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Question 3 22 marks Let 0

Question 3 22 marks Let 0 < a < b. Consider the functional Sly] = (a) Show that the EulerLagrange equation for Sly] may be written as (12 y 5 dy dx2 a: nr (b) Show that S[y] is invariant under the scale transformation cur ,By, where a and /3 are constants satisfying a2 = 1. (c) Deduce that a first-integral of Sly] is 4:r5yyt + (y'2 -h {!J3) where c is constant, stating any theorems that you use. (d) Let A and B be constants. By direct substitution or otherwise, show that the functions 2 and Y2 = both give solutions of the first-integral for any A and B, but that only one of and Y2 satisfies the EulerLagrange equation for arbitrary