (p. Her e o f cours e { —} denotes th e expressio n in th e integran d o f (3) . Thi s resul t i s th e precis e analo g o f (1 ) i n Section II , an d w e would lik e to obtai n th e resul t simila r t o (2) tha t is, we would lik e to conclud e tha t { —} itself i s equal t o zero . At thi s poin t w e use a rathe r eas y fac t abou t integration . How - ever, thi s i s o f suc h importan t historica l significanc e tha t i t actuall y has th e rathe r dauntin g name , Th e Fundamenta l Lemm a o f th e Calculus o f Variations . Thi s assert s tha t i f (4 ) i s valid fo r al l tes t functions (/? , the n { —} = 0 . Th e proo f depend s somewha t o n th e assumptions. Th e easies t versio n assume s tha t th e functio n { —} is continuous. The n i f i t i s nonzer o a t som e poin t XQ D, sa y i t i s positive, the n b y continuit y i t remain s positiv e i n a neighborhoo d o f XQ. The n w e ca n selec t a tes t functio n (p whic h i s 0 on D , whic h is positive a t XQ, an d whic h i s identically zer o outsid e a small neigh - borhood o f XQ. The n { }p i s a continuou s nonnegativ e functio n o n D whic h i s positive i n a neighborhoo d o f x$, S O that o f cours e / {-}(pdx 0 . JD This contradict s (4) . Thus w e conclude tha t i f u i s a critical poin t fo r / , the n du ^ dxi \du Xi J This resul t i s calle d th e Euler-Lagrang e equation . Notic e tha t i t is a necessary and sufficient conditio n fo r u to be a critical point fo r / . It i s therefore a necessary conditio n fo r / t o attai n a n extreme value at zz , but nothin g i n th e argumen t woul d eve n hin t tha t i t woul d b e a sufficien t conditio n (an d i t i s not). Be carefu l o f th e strang e notatio n dg/du Xi . Th e functio n g de - pends o n x, u, an d n othe r independen t rea l variables. Thu s dg/du Xi is the partia l derivativ e o f g with respec t t o th e argumen t tha t occu - pies th e slo t wher e w e have inserte d u x .
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