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Hamel’s Formalism and Variational Integrators

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Geometry, Mechanics, and Dynamics

Part of the book series: Fields Institute Communications ((FIC,volume 73))

Abstract

Hamel’s formalism is a representation of Lagrangian mechanics obtained by measuring the velocity components relative to a frame that generically is not induced by configuration coordinates. The use of this formalism often leads to a simpler representation of dynamics. Utilizing the variational discretization approach, this paper develops a discrete Hamel’s formalism with applications to nonholonomic integrators.

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Notes

  1. 1.

    If Q is a Lie group, this formula is derived in Bloch, Krishnaprasad, Marsden, and Ratiu\ [4].

  2. 2.

    Constraints are nonholonomic if and only if they cannot be rewritten as position constraints.

  3. 3.

    For a noncommutative symmetry group, L depends on \((s,\dot{s})\) through the combination \(s^{-1}\dot{s}\).

  4. 4.

    The general noncommutative setting is not studied in this paper and will be the subject of a future publication.

  5. 5.

    The stability analysis of relative equilibria of nonholonomic systems has a long history, starting form the results of Walker [38] and Routh [36]; see [39] for some of this history and for the energy-momentum method for nonholonomic systems.

  6. 6.

    Equations (34) were derived in [25, 26] without the use of the discrete Hamel formalism.

  7. 7.

    See [33, 37], and [2] for the Suslov top.

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Acknowledgements

The authors would like to thank Jerry Marsden for his inspiration and helpful discussions in the beginning of this work, and the reviewers for useful remarks. The research of DVZ was partially supported by NSF grants DMS-0604108, DMS-0908995 and DMS-1211454. The research of KRB was partially supported by NSF grants DMS-0604108 and DMS-0908995. DVZ would like to acknowledge support and hospitality of Mathematisches Forschungsinstitut Oberwolfach, the Fields Institute, and the Beijing Institute of Technology, where a part of this work was carried out.

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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Kenneth R. Ball & Dmitry V. Zenkov

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  1. Kenneth R. Ball
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Correspondence to Dmitry V. Zenkov .

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Editors and Affiliations

  1. Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada

    Dong Eui Chang

  2. Department of Mathematics, Imperial College London, London, United Kingdom

    Darryl D. Holm

  3. Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

    George Patrick

  4. Section de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Tudor Ratiu

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Ball, K.R., Zenkov, D.V. (2015). Hamel’s Formalism and Variational Integrators. In: Chang, D., Holm, D., Patrick, G., Ratiu, T. (eds) Geometry, Mechanics, and Dynamics. Fields Institute Communications, vol 73. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2441-7_20

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