Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations

TL;DR

This paper develops global Helmholtz conditions using semi-basic 1-forms to determine when a semispray is locally Lagrangian, linking classical multiplier matrix methods with modern geometric approaches.

Contribution

It introduces a new geometric framework using semi-basic 1-forms to characterize the inverse problem of the calculus of variations, extending classical Helmholtz conditions.

Findings

Helmholtz conditions expressed via semi-basic 1-forms

Relation between these conditions and classical multiplier matrix formulation

Special cases for 1-homogeneous and 0-homogeneous semi-basic 1-forms

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.