Symmetries and Dissipation Laws on Contact Systems

In this article, we focus on the formulation of dissipative mechanical systems through contact Hamiltonian systems. Different forms of symmetry of a contact dynamical system (geometric, dynamic, and gage) are defined to, in the realm of Noether, find their corresponding dissipated quantities. We also address the existence of dissipated quantities associated with a general vector field X on \(TQ\times \mathbb {R},\) $TQ\times \mathbb{R},$ focusing on the case where its contact Hamiltonian function is dissipative.