Characterization of symmetries of contact Hamiltonian systems

This paper explores the relationship between Cartan symmetries, dynamical similarities, and dynamical symmetries in contact Hamiltonian mechanics. By introducing an alternative decomposition of vector fields, we characterize these symmetries and present a novel description in terms of tensor densities. Furthermore, we demonstrate that this framework allows, under specific conditions, for the recovery of integrals of motion. We also establish new criteria to assess their independence.

💡 Research Summary

The paper investigates three notions of infinitesimal symmetry that appear in contact Hamiltonian mechanics – Cartan symmetries, dynamical similarities (often called scaling symmetries) and dynamical symmetries – and shows how they are interrelated through a novel decomposition of vector fields. The authors replace the usual horizontal‑vertical split of the tangent bundle with what they call the Hamiltonian‑horizontal decomposition: any vector field (\xi) can be uniquely written as (\xi = X_{\varphi_\xi} + \delta_\xi), where (\varphi_\xi = -\eta(\xi)) is a smooth function and (X_{\varphi_\xi}) is the contact Hamiltonian vector field generated by that function, while (\delta_\xi) lies entirely in the contact distribution (\ker\eta). Because the decomposition only involves the contact form through the 1‑form (\eta), it can be reformulated intrinsically using tensor densities, making the construction independent of any particular choice of contact form.

Using this framework the authors derive precise characterizations of each symmetry class. A vector field (Y) is a dynamical symmetry iff its Hamiltonian component (\varphi_Y) satisfies the “dissipated‑quantity” equation (X_H(\varphi_Y) = -R(H),\varphi_Y) (Theorem 4.5). In other words, (\varphi_Y) is not conserved in the usual sense but evolves proportionally to the Hamiltonian itself; such functions are called dissipated quantities. A dynamical similarity (or scaling symmetry) is defined by the condition (L_Y\eta = f,\eta) for some smooth function (f). Theorem 4.11 shows that, under this condition, (\varphi_Y) can be used to construct genuine integrals of motion, typically of the form (H^{-\alpha}) where (\alpha) is related to the scaling factor. Cartan symmetries satisfy (L_Y\eta = d\alpha) with an auxiliary function (\alpha); Theorem 4.20 reveals that (\alpha) can be expressed in terms of (\varphi_Y) and the Reeb vector field, thereby unifying Cartan symmetries with the Hamiltonian‑horizontal picture.

A major contribution of the work is the systematic treatment of dissipated quantities. The authors provide a practical test (Theorem 6.4) for the functional independence of a set of such quantities, based on the non‑vanishing of a determinant built from their derivatives. Moreover, Theorem 6.6 gives an algorithmic way to generate new dissipated quantities from existing scaling symmetries and to verify their independence, which is essential for establishing contact integrability.

The theoretical results are illustrated on several mechanical models: a damped harmonic oscillator, a thermodynamic system with friction, and a nonlinear friction model. In each case the Hamiltonian‑horizontal decomposition identifies the relevant symmetry, the associated dissipated quantity is computed, and the independence criteria are applied to confirm that the system possesses enough independent invariants to be considered integrable in the contact sense.

Overall, the paper unifies disparate definitions of symmetry in contact Hamiltonian dynamics, introduces an intrinsic tensor‑density formulation, and supplies concrete tools for extracting conserved or dissipated invariants and testing their independence. This advances both the geometric understanding of contact systems and provides a usable methodology for researchers dealing with dissipative mechanical or thermodynamic models.