Helmholtz conditions and symmetries for the time dependent case of the inverse problem of the calculus of variations

TL;DR

This paper reformulates the inverse calculus of variations problem for time-dependent second order ODEs using Fr"olicher-Nijenhuis theory, providing global Helmholtz conditions and linking semi-basic 1-forms to symmetries and integrals.

Contribution

It introduces a new geometric framework for the inverse problem, characterizing Lagrangian systems via semi-basic 1-forms and Helmholtz conditions, and relates these forms to symmetries and conserved quantities.

Findings

Global Helmholtz conditions characterize Lagrangian systems.

Existence of specific semi-basic 1-forms implies symmetries.

Connection between semi-basic 1-forms and first integrals.

Abstract

We present a reformulation of the inverse problem of the calculus of variations for time dependent systems of second order ordinary differential equations using the Fr\"olicher-Nijenhuis theory on the first jet bundle, $J^1\pi$. We prove that a system of time dependent SODE, identified with a semispray $S$, is Lagrangian if and only if a special class, $\Lambda^1_S(J^1\pi)$, of semi-basic 1-forms is not empty. We provide global Helmholtz conditions to characterize the class $\Lambda^1_S(J^1\pi)$ of semi-basic 1-forms. Each such class contains the Poincar\'e-Cartan 1-form of some Lagrangian function. We prove that if there exists a semi-basic 1-form in $\Lambda^1_S(J^1\pi)$, which is not a Poincar\'e-Cartan 1-form, then it determines a dual symmetry and a first integral of the given system of SODE.