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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.
@article{COCV_2004__10_4_526_0,
author = {Altafini, Claudio},
title = {Reduction by group symmetry of second order variational problems on a semidirect product of {Lie} groups with positive definite riemannian metric},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {526--548},
publisher = {EDP-Sciences},
volume = {10},
number = {4},
year = {2004},
doi = {10.1051/cocv:2004018},
mrnumber = {2111078},
zbl = {1072.49001},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2004018/}
}
TY - JOUR AU - Altafini, Claudio TI - Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite riemannian metric JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2004 SP - 526 EP - 548 VL - 10 IS - 4 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv:2004018/ DO - 10.1051/cocv:2004018 LA - en ID - COCV_2004__10_4_526_0 ER -
%0 Journal Article %A Altafini, Claudio %T Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite riemannian metric %J ESAIM: Control, Optimisation and Calculus of Variations %D 2004 %P 526-548 %V 10 %N 4 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv:2004018/ %R 10.1051/cocv:2004018 %G en %F COCV_2004__10_4_526_0
Altafini, Claudio. Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite riemannian metric. ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 4, pp. 526-548. doi: 10.1051/cocv:2004018
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