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Consider the following particular case of the inverse problem of the calculus of variations: given a system of second-order equations $$ \ddot{q}^i = f^i(q, \dot{q}, t), \quad i = 1, \dots, n, \label{...

I was recently looking into generalisations of the brachistochrone problem: for example, in this article the authors study the brachistochrone with Amontons-Coulomb friction where a bead slides along ...

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 ...

Let's assume we have a continuous and finite scalar function $f(x,y)$ over the $xy$ plane ($\mathbb{R}^{2}$) and this function is to be integrated over a bounded area (surface) $A\subset\mathbb{R}^{2}...

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 $xy$ ...