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Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 4

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 6 part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Solution 48.7 C1 Extremals. Without loss of generality we take the vertical line to be the y axis. We will consider x1 y1 1. With ds 1 y 2 dx we extremize the integral r . y y 2 dx. Jo Since the Lagrangian is independent of x we know that the Euler differential equation has a first integral. dx F Fy 0 V Fyy F - Fy 0 d d y Fy - F 0 dx y Fy F const For the given Lagrangian this is y y V xo vW1 y 2 const V1 y 2 y 2Vy Vy 1 y 2 constV1 y 2 ựỹ const 1 y 2 y const is one solution. To find the others we solve for y and then solve the differential equation. y a 1 y 2 _ y a y . V a dx J dy V y a 2094 x b 2 J a y a x2 bx b2 y 4a 2a 4a The natural boundary condition is Fy lx 0 ự yy o 0 a 1 y 2 x 0 y 0 0 The extremal that satisfies this boundary condition is y x a 4a Now we apply y x1 y1 to obtain for y1 x1. The value of the integral is x1 x2 12a2 12a3 2 By denoting y1 cx1 c 1 we have 1 a 2 cx1 xn c2 1 The values of the integral for these two values of a are v 2 xi 3 2 1 3c2 3 c2 1 3 c ực2 1 3 2 .