The inverse problem in the calculus of variations: new developments

We deal with the problem of determining the existence and uniqueness of Lagrangians for systems of n second order ordinary differential equations. A number of recent theorems are presented, using exterior differential systems theory (EDS). In particular, we indicate how to generalise Jesse Douglas’s famous solution for n = 2. We then examine a new class of solutions in arbitrary dimension n and give some non-trivial examples in dimension 3.

A VARIATIONAL PROBLEM FOR CURVES ON FINSLER SURFACES

We study the variational problem for $N$-parallel curves on a Finsler surface by means of exterior differential systems using Griffiths’ method. We obtain the conditions when these curves are extremals of a length functional and write the explicit form of Euler–Lagrange equations for this type of variational problem.