The Cartan form for constrained Lagrangian systems and the nonholonomic Noether theorem

TL;DR

This paper develops a general framework using Cartan forms to analyze conservation laws and symmetries in nonholonomic mechanical systems, extending Noether's theorem to these constrained systems.

Contribution

It introduces a comprehensive Cartan form approach to relate symmetries and first integrals in nonholonomic systems, generalizing existing theories.

Findings

Established the most general relations between symmetries and first integrals for nonholonomic systems.

Extended the nonholonomic Noether theorem within the Cartan formalism.

Applied the formalism to Riemannian submanifolds and quadratic first integrals.

Abstract

This paper deals with conservation laws for mechanical systems with nonholonomic constraints. It uses a Lagrangian formulation of nonholonomic systems and a Cartan form approach. We present what we believe to be the most general relations between symmetries and first integrals. We discuss the so-called nonholonomic Noether theorem in terms of our formalism, and we give applications to Riemannian submanifolds, to Lagrangians of mechanical type, and to the determination of quadratic first integrals.