Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite Riemannian metric

Claudio Altafini

ESAIM: Control, Optimisation and Calculus of Variations, 10(4):526-548, 2004.

For an invariant Lagrangian equal to kinetic energy and defined on a semidirect product of Lie groups, the variational problems can be reduced using the group symmetry. Choosing the Riemannian connection of a positive definite metric tensor, instead of any of the canonical connections for the Lie group, simplifies the reduction of the variations but complicates the expression for the Lie algebra valued covariant derivatives. The origin of the discrepancy is due to the semidirect product structure, which implies that the Riemannian exponential map and the Lie group exponential map do not coincide. The consequence is that the reduced equations contain more terms than the original ones. The reduced Euler-Lagrange equations are well-known under the name of Euler-Poincare' equations. We treat in a similar way the reduction of second order variational problems corresponding to geometric splines on the Lie group. Here the problems connected with the semidirect structure are emphasized and a number of extra terms is appearing in the reduction. If the Lagrangian corresponds to a fully actuated mechanical system, then the resulting necessary condition can be expressed directly in terms of the control input. As an application, the case of a rigid body on the Special Euclidean group is considered.

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