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arXiv · 1412.1810

New progress in the inverse problem in the calculus of variations

Thoan Do·Geoff Prince
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Abstract

We present a new class of solutions for the inverse problem in the calculus of variations in arbitrary dimension $n$. This is the problem of determining the existence and uniqueness of Lagrangians for systems of $n$ second order ordinary differential equations. We also provide a number of new theorems concerning the inverse problem using exterior differential systems theory (EDS). Concentrating on the differential step of the EDS process, our new results provide a significant advance in the understanding of the inverse problem in arbitrary dimension. In particular, we indicate how to generalise Jesse Douglas's famous solution for $n=2$. We give some non-trivial examples in dimensions 2,3 and 4. We finish with a new classification scheme for the inverse problem in arbitrary dimension.

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Thoan Do, Geoff Prince. 2014-12-04. New progress in the inverse problem in the calculus of variations. https://doi.org/10.1016/j.difgeo.2016.01.005

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