Characterization of symmetries of contact Hamiltonian systems
Federico Zadra, Marcello Seri
TL;DR
This work addresses how Cartan symmetries, dynamical similarities, and dynamical symmetries interrelate in contact Hamiltonian systems. It introduces a Hamiltonian–horizontal decomposition and an intrinsic tensor-density description that ties symmetry classes to Jacobi structure, enabling the recovery of dissipated quantities and, under certain conditions, integrals of motion. Key results include necessary and sufficient conditions for dynamical and Cartan symmetries, the role of scaling in yielding conserved/dissipated quantities, and practical tools for assessing independence and constructing new integrals, all illustrated through mechanical examples such as damped oscillators and dissipative particles. The framework provides new geometric and algebraic tools to study the dynamics and integrability of contact Hamiltonian systems, with potential implications for dissipative mechanics and beyond.
Abstract
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.