Frank Jone s Then th e correspondin g Euler-Lagrang e equatio n i s |.( u ^^)-|(A (tt vrT^))=o. This simplifie s t o 1 + u 2 - uu" = 0 . This equatio n i s fairly easil y integrated , resultin g i n u(x) = — cosh(Ar + B), where A 0 and B ar e constant s o f integration . B. Geodesies . W e just mentio n thi s i n passing . Give n a smoot h surface i n M 3 , ther e ar e calculu s formula s fo r th e ar c lengt h o f a curve lying on the surface . Th e problem o f minimizing th e ar c lengt h of a curv e o n th e give n surfac e whic h connect s tw o give n point s i s a calculus-of-variation s problem . Ther e i s a correspondin g Euler - Lagrange equation , whic h i s actuall y a syste m o f tw o ordinar y dif - ferentiations, eac h o f second order , i n whic h th e unknown s represen t the coordinate s o f a point o n the curve . In th e elementar y cas e o f the (x,?/)-plan e itself , a curv e give n a s x = x(s), y = y(s) ha s lengt h 6 J V^' 2 + y' 2 ds. J a This does not quit e fit th e Euler-Lagrang e scenario , but th e variatio n idea o f Sectio n II I lead s t o a n equatio n ' ' = 0. / J a Jt ^/{x'{s) + V(s))2 + (y'(s) + tip(s))2ds t=0 That is , L b °W + Wds = 0. a v/x' 2 + y' 2 Here p(s) an d ip(s) ar e arbitrary , excep t the y ar e zer o a t th e end - points. Integratin g b y part s the n give s

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