Find the two coupled Euler-Lagrange equations giving the stationarypath of the functionalS[v₁.92] = [ da [y?² + 2y2 + (2m2 + v⁄2)²].

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Answered: Find the two coupled Euler-Lagrange…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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- 11) Calculate the Jacobian, J, for the change of variables x = u cos(0) – v sin(e) and y = usin(0) + v cos(0).
Exercise 6.7 (E) Show that in polar coordinates the functional ·b S[y] = [" da √/x² + y² √/1+y'(a)²_becomes_S[r] = " de r Va and that the resulting Euler-Lagrange equation is 3 dr d²r d0² r de Hence show that equations for the stationary paths are 2 - 2r=0 which can be written as d² d0² r² + r² (0)² (-) + - 0. = 1 = A cos 20+ B sin 20 or A (x²- y²) + 2Bxy = 1, p2 where A and B are constants and 0 ≤ 0 < π.
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Find the two coupled Euler-Lagrange equations giving the stationary path of the functional S[y₁,92] = [ da [y² + 2y² + (2y1 + 32)²].