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On the Completeness of Trajectories for Some Mechanical Systems

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Geometry, Mechanics, and Dynamics

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

Abstract

The classical tools which ensure the completeness of both, vector fields and second order differential equations for mechanical systems, are revisited. Possible extensions in three directions are discussed: infinite dimensional Banach (and Hilbert) manifolds, Finsler metrics and pseudo-Riemannian spaces, the latter including links with some relativistic spacetimes. Special emphasis is taken in the cleaning up of known techniques, the statement of open questions and the exploration of prospective frameworks.

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Notes

  1. 1.

    In particular, our Banach manifolds will be always regular and, so, some difficulties pointed out by Palais in [53] (see Sect. 2 including the Appendix therein), will not apply. The central role of paracompactness from the topological viewpoint is stressed in Figure 1. Notice that, as a difference with the finite dimensional case, second countability does not imply paracompactness (see for example [46], [42, Sect. 27.6] or [53]).

  2. 2.

    By the same reason that neither is the absolute value function on \(\mathbb{R}\). Moreover, at least in the finite-dimensional case, the square of a norm is smooth at 0 if and only if the norm comes from a scalar product [67, Prop. 4.1].

  3. 3.

    Consistently, paracompactness can be deduced from the hypothesis of metrizability (or even just from pseudo-metrizability, see [2, Lemma 5.515]).

  4. 4.

    This can be rephrased as a bound of the spectrum of S, see [44, Th. 3.10].

  5. 5.

    These improvements can be also extended to other contexts, as the completeness of certain Finsler metrics in [21].

  6. 6.

    According to Palais [52, Defn. 4.1] (and taking into account Moore’s modification [49, p. 50]), a pseudo-gradient for a function V on an open subset U is a locally Lipschitz vector field X such that \(\epsilon ^{2}F_{p}(X_{p})^{2} \leq \parallel \mathit{dV}_{p} \parallel \leq \epsilon ^{-2}\mathit{dV}_{p}(X_{p})\) for all p ∈ U.

  7. 7.

    However, standard Finsler metrics are usually allowed to be non-reversible, see Remark 8.

  8. 8.

    Explicit examples by Kundt [39], Geroch [32, p. 531] and Beem [5] showed the full logical independence among spacelike, timelike and lightlike geodesic completeness.

  9. 9.

    This case is interesting also for the classification of flat compact Lorentzian manifolds, which are called then standard, see [38].

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Acknowledgements

Partially supported by Spanish grants with Feder funds P09-FQM-4496 (J. Andalucía) and MTM2013-47828-C2-1-P (Mineco).

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  1. Facultad de Ciencias, Departamento de Geometría y Topología, Universidad de Granada, Campus Fuentenueva s/n, E18071, Granada, Spain

    Miguel Sánchez

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  1. Miguel Sánchez
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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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Sánchez, M. (2015). On the Completeness of Trajectories for Some Mechanical Systems. 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_15

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