Invariant measures for contact Hamiltonian systems: symplectic sandwiches with contact bread

Abstract

We prove that, under some natural conditions, Hamiltonian systems on a contact manifold $C$ can be split into a Reeb dynamics on an open subset of $C$ and a Liouville dynamics on a submanifold of $C$ of codimension 1. For the Reeb dynamics we find an invariant measure. Moreover, we show that, under certain completeness conditions, the existence of an invariant measure for the Liouville dynamics can be characterized using the notion of a symplectic sandwich with contact bread.

arXiv comment21 pages, no figures, it matches the final version accepted in JPA

References in corpus (20)

• Contact Hamiltonian Mechanics
• Contact Hamiltonian Systems
• Cosymplectic and contact structures to resolve time-dependent and dissipative hamiltonian systems
• New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries
• Contact manifolds and dissipation, classical and quantum
• Infinitesimal symmetries in Contact Hamiltonian systems
• A contact geometry framework for field theories with dissipation
• A thermostat algorithm generating target ensembles
• Contact variational integrators
• From an Action Principle for Action-dependent Lagrangians toward non-conservative Gravity: accelerating Universe without dark energy
• Liouville's Theorem and the canonical measure for nonconservative systems from contact geometry
• Completely Integrable Contact Hamiltonian Systems and Toric Contact Structures on $S^2\times S^3$
• Dynamical Similarity
• Remarks on Contact and Jacobi Geometry
• Unimodularity and preservation of volumes in nonholonomic mechanics
• Scalar Fields and the FLRW Singularity
• Numerical integration in celestial mechanics: a case for contact geometry
• Extended Hamilton-Jacobi theory, contact manifolds and integrability by quadratures
• Covariant Variational Evolution and Jacobi Brackets: Fields
• Covariant Variational Evolution and Jacobi Brackets: Particles

Cited by in corpus (12)

• Multicontact formulation for non-conservative field theories
• Hamilton-Jacobi theory and integrability for autonomous and non-autonomous contact systems
• Higher-order contact mechanics
• Nonautonomous k-contact field theories
• Generalized nonconservative gravitational field equations from Herglotz action principle
• Herglotz Action for Homogeneous Cosmology
• Inverse problem and equivalent contact systems
• Practical Introduction to Action-Dependent Field Theories
• Contact Lie systems on Riemannian and Lorentzian spaces: from scaling symmetries to curvature-dependent reductions
• Invariant measures for some dissipative systems from the Jacobi last multiplier
• Constrained Lagrangian dissipative contact dynamics
• Equilibria and bifurcations in contact dynamics