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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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