Question

(a) Show that, if y satisfies the Euler-Lagrange equation associated with the integral 2. qy2) dx, I = (6) where p() and q(x) are known functions, then I has the value 12 (b) Show that, if y satisfies the Euler-Lagrange equation associated with (6) and if z(x) is an arbitrary differentiable function for which z(x)z(r2) = 0, (7) 1 then (yyz)da= 0. + Hence show that by replacing y in (6) by the function (y + z), where the condition (7) ensures that y and (yz) have the same end-points, the value of I in increased by the nonnegative amount q222) dx Thus it is shown that the extremizing function y renders (6) an absolute minimum with respect to differen- tiable functions assuming the required end-point values yi and y2

Answer 1

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