A Geometric Characterization of Certain First Integrals for Nonholonomic Systems with Symmetries

TL;DR

This paper provides a geometric framework for understanding first integrals in nonholonomic systems with symmetries, introducing the concept of $\

Contribution

It introduces the $\

Findings

Predicts the number of gauge momenta in symmetric nonholonomic systems.

Unifies previous approaches to first integrals via the $\

paper_type":"theoretical"}}

Abstract

We study the existence of first integrals in nonholonomic systems with symmetry. First we define the concept of $\mathcal{M}$-cotangent lift of a vector field on a manifold $Q$ in order to unify the works [Balseiro P., Arch. Ration. Mech. Anal. 214 (2014), 453-501, arXiv:1301.1091], [Fass\`o F., Ramos A., Sansonetto N., Regul. Chaotic Dyn. 12 (2007), 579-588], and [Fass\`o F., Giacobbe A., Sansonetto N., Rep. Math. Phys. 62 (2008), 345-367]. Second, we study gauge symmetries and gauge momenta, in the cases in which there are the symmetries that satisfy the so-called vertical symmetry condition. Under such condition we can predict the number of linearly independent first integrals (that are gauge momenta). We illustrate the theory with two examples.