Exactness of Lepage 2-forms and globally variational differential equations

TL;DR

This paper investigates the conditions under which Lepage 2-forms are exact, enabling the construction of global variational principles for certain differential systems, including geodesic equations in geometry.

Contribution

It establishes that locally variational systems with homogeneous functions of degree not equal to 0 or 1 are globally variational and introduces a new method to find global Lagrangians.

Findings

Homogeneous functions of degree c ≠ 0,1 lead to globally variational systems.

A constructive method for global Lagrangians is developed.

Geodesic equations in Riemann and Finsler geometry are included.

Abstract

The exactness equation for Lepage 2-forms, associated with variational systems of ordinary differential equations on smooth manifolds, is analyzed with the aim to construct a concrete global variational principle. It is shown that locally variational systems defined by homogeneous functions of degree $c \neq 0, 1$ are automatically globally variational. A new constructive method of finding a global Lagrangian is described for these systems, which include for instance the geodesic equations in Riemann and Finsler geometry.