Motivated by the problem of longitudinal data assimilation, e.g., in the registration of a sequence of images, we develop the higher-order framework for Lagrangian and Hamiltonian reduction by symmetry in geometric mechanics. In particular, we obtain the reduced variational principles and the associated Poisson brackets. The special case of higher order Euler-Poincaré and Lie-Poisson reduction is also studied in detail.
H. Cendra, J. E. Marsden and T. S. Ratiu. Lagrangian Reduction by Stages, Memoirs of the Amer. Math. Soc., 152(722) (2001), 1–117.MathSciNetGoogle Scholar
H. Cendra, J. E. Marsden, S. Pekarsky and T. S. Ratiu. Variational principles for Lie-Poisson and Hamilton-Poincaré equations. Mosc. Math. J., 3(3) (2003), 833–867.zbMATHMathSciNetGoogle Scholar
L. Colombo and D. Martín de Diego. On the geometry of higher-order variational problems on Lie groups, preprint available at <http://arxiv.org/pdf/1104.3221v1> (2011).
P.E. Crouch and F. Silva Leite. The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces, Journal of Dynamical and Control Systems, 1(2) (1995), 177–202.CrossRefzbMATHMathSciNetGoogle Scholar
M. de Leon and P. R. Rodrigues. Generalized Classical Mechanics and Field Theory, North-Holland Mathematics Studies, 112 (1985).Google Scholar
M. de Leon, P. Pitanga and P. R. Rodrigues. Symplectic reduction of higher order Lagrangian systems with symmetry, J. Math. Phys., 35(12) (1994), 6546–6556.CrossRefzbMATHMathSciNetGoogle Scholar
F. Gay-Balmaz, D. D. Holm, D. Meier, T. S. Ratiu and F.-X. Vialard. Invariant higher-order variational problems. Comm. Math. Phys., to appear. Preprint available at <http://arxiv.org/pdf/1012.5060> (2010).
D. D. Holm, J. E. Marsden and T. S. Ratiu. The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. in Math., 137 (1998), 1–81.CrossRefzbMATHMathSciNetGoogle Scholar
I. I. Hussein and A. M. Bloch. Dynamic interpolation on Riemannian manifolds: an application to interferometric imaging, Proceedings of the 2004 American Control Conference, 1 (2004), 685–690.Google Scholar
S. Sternberg. Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field, Proc. Nat. Acad. Sci., 74 (1977), 5253–5254.CrossRefzbMATHMathSciNetGoogle Scholar