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

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.

Figures (8)

• Figure 1: Geometric structure of contact Hamiltonian dynamics. The plane represents the contact distribution $\zeta=\ker(\alpha)$, while the vertical arrow is the Reeb vector field $R=\partial_s$, which generates the non-conservative (dissipative) component of the dynamics. The blue arrow shows the tangential component $X_H^{\zeta}$ of the contact Hamiltonian vector field, lying entirely within $\zeta$, and the tilted black arrow depicts the full contact vector field $X_H$, whose vertical component encodes the irreversible contribution $R(H)$. This schematic illustrates the decomposition of contact dynamics into conservative and dissipative directions.
• Figure 2: Phase portrait of the harmonic oscillator illustrating the difference between conservative Hamiltonian dynamics ($\lambda = 0$) and contact dissipative dynamics with linear dissipation $\Gamma'(s)=\lambda>0$. The solid curves correspond to the Hamiltonian flow, which preserves the level sets of the mechanical nergy$H_0=\tfrac{1}{2}(p^2+q^2)$ and therefore yields closed orbits in the $(q,p)$-plane. The dashed curves correspond to the contact Hamiltonian flow, for which the momentum evolution $p\dot{} = -q - \lambda p$ induces exponential decay of mechanical energy. As predicted by the contact Hamiltonian equations, trajectories spiral toward the origin, demonstrating the geometric effect of dissipation generated by the Reeb component $R(H)=\lambda$.
• Figure 3: Exponential decay of the momentum map $J_{\xi}(t)$ under contact Hamiltonian dynamics. For a strictly invariant group action and a separated contact Hamiltonian $H(q,p,s)=H_{0}(q,p)+\Gamma(s)$, the evolution of the momentum map satisfies $\dot{J}_{\xi} = -\,\Gamma'(s)\,J_{\xi}$, yielding the explicit solution $J_{\xi}(t)=J_{\xi}(0)e^{-\lambda t}$ with $\lambda=\Gamma'(s)$. The solid curve ($\lambda=0$) represents the conservative Hamiltonian case, for which momenta are conserved. The dashed curves ($\lambda>0$) illustrate the strictly dissipative contact case, where the Reeb contribution $R(H)=\lambda$ drives exponential decay of all momentum components, reflecting the loss of symmetry-induced conservation laws.
• Figure 4: Trajectories of the dissipative rigid body dynamics $\dot M = M\times\Omega - \gamma M$ with $\Omega = I^{-1}M$ are shown in $(M_1,M_2,M_3)$-space for initial conditions near each principal axis. Solutions starting near the smallest and largest inertia axes remain close to those directions as they decay toward the origin, while trajectories near the intermediate axis peel away and migrate toward a stable axis, illustrating the classical stability structure of rigid-body rotation. Dissipation drives all trajectories toward the origin, reflecting the monotone decay of the rotational kinetic energy $H_0 = \tfrac{1}{2} M\cdot I^{-1}M$ under the contact-type damping term $-\gamma M$.
• Figure 5: Equilibria of the dissipative rigid body $\dot M = M\times\Omega - \gamma M$ with $\Omega = I^{-1}M$ lie on the principal axes of inertia. The smallest and largest inertia axes ($M_1$ and $M_3$) correspond to stable equilibria, while the intermediate axis ($M_2$) yields an unstable equilibrium, consistent with the classical rigid-body stability structure. The diagram shows stable (green) and unstable (red) equilibrium points on the momentum sphere, clearly illustrating the effect of the contact-type damping term on the qualitative dynamics.