1. Beltrami identity Consider the integral functional $J [y] = ?_{x_{1}}^{x_{2}} d x F (x; y (x), y^{?} (x)) .$ Show that when the integrand does not depend explicitly on $x$ , then $F ? y^{?} (x) \frac{? F}{? y ^{?}} = C$ where $C$ is an undetermined constant. Hint: Do not start from scratch. Use the Euler-Lagrange equation, and use that $\frac{d F}{d x} = \frac{? F}{? x} + \frac{? y}{? x} \frac{? F}{? y} + \frac{? y ^{?}}{? x} \frac{? F}{? y ^{?}}$

Question

1. Beltrami identity Consider the integral functional

J [y] = ?_{x_{1}}^{x_{2}} d x F (x; y (x), y^{?} (x)) .

Show that when the integrand does not depend explicitly on

x

, then

F ? y^{?} (x) \frac{? F}{? y ^{?}} = C

where

C

is an undetermined constant. Hint: Do not start from scratch. Use the Euler-Lagrange equation, and use that

\frac{d F}{d x} = \frac{? F}{? x} + \frac{? y}{? x} \frac{? F}{? y} + \frac{? y ^{?}}{? x} \frac{? F}{? y ^{?}}

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Expert Answer to -  1. Beltrami identity Consider the integral functional J[y]=x1x2dxF(x;y(x),y(x)). Sho

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Solution for -  1. Beltrami identity Consider the integral functional J[y]=x1x2dxF(x;y(x),y(x)). Sho

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This an additional answer to -  1. Beltrami identity Consider the integral functional J[y]=x1x2dxF(x;y(x),y(x)). Sho

(Solved): 1. Beltrami identity Consider the integral functional J[y]=x1x2dxF(x;y(x),y(x)). Sho ...