Journal of Nonlinear Science

, Volume 22, Issue 4, pp 553–597 | Cite as

Invariant Higher-Order Variational Problems II

  • François Gay-Balmaz
  • Darryl D. HolmEmail author
  • David M. Meier
  • Tudor S. Ratiu
  • François-Xavier Vialard


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.


Hamilton’s principle Other variational principles Constrained dynamics Higher-order theories Optimal control problems involving partial differential equations 

Mathematics Subject Classification

70H25 70H30 70H45 70H50 49J20 



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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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • François Gay-Balmaz
    • 1
  • Darryl D. Holm
    • 2
    Email author
  • David M. Meier
    • 2
  • Tudor S. Ratiu
    • 3
  • François-Xavier Vialard
    • 4
  1. 1.Laboratoire de Météorologie DynamiqueÉcole Normale Supérieure/CNRSParisFrance
  2. 2.Department of MathematicsImperial CollegeLondonUK
  3. 3.Section de Mathématiques and Bernoulli CenterÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland
  4. 4.Centre de Recherche en Mathématiques de la DécisionUniversité Paris-DauphineParisFrance

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