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], [Fassò F., Ramos A., Sansonetto N., Regul. Chaotic Dyn.12 (2007), 579–588], and [Fassò 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.
Keywords:nonholonomic systems; Lie group symmetries; first integrals; gauge symmetries and gauge momenta.