Find the (generalized Euler) equation which a function y(x) must satisify in order that the action(adsbygoogle = window.adsbygoogle || []).push({});

[tex] A = \int_{x_{1}}^{x_{2}} L(y(x),y'(x,),y''(x),x) dx [/tex] [/tex]

be stationary that is [itex] \delta A = 0 [/itex] for arbitrary variations [itex] \delta y(x) [/itex] such taht

[tex] \delta y(x) = \frac{d}{dx} \delta y(x)=0 \mbox{when} \ x=x_{1},x_{2} [/tex]

Hint: [tex] \delta A = \int_{x_{1}}^{x_{2}} \left(\frac{\partial L}{\partial y} \delta y + \frac{\partial L}{\partial y'} \delta y'+ \frac{\partial L}{\partial y''} \delta y'' \right) dx, \mbox{where} \ \delta y' = \frac{d}{dx} \delta y \mbox{ and } \delta y'' = \frac{d}{dx} \delta y' = \frac{d^2}{dx^2} \delta y [/tex]

by generalized Euler equation does the question ask the Euler Langrange equation?

well in the hint the first term is zero...

so we have

[tex] \delta A = \int_{x_{1}}^{x_{2}} \left(\frac{\partial L}{\partial y} \delta y + \frac{\partial L}{\partial y'} \delta y'+ \frac{\partial L}{\partial y''} \delta y'' \right) dx = \int_{x_{1}}^{x_{2}} \left(\frac{\partial L}{\partial y} \frac{d}{dx} \delta y + \frac{\partial L}{\partial y''} \frac{d}{dx} \delta y'\right) dx = 0 [/tex]

integration by parts so

[tex] \left[ {\frac{\partial L}{\partial y} \delta y} \right]_{x_{1}}^{x_{2}} - \left[\frac{\partial^2 L}{\partial y \partial x} \delta y} \right]_{x_{1}}^{x_{2}} + \left[ \frac{\partial L}{\partial y''} \delta y'} \right]_{x_{1}}^{x_{2}} - \left[ \frac{\partial^2 L}{\partial y'' \partial x} \delta y'} \right]_{x_{1}}^{x_{2}} = 0 [/tex]

not too sure about whrer this is going...

Will continued working of thie problem yield the EUler Lagrange equations?

Please help!

Thank you

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# Homework Help: Find the equation

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