Invariant measures for some dissipative systems from the Jacobi last multiplier

TL;DR

This paper derives explicit invariant phase-space measures for dissipative systems using Jacobi last multipliers, extending classical Hamiltonian volume preservation to more general systems.

Contribution

It provides new explicit formulas for invariant measures in dissipative systems like conformal, contact, and Liéard systems, broadening the scope of phase-space volume analysis.

Findings

Explicit invariant measures for dissipative systems derived

Application of Jacobi last multipliers to non-conservative systems

Extension of Liouville's theorem to certain dissipative dynamics

Abstract

Hamiltonian dynamics describing conservative systems naturally preserves the standard notion of phase-space volume, a result known as the Liouville's theorem which is central to the formulation of classical statistical mechanics. In this paper, we obtain explicit expressions for invariant phase-space measures for certain (generally dissipative) mechanical systems, namely, systems described by conformal vector fields on symplectic manifolds that are cotangent bundles, contact Hamiltonian systems, and systems of the Li\'enard class. The latter class of systems can be described by certain generalized conformal vector fields on the cotangent bundle of the configuration space. The computation of the invariant measures is achieved by making use of the formalism of Jacobi last multipliers.