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I am looking for an explicit (preferably simple) example of an ODE with time-independent coefficients in $\mathbb{R}^3$ such that there does not exist an Euler-Lagrange equation $$\frac{\partial L}{\...

Some of the second order ODE can be considered as Euler-Lagrange equations for an appropriate Lagrangian. However this is true not for arbitrary second order equation. But some of important equations ...

I am having difficulty formulating a problem, which involves optimizing a contour shape, into a well-posed variational form that would give a reasonable answer. Within a bounded region on the $xy$ ...