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Hamel’s Formalism for Infinite-Dimensional Mechanical Systems

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Abstract

In this paper, we introduce Hamel’s formalism for infinite-dimensional mechanical systems and in particular consider its applications to the dynamics of nonholonomically constrained systems. This development is a nontrivial extension of its finite-dimensional counterpart. The analysis is applied to several continuum mechanical systems of interest, including coupled systems and systems with infinitely many constraints.

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Notes

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

  2. The map i satisfies the following property: A map \(f:N\rightarrow Q\) is smooth if and only if \(i\circ f:N\rightarrow M\) is smooth. Note that Q is usually not a submanifold of M. See Kriegl and Michor (1997) for details.

  3. Here, U can be thought of as an open subset of a convenient space.

  4. Each vector from \(T_q M\) can be represented this way.

  5. If \(\frac{\delta L}{\delta q}-\frac{d}{\mathrm{d}t}\frac{\delta L}{\delta \dot{q}}\ne 0\) at some \(t_0\), and \(i_{*}T_{q(t_0)} Q\) is dense in \(T_{q(t_0)} M\), there exists \(X\in T_{q(t_0)} Q\) such that \(\big \langle \frac{\delta L}{\delta q}-\frac{d}{\mathrm{d}t}\frac{\delta L}{\delta \dot{q}},X\big \rangle >0\). Using continuity, it is straightforward to construct a variation of the curve q(t) for which \(\int _a^b \big \langle \frac{\delta L}{\delta q}-\frac{d}{\mathrm{d}t}\frac{\delta L}{\delta \dot{q}},\delta q(t)\big \rangle \,\mathrm{d}t>0\), which is a contradiction.

  6. Nonsplitting closed subspaces already exist in Banach spaces; for more information on splitting subspaces, see Domański and Mastyło (2007) and references therein. The continuity of \(\pi ^{{\mathcal {D}}}\) in a Banach space is a consequence of the closed graph theorem. For more spaces with this property see Jarchow (1981).

  7. This holds if \(T_q M\) is a Fréchet space and A(q) has closed range, see Jarchow (1981) for details.

  8. Unlike the regular sleigh, for which \(\dot{\theta }= \mathrm {const}\).

  9. According to Kriegl and Michor (1997), all finite-dimensional Lie groups and all known infinite-dimensional Lie groups are regular.

  10. Here and below, the subscripts ‘c’ and ‘s’ stand for ‘convective’ and ‘spatial’, respectively.

  11. Often, \({\partial l}/{\partial \Omega }\) is viewed as an independent variable in these equations. See Cendra et al. (2001a; b), and Bloch et al. (2009) for the history, motivation, and finite-dimensional version of these equations.

  12. The orthogonal complement may not exist in nonHilbert spaces.

  13. This is called the dimension assumption in the finite-dimensional setting, see Bloch et al. (1996a) for details.

  14. If \(\mathcal S_q = \{0\}\), a set of nonholonomic constraints is said to be purely kinematic.

  15. Note that the intersection of two splitting subspaces may fail to be splitting already in a Banach space. For the intersection of two subspaces to be splitting, additional assumptions are necessary. According to Bill Johnson, asking that two subspaces are norm one complemented and the space itself is uniformly convex is sufficient. See http://mathoverflow.net/questions/85492/intersection-of-complemented-subspaces-of-a-banach-space for details.

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Acknowledgments

We would like to thank Professors Yongxin Guo, Francois Gay-Balmaz, Vakhtang Putkaradze, and Tudor Ratiu for valuable discussions, and the reviewers for helpful remarks.

The research of AMB was partially supported by NSF Grants DMS-1207893, DMS-1613819, INSPIRE-1363720, and the Simons Foundations. The research of DS was partially supported by the China Scholarship Council. DS wishes to thank support and hospitality of North Carolina State University during his visit. The research of YBK was partially supported by NDSEG Fellowship. The research of DVZ was partially supported by NSF grants DMS-0908995 and DMS-1211454. DVZ would like to acknowledge support and hospitality of the Beijing Institute of Technology, where a part of this work was carried out.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Donghua Shi

  2. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Yakov Berchenko-Kogan

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

    Dmitry V. Zenkov

  4. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Anthony M. Bloch

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  1. Donghua Shi
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  2. Yakov Berchenko-Kogan
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  4. Anthony M. Bloch
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Correspondence to Donghua Shi.

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Communicated by Paul Newton.

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Shi, D., Berchenko-Kogan, Y., Zenkov, D.V. et al. Hamel’s Formalism for Infinite-Dimensional Mechanical Systems. J Nonlinear Sci 27, 241–283 (2017). https://doi.org/10.1007/s00332-016-9332-7

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