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The inverse problem in the calculus of variations: new developments

Thoan Do; Geoff Prince — 2021

Communications in Mathematics

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.

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