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

Published online by Cambridge University Press:  15 October 2004

Claudio Altafini
Claudio Altafini*
Affiliation:
SISSA-ISAS, International School for Advanced Studies, via Beirut 2-4, 34014 Trieste, Italy; altafini@sissa.it.
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Abstract

For a Riemannian structure on a semidirect product of Lie groups, the variational problems can be reduced using the group symmetry. Choosing the Levi-Civita 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 in 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 look more complicated than the original ones. The main scope of this paper is to treat the reduction of second order variational problems (corresponding to geometric splines) on such semidirect products of Lie groups. Due to the semidirect structure, a number of extra terms appears in the reduction, terms that are calculated explicitely. The result is used to compute the necessary conditions of an optimal control problem for a simple mechanical control system having invariant Lagrangian equal to the kinetic energy corresponding to the metric tensor. As an example, the case of a rigid body on the Special Euclidean group is considered in detail.

Keywords

Lie groupsemidirect productsecond order variational problemsreductiongroup symmetrygeometric splinesoptimal control.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 4 , October 2004 , pp. 526 - 548
Copyright
© EDP Sciences, SMAI, 2004

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