# On a global Lagrangian construction for ordinary variational equations on 2-manifolds

## TL;DR

This paper presents a constructive method to find global Lagrangians for second-order ODEs on 2-manifolds, extending Takens' sheaf-theoretic result with explicit solutions using cohomology and Lepage forms.

## Contribution

It introduces a new constructive approach for obtaining global Lagrangians on 2-manifolds, based on solving exactness equations for Lepage 2-forms.

## Key findings

- Explicit method for global Lagrangian construction
- Applications to geometry and mechanics examples
- Extension of Takens' theorem to practical computation

## Abstract

Locally variational systems of differential equations on smooth manifolds, having certain de Rham cohomology group trivial, automatically possess a global Lagrangian. This important result due to Takens is, how-ever, of sheaf-theoretic nature. A new constructive method of finding a global Lagrangian for second-order ODEs on 2-manifolds is described on the basis of solvability of exactness equation for Lepage 2-forms, and the top-cohomology theorems. Examples from geometry and mechanics are discussed.