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Conservation of ‘Moving’ Energy in Nonholonomic Systems with Affine Constraints and Integrability of Spheres on Rotating Surfaces

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Abstract

Energy is in general not conserved for mechanical nonholonomic systems with affine constraints. In this article we point out that, nevertheless, in certain cases, there is a modification of the energy that is conserved. Such a function is the pull-back of the energy of the system written in a system of time-dependent coordinates in which the constraint is linear, and for this reason will be called a ‘moving’ energy. After giving sufficient conditions for the existence of a conserved, time-independent moving energy, we point out the role of symmetry in this mechanism. Lastly, we apply these ideas to prove that the motions of a heavy homogeneous solid sphere that rolls inside a convex surface of revolution in uniform rotation about its vertical figure axis, are (at least for certain parameter values and in open regions of the phase space) quasi-periodic on tori of dimension up to three.

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Notes

  1. L. García-Naranjo (private communication) and the referee pointed out to us that this mechanism may be used to explain the existence of an integral of motion of the Veselova system which was found by Fedorov (1989), Fedorov and Jovanović (2004); some remarks on this can also be found in the article (Borisov et al. 2015) that we mention in Remark ii below.

  2. They are in fact invariant under an action of \(\mathrm {SO(3)}\times S^1\), but this is not used in the quoted references.

  3. Here and in the sequel we prefer writing the old coordinates as functions of the new. Therefore, the pull-back under the change ‘old \(\rightarrow \) new’ coordinates, that we have mentioned above, becomes the push-forward under the change ‘new \(\rightarrow \) old’ coordinates, and vice versa.

  4. A large class of time-dependent holonomic constraints for systems of N material points can be modeled in this way. After the choice of a reference frame, that provides a (time-dependent) identification of physical 3-space with \(\mathbb {R}^{3}\), a time-dependent holonomic constraint is given by a time-dependent embedding of a manifold Q into the configuration space \((\mathbb {R}^{3})^N\) of the unconstrained system, and the Lagrangian is the restriction of the Lagrangian of the unconstrained system to the resulting time-dependent, extended submanifold.

  5. A fact which is well known in the theory of time-dependent canonical transformations.

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Acknowledgments

We thank Enrico Pagani for a useful conversation and Larry Bates for suggesting the term ‘moving’ energy. We also thank the anonymous referee for her/his useful comments.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121, Padua, Italy

    Francesco Fassò & Nicola Sansonetto

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  1. Francesco Fassò
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  2. Nicola Sansonetto
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Correspondence to Francesco Fassò.

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Communicated by Anthony Bloch.

Nicola Sansonetto: Supported by the research project Symmetries and integrability of nonholonomic mechanical systems of the University of Padova.

This work is part of the research projects Symmetries and integrability of nonholonomic mechanical systems of the University of Padova and PRIN Teorie geometriche e analitiche dei sistemi Hamiltoniani in dimensioni finite e infinite.

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Fassò, F., Sansonetto, N. Conservation of ‘Moving’ Energy in Nonholonomic Systems with Affine Constraints and Integrability of Spheres on Rotating Surfaces. J Nonlinear Sci 26, 519–544 (2016). https://doi.org/10.1007/s00332-015-9283-4

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  • DOI: https://doi.org/10.1007/s00332-015-9283-4

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