Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations
Differential Geometry
2009-08-12 v1 Mathematical Physics
math.MP
Abstract
We use Fr\"olicher-Nijenhuis theory to obtain global Helmholtz conditions, expressed in terms of a semi-basic 1-form, that characterize when a semispray is locally Lagrangian. We also discuss the relation between these Helmholtz conditions and their classic formulation written using a multiplier matrix. When the semi-basic 1-form is 1-homogeneous (0-homogeneous) we show that two (one) of the Helmholtz conditions are consequences of the other ones. These two special cases correspond to two inverse problems in the calculus of variation: Finsler metrizability for a spray, and projective metrizability for a spray.
Keywords
Cite
@article{arxiv.0903.1169,
title = {Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations},
author = {Ioan Bucataru and Matias F. Dahl},
journal= {arXiv preprint arXiv:0903.1169},
year = {2009}
}