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
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.