Contact Lagrangian systems subject to impulsive constraints

TL;DR

This paper develops a geometric framework for contact Lagrangian systems with impulsive forces and nonholonomic constraints, introducing new equations and theorems to analyze energy changes and system dynamics.

Contribution

It introduces the Herglotz equations for constrained contact systems and a Carnot-type theorem for impulsive forces, expanding the theoretical understanding of such systems.

Findings

Derived the dynamics using projectors and Riemannian metrics.

Formulated Herglotz equations for nonholonomic constraints.

Provided a Carnot-type theorem characterizing energy changes.

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 surface and a rolling sphere on a non-uniform surface, both with dissipation.