Consider the functional บ S[y] = √ d dx F(x,y,y), y(a) = A, where the right-hand end point v is determined so that the path lies on the curve 7(x, y) = 0, i.e. t(v, y(v)) = 0, and a and A are constants. Consider a perturbed path y₁ = y + ch, where h is an admissible perturbation, i.e. 0 and T(Ve, Ye(Ve)) = 0, where v₁ = v + €§ +0(€²) is the = satisfying h(a) perturbed value of v. Show that to first order in €, Ye(ve) = y(v) + €(y' (v) § + h(v)) + O(€²), and hence that at (x, y) = (v, y(v)), §(Tz+y' (v) Ty) + h(v) Ty = 0.

Consider the functional บ S[y] = √ d dx F(x,y,y), y(a) = A, where the right-hand end point v is determined so that the path lies on the curve 7(x, y) = 0, i.e. t(v, y(v)) = 0, and a and A are constants. Consider a perturbed path y₁ = y + ch, where h is an admissible perturbation, i.e. 0 and T(Ve, Ye(Ve)) = 0, where v₁ = v + €§ +0(€²) is the = satisfying h(a) perturbed value of v. Show that to first order in €, Ye(ve) = y(v) + €(y' (v) § + h(v)) + O(€²), and hence that at (x, y) = (v, y(v)), §(Tz+y' (v) Ty) + h(v) Ty = 0.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section: Chapter Questions

Problem 12T

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