New progress in the inverse problem in the calculus of variations

Thoan Do, Geoff Prince

Published 2014-12-04Version 1

We present a new class solutions for the inverse problem in the calculus of variations in arbitrary dimension n. We also provide a number of new theorems concerning the inverse problem using exterior differential systems theory. Our new techniques provide a significant advance in the understanding of the inverse problem in arbitrary dimension. We show that when the eigenvalues of a certain curvature tensor are distinct and with n-1 integrable eigen-distributions, the corresponding differential equations are variational only if the non-integrable eigenspace has a certain geometric property. The resulting Lagrangians depend on n-1 functions each of 2 variables. We give some non-trivial examples in dimension 3.