Contact Lagrangian systems subject to impulsive constraints

Abstract

We describe geometrically contact Lagrangian systems under impulsive forces and constraints, as well as instantaneous nonholonomic constraints which are not uniform along the configuration space. In both situations, the vector field describing the dynamics of a contact Lagrangian system is determined by defining projectors to evaluate the constraints by using a Riemannian metric. In particular, we introduce the Herglotz equations for contact Lagrangian systems subject to instantaneous nonholonomic constraints. Moreover, we provide a Carnot-type theorem for contact Lagrangian systems subject to impulsive forces and constraints, which characterizes the changes of energy due to contact-type dissipation and impulsive forces. We illustrate the applicability of the method with practical examples, in particular, a rolling cylinder on a springily plane and a rolling sphere on a non-uniform plane, both with dissipation.

Generalized virial theorem for contact Hamiltonian systems

“…However, unlike their symplectic counterparts, Hamiltonian dynamics on a contact manifold is neither volume preserving, nor does it conserve the corresponding Hamiltonian function along the evolution. Consequently, contact Hamiltonian dynamics has found applications in describing dissipative mechanical systems [17][18][19][20][21][22][23][24], as well as thermostat problems where the system interacts with an environment [25]. It has also found applications in reversible [26][27][28][29][30][31][32][33][34][35] as well as irreversible thermodynamics [36][37][38].…”