Conserved quantities and Hamiltonization of nonholonomic systems

  • Paula Balseiro

    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
  • Luis P. Yapu

    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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Abstract

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.

Cite this article

Paula Balseiro, Luis P. Yapu, Conserved quantities and Hamiltonization of nonholonomic systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 1, pp. 23–60

DOI 10.1016/J.ANIHPC.2020.05.003