A variational principle for volume-preserving dynamics

TL;DR

This paper introduces a variational framework for describing volume-preserving (Liouville) dynamics on manifolds, using a maximal degree variational principle to characterize integral manifolds of the Liouville vector field.

Contribution

It presents a novel variational principle specifically tailored for Liouville vector fields, providing explicit formulas and coordinate-based descriptions.

Findings

Derives explicit formulae for the variational principle

Characterizes integral manifolds of volume-preserving flows

Provides a coordinate-based approach for practical applications

Abstract

We provide a variational description of any Liouville (i.e. volume preserving) autonomous vector fields on a smooth manifold. This is obtained via a ``maximal degree'' variational principle; critical sections for this are integral manifolds for the Liouville vector field. We work in coordinates and provide explicit formulae.