Liouville structures

TL;DR

This paper provides a detailed definition and analysis of Liouville structures, emphasizing their importance in variational formulations of physical theories and illustrating their properties with examples from mechanics.

Contribution

It offers a precise definition of Liouville structures, explores their properties, and presents examples relevant to mechanics, highlighting their unique role beyond cotangent fibrations.

Findings

Liouville structures are isomorphic to cotangent vector fibrations.

They cannot be replaced by cotangent fibrations in physical theories.

Examples demonstrate their application in variational mechanics.

Abstract

A 'Liouville structure' is a structure isomorphic to a cotangent vector fibration. A Liouville structure is an essential ingredient of every variational formulation of a physical theory. For reasons of interpretation the Liouville structure can not be replaced by the corresponding cotangent fibration. We give a precise definition of Liouville structures, study their properties, and give examples used in variational formulations of mechanics.