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Write the Euler-Lagrange equations for $L(x,y,z,y',z',y'',z'',y''',z''',...,y^{(k)},z^{(k)})\,$

The equations will be written with the same pattern, as a system for more than one function.

$\frac{\partial L}{\partial y} - \frac{d}{dx}\frac{\partial L}{\partial y'} + \frac{d^2}{dx^2}\frac{\partial L}{\partial y''} - \frac{d^3}{dx^3}\frac{\partial L}{\partial y'''} +...+(-1)^k\frac{d^k}{dx^k}\frac{\partial L}{\partial y^{(k)}}=0\,$

$\frac{\partial L}{\partial z} - \frac{d}{dx}\frac{\partial L}{\partial z'} + \frac{d^2}{dx^2}\frac{\partial L}{\partial z''} - \frac{d^3}{dx^3}\frac{\partial L}{\partial z'''} +...+(-1)^k\frac{d^k}{dx^k}\frac{\partial L}{\partial z^{(k)}}=0\,$

Calculus of Variations

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