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 (2010)

  • Volume: 10, Issue: 4, page 526-548
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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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 10.4 (2010): 526-548. <http://eudml.org/doc/90741>.

@article{Altafini2010,
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. },
author = {Altafini, Claudio},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = { Lie group; semidirect product; second order variational problems; reduction; group symmetry; geometric splines; optimal control.; Lie group; second order variational problems; optimal control},
language = {eng},
month = {3},
number = {4},
pages = {526-548},
publisher = {EDP Sciences},
title = {Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite Riemannian metric},
url = {http://eudml.org/doc/90741},
volume = {10},
year = {2010},
}

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
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 4
SP - 526
EP - 548
AB - 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.
LA - eng
KW - Lie group; semidirect product; second order variational problems; reduction; group symmetry; geometric splines; optimal control.; Lie group; second order variational problems; optimal control
UR - http://eudml.org/doc/90741
ER -

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