Symmetry reduction and reconstruction in contact geometry and Lagrange-Poincar\'e-Herglotz equations

TL;DR

This paper develops a reduction framework for contact Lagrangian systems with symmetries, deriving explicit reduced equations called Lagrange-Poincare-Herglotz equations, and illustrates the approach with examples.

Contribution

It introduces a novel reduction method for contact systems using symmetry and quasi-velocities, leading to explicit coordinate expressions of the reduced equations.

Findings

Explicit coordinate expressions for reduced equations

Application to example systems demonstrating the framework

Enhanced understanding of symmetry reduction in contact geometry

Abstract

In this paper, we investigate the reduction process of a contact Lagrangian system whose Lagrangian is invariant under a group of symmetries. We give explicit coordinate expressions of the resulting reduced differential equations, the so-called Lagrange-Poincare-Herglotz equations. Our framework relied on the associated Herglotz vector field and its projected vector field, and the use of well-chosen quasi-velocities. Some examples are also discussed.