Variational principle and phase space measure in non-canonical coordinates

TL;DR

This paper derives non-canonical equations of motion from a variational principle, explores the geometry of phase space under non-canonical transformations, and discusses implications for invariant measures and entropy.

Contribution

It provides a covariant expression for phase space measure and entropy in non-canonical coordinates based on a variational principle.

Findings

Invariant measure derived from non-canonical transformations

Phase space geometry is non-trivial despite incompressible dynamics

Covariant entropy expression established in non-canonical coordinates

Abstract

Non-canonical equations of motion are derived from a variational principle written in symplectic form. The invariant measure of phase space and the covariant expression for the entropy are derived from non-canonical transformations of coordinates. This shows that the geometry of non-canonical phase space is non trivial even if dynamics has no compressibility.