Geometric Formulation of Combined Conservative Dissipative Mechanics via Contact Hamiltonian Dynamics Symmetries, Reduction, and Variational Integrators

TL;DR

This paper introduces a geometric framework using contact Hamiltonian dynamics to model, analyze, and numerically simulate systems that combine conservative and dissipative behaviors, with applications to rigid body dynamics.

Contribution

It develops a unified contact geometric formulation for mixed conservative-dissipative systems, including new integrators that preserve geometric structure and accurately model energy decay.

Findings

Explicit laws for dissipation effects on symmetry and momentum

Construction of a second-order contact variational integrator

Numerical experiments showing accurate energy decay and geometric consistency

Abstract

We develop a unified geometric framework for mechanical systems that combine conservative and dissipative dynamics by formulating them on contact manifolds. Within this setting, we identify the Reeb vector field as the intrinsic generator of irreversibility and derive explicit laws describing how dissipation modifies symmetry reduction and momentum evolution. As a concrete application, we construct the contact Hamiltonian formulation of the rigid body with isotropic and anisotropic damping, classify all equilibrium configurations, and analyze their stability. Building on this continuous formulation, we design a second-order structure preserving contact variational integrator obtained by a symmetric splitting of kinetic, potential, and dissipative components. Numerical experiments for representative dissipative systems demonstrate accurate energy decay, geometric consistency, and recovery of the symplectic Verlet scheme in the conservative limit. The proposed framework provides a coherent connection between the geometry of contact dynamics, physical irreversibility, and numerically stable integration, offering new tools for the analysis of mixed conservative dissipative mechanical systems.