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Hamiltonization of Solids of Revolution Through Reduction

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Abstract

In this paper, we study the relation between conserved quantities of nonholonomic systems and the hamiltonization problem employing the geometric methods of Balseiro (Arch Ration Mech Anal 214:453–501, 2014) and Balseiro and Garcia-Naranjo (Arch Ration Mech Anal 205(1):267–310, 2012). We illustrate the theory with classical examples describing the dynamics of solids of revolution rolling without sliding on a plane. In these cases, using the existence of two conserved quantities we obtain, by means of gauge transformations and symmetry reduction, genuine Poisson brackets describing the reduced dynamics.

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Notes

  1. If \(\pi \) is a bivector field on \({\mathcal {M}}\), the Schouten bracket \([\pi ,\pi ]\) is a 3-vector field such that, for \(f,g,h\in C^\infty ({\mathcal {M}})\), \(\tfrac{1}{2}[\pi ,\pi ](\text {d}f,\text {d}g,\text {d}h) = \text {cyclic} [\, \{f,\{g,h\}\}\, ] \), where \(\{\cdot ,\cdot \}\) is the bracket associated with \(\pi \) (and where “\(\text {cyclic} [\, \cdot \, ]\)” denotes the cyclic sum), see Marsden and Ratiu (2002).

  2. We are denoting by \(\mathbf{X}, \partial _{a_1}, \partial _{a_2}\) vector fields on Q and also vector fields on \({\mathcal {M}}\). When it is clear from the context, we will also denote by \(\varvec{\lambda }\) and \(\epsilon ^1, \epsilon ^2\) the 1-forms on \({\mathcal {M}}\) that are the pull backs of the corresponding 1-forms on Q.

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Acknowledgements

I thank CNPq (Brazil) for supporting this project. I am grateful to Richard Cushman, Jedrzej Sniatycki, Larry Bates, Nicola Sansonetto and Alessia Mandini for stimulating conversations. I thank Dmitry Zenkov for the invitation to the CMS meeting in Edmonton in July 2016, where part of this work was presented. I am especially indebted to Luis Garcia-Naranjo for inspiring discussions, particularly concerning the symmetry group for the Routh sphere (Sect. 4.1) ; his joint work with García-Naranjo and Montaldi (2016) contains results related to ours, but independently obtained, where the horizontal gauge momenta become Casimirs of an alternative reduced bracket.

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  1. Universidade Federal Fluminense, Instituto de Matemática e Estatística, Rua Professor Marcos Waldemar de Freitas Reis, s/n, Bloco H - Campus do Gragoatá, Niterói, Rio de Janeiro, 24020-140, Brazil

    Paula Balseiro

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  1. Paula Balseiro
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Communicated by Anthony Bloch.

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Balseiro, P. Hamiltonization of Solids of Revolution Through Reduction. J Nonlinear Sci 27, 2001–2035 (2017). https://doi.org/10.1007/s00332-017-9394-1

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