Invariant measures for some dissipative systems from the Jacobi last multiplier

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 Liénard 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.