 
Summary: REDUCTION BY GROUP SYMMETRY OF VARIATIONAL
PROBLEMS ON A SEMIDIRECT PRODUCT OF LIE GROUPS
WITH POSITIVE DEFINITE RIEMANNIAN METRIC
CLAUDIO ALTAFINI
Abstract. 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 EulerLagrange equations are wellknown under the name of Euler
PoincarŽe 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.
Key words. Lie group, semidirect product, second order variational problems, reduction, group
symmetry, optimal control
