New progress in the inverse problem in the calculus of variations

TL;DR

This paper advances the understanding of the inverse problem in the calculus of variations for systems of second order ODEs in any dimension, introducing new theorems, solutions, and a classification scheme using exterior differential systems theory.

Contribution

It introduces a new class of solutions, new theorems, and a classification scheme for the inverse problem in arbitrary dimensions, generalizing previous solutions.

Findings

New solutions for the inverse problem in arbitrary dimensions

New theorems using exterior differential systems theory

Examples in dimensions 2, 3, and 4

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.