6 Frank Jone s V. Severa l classica l example s In thi s las t portio n o f the tal k w e shall presen t a variet y o f beautifu l ancient example s o f the us e of the Euler-Lagrang e equation . A. Minima l surfaces . Give n a close d curv e i n IR 3 , th e proble m i s to tr y t o find a surfac e o f minima l are a whic h "fill s in " th e give n curve. Le t u s restric t attentio n t o th e followin g situation . Give n a domain D i n the (x,y)-plan e wit h boundar y 9D , w e assume that th e given curv e ha s th e for m (x,?/ , 7(^,2/)) fo r (x,y) G dD, an d tha t w e seek a correspondin g surfac e o f minima l are a whic h ha s th e explici t description z = u(x,y), wher e u i s th e unknow n function . Vecto r calculus give s us th e are a o f the surfac e i n th e for m Jl + u 2 x + u 2 y dxdy. This i s a perfect set-u p fo r Euler-Lagrange , an d th e equatio n w e ob- tain i s -JK^v/i+""+"5)-l(4yi+^+»o=o- In othe r words , 5 I ( , Ux ) + w ( 1 Uy ) = °- This equatio n i s often calle d the minimal surface equation. Perform - ing the indicate d derivative s put s i t int o th e for m (5') ( 1 + U 2 y )UXX - 2U x UyUXy + ( 1 + U^Uyy = 0 . We now give three examples of solutions of this interesting partia l differential equation notic e that w e are effectively ignorin g the initia l desire t o minimiz e surfac e are a filling in a curve . Example 1 (Plane) . Clearly , u(x,y) = Ax + By + C i s a solutio n of the minima l surfac e equation , whic h i s significant, bu t no t al l tha t interesting. / /
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