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Invariant Higher-Order Variational Problems II

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Abstract

Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with normal metrics. The prime examples of such object manifolds are the symmetric spaces. We characterize the class of cubics on object manifolds that can be lifted horizontally to cubics on the group of transformations. Conversely, we show that certain types of non-horizontal geodesic on the group of transformations project to cubics. Finally, we apply second-order Lagrange–Poincaré reduction to the problem of Riemannian cubics on the group of transformations. This leads to a reduced form of the equations that reveals the obstruction for the projection of a cubic on a transformation group to again be a cubic on its object manifold.

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Acknowledgements

We thank B. Doyon, P. Michor, L. Noakes and A. Trouvé for encouraging comments and insightful remarks during the course of this work. We also thank the referees for encouragement and informative comments. In particular, we thank one of the referees for directing us to Krakowski (2002). DDH, DMM and FXV are grateful for partial support by the Royal Society of London Wolfson Research Merit Award and the European Research Council Advanced Grant. FGB has been partially supported by a “Projet Incitatif de Recherche” contract from the Ecole Normale Supérieure de Paris. TSR was partially supported by Swiss NSF grant 200020-126630 and by the government grant of the Russian Federation for support of research projects implemented by leading scientists, Lomonosov Moscow State University under the agreement No. 11.G34.31.0054.

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Authors and Affiliations

  1. Laboratoire de Météorologie Dynamique, École Normale Supérieure/CNRS, Paris, France

    François Gay-Balmaz

  2. Department of Mathematics, Imperial College, London, SW7 2AZ, UK

    Darryl D. Holm & David M. Meier

  3. Section de Mathématiques and Bernoulli Center, École Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland

    Tudor S. Ratiu

  4. Centre de Recherche en Mathématiques de la Décision, Université Paris-Dauphine, Paris, France

    François-Xavier Vialard

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  1. François Gay-Balmaz
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  2. Darryl D. Holm
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  3. David M. Meier
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  4. Tudor S. Ratiu
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  5. François-Xavier Vialard
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Correspondence to Darryl D. Holm.

Additional information

Communicated by Anthony Bloch.

Fondly remembering our late friend Jerry Marsden.

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Gay-Balmaz, F., Holm, D.D., Meier, D.M. et al. Invariant Higher-Order Variational Problems II. J Nonlinear Sci 22, 553–597 (2012). https://doi.org/10.1007/s00332-012-9137-2

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