Calculus o f Variation s 3 While al l o f thi s shoul d b e ver y familia r t o you , ther e ar e som e important observation s tha t w e need t o stress : A point x satisfyin g (2 ) i s said t o b e a critical point fo r / . Jus t a s in the case n = 1 , any extreme point fo r / i s a critical point, bu t no t conversely . Th e point x + th use d in the analysis is said to be a variation of th e fixed poin t x. Th e ide a i s that \t\ i s small, an d i t i s only the limiting behavior of f(x+th) a s t 0 that interest s us. W e procee d fro m (1 ) t o (2 ) b y makin g judicious choice s o f h. III. Th e domai n i s infinite-dimensiona l Now w e com e t o th e actua l situatio n o f interes t i n th e calculu s o f variations. W e tr y t o analyz e th e critica l behavio r o f a real-value d function / whos e domai n i s a certai n spac e o f real-value d functions . Thus, fo r eac h suc h functio n u ther e i s a corresponding valu e f(u). We assume tha t / attain s a n extrem e valu e at a certain functio n u. The n fo r an y functio n cp tha t i s "admissible " i n som e sense , w e consider th e variation u + tip o f th e functio n u. The n f(u + t(p) has a n extrem e valu e a t t = 0 , s o w e conclud e tha t th e "directiona l derivative" D v f(u) = f t (f(u + tp))\ t=0 must equa l zero . Thu s w e are le d to th e definitio n tha t u i s a critical "point" fo r / i f D^f(u) = 0 fo r al l admissible function s tp. This i s about a s far a s we can go without specifyin g jus t wha t sor t of function / is . I hav e chose n t o demonstrat e th e idea s wit h th e mos t well-known situation . IV. Th e Euler-Lagrang e scenari o This i s a situatio n tha t i s encountere d i n a tremendou s variet y o f circumstances. Withou t bein g too specific abou t hypotheses , suppos e
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