Conserved quantities and Hamiltonization of nonholonomic systems

Balseiro, Paula; Yapu, Luis P.

doi:10.1016/j.anihpc.2020.05.003

Balseiro, Paula ¹ ; Yapu, Luis P.¹

¹ Universidade Federal Fluminense, Instituto de Matemática e Estatística, Rua Prof. Marcos Waldemar de Freitas Reis S/N (Campus do Gragoatá), CEP 24210-201, Niteroi, Rio de Janeiro, Brazil

Annales de l'I.H.P. Analyse non linéaire, janvier – février 2021, Tome 38 (2021) no. 1, pp. 23-60.

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Résumé

This paper studies hamiltonization of nonholonomic systems using geometric tools, building on [1,5]. The main novelty in this paper is the use of symmetries and suitable first integrals of the system to explicitly define a new bracket on the reduced space that codifies the nonholonomic dynamics and carries, additionally, an almost symplectic foliation (determined by the common level sets of the first integrals); in particular cases of interest, this new bracket is a Poisson structure that hamiltonizes the system. Our construction of the new bracket is based on a gauge transformation of the nonholonomic bracket by a global 2-form that we explicitly describe. We study various geometric features of the reduced brackets and apply our formulas to obtain a geometric proof of the hamiltonization of a homogeneous ball rolling without sliding in the interior side of a convex surface of revolution.

DOI : 10.1016/j.anihpc.2020.05.003

Mots-clés : Nonholonomic systems, Hamiltonization, Geometric mechanics, Poisson brackets

Affiliations des auteurs :

Balseiro, Paula ¹ ; Yapu, Luis P. ¹

¹ Universidade Federal Fluminense, Instituto de Matemática e Estatística, Rua Prof. Marcos Waldemar de Freitas Reis S/N (Campus do Gragoatá), CEP 24210-201, Niteroi, Rio de Janeiro, Brazil

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Balseiro, Paula; Yapu, Luis P. Conserved quantities and Hamiltonization of nonholonomic systems. Annales de l'I.H.P. Analyse non linéaire, janvier – février 2021, Tome 38 (2021) no. 1, pp. 23-60. doi : 10.1016/j.anihpc.2020.05.003. http://www.numdam.org/articles/10.1016/j.anihpc.2020.05.003/

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