Contracting Forced Lagrangian and Contact Lagrangian Systems: application to nonholonomic systems with symmetries

TL;DR

This paper develops criteria to identify contracting behavior in mechanical systems on Riemannian manifolds, demonstrating stability implications for dissipative, contact, and nonholonomic systems.

Contribution

It introduces a sufficient condition for contraction in general Riemannian manifolds and applies it to analyze stability in specific mechanical systems with symmetries.

Findings

Dissipative forced mechanical systems are shown to be contracting.

Contracting systems exhibit stability properties.

Application to contact and nonholonomic systems confirms stability results.

Abstract

In this paper we address the problem of identifying contracting systems among dynamical systems appearing in mechanics. First, we introduce a sufficient condition to identify contracting systems in a general Riemannian manifold. Then, we apply this technique to establish that a particular type of dissipative forced mechanical system is contracting, while stating immediate consequences of this fact for the stability of these systems. Finally, we use the previous results to study the stability of particular types of Contact and Nonholonomic systems.