### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 018, 14 pages      arXiv:1510.08314      https://doi.org/10.3842/SIGMA.2016.018
Contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti

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

Paula Balseiro a and Nicola Sansonetto b
a) Universidade Federal Fluminense, Instituto de Matemática, Rua Mario Santos Braga S/N, 24020-140, Niteroi, Rio de Janeiro, Brazil
b) Università degli Studi di Padova, Dipartimento di Matematica, via Trieste 64, 35121 Padova, Italy

Received October 29, 2015, in final form February 12, 2016; Published online February 21, 2016

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.

Key words: nonholonomic systems; Lie group symmetries; first integrals; gauge symmetries and gauge momenta.

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