A variational problem on Stiefel manifolds

Anthony M. Bloch, Peter E. Crouch, Amit Sanyal

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

In their paper on discrete analogues of some classical systems such as the rigid body and the geodesic flow on an ellipsoid, Moser and Veselov introduced their analysis in the general context of flows on Stiefel manifolds. We consider here a general class of continuous time, quadratic cost, optimal control problems on Stiefel manifolds, which in the extreme dimensions again yield these classical physical geodesic flows. We have already shown that this optimal control setting gives a new symmetric representation of the rigid body flow and in this paper we extend this representation to the geodesic flow on the ellipsoid and the more general Stiefel manifold case. The metric we choose on the Stiefel manifolds is the same as that used in the symmetric representation of the rigid body flow and that used by Moser and Veselov. In the extreme cases of the ellipsoid and the rigid body, the geodesic flows are known to be integrable. We obtain the extremal flows using both variational and optimal control approaches and elucidate the structure of the flows on general Stiefel manifolds.

Original languageEnglish (US)
Article number002
Pages (from-to)2247-2276
Number of pages30
JournalNonlinearity
Volume19
Issue number10
DOIs
StatePublished - Oct 1 2006
Externally publishedYes

Fingerprint

Stiefel Manifold
Variational Problem
Geodesic Flow
Rigid Body
Ellipsoid
rigid structures
optimal control
ellipsoids
Optimal Control
Extremes
Optimal Control Problem
Continuous Time
Choose
Costs
Analogue
Metric
analogs
costs

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

Cite this

A variational problem on Stiefel manifolds. / Bloch, Anthony M.; Crouch, Peter E.; Sanyal, Amit.

In: Nonlinearity, Vol. 19, No. 10, 002, 01.10.2006, p. 2247-2276.

Research output: Contribution to journalArticle

Bloch, Anthony M. ; Crouch, Peter E. ; Sanyal, Amit. / A variational problem on Stiefel manifolds. In: Nonlinearity. 2006 ; Vol. 19, No. 10. pp. 2247-2276.
@article{384b396c8d9c497fa9dc9142f8343d81,
title = "A variational problem on Stiefel manifolds",
abstract = "In their paper on discrete analogues of some classical systems such as the rigid body and the geodesic flow on an ellipsoid, Moser and Veselov introduced their analysis in the general context of flows on Stiefel manifolds. We consider here a general class of continuous time, quadratic cost, optimal control problems on Stiefel manifolds, which in the extreme dimensions again yield these classical physical geodesic flows. We have already shown that this optimal control setting gives a new symmetric representation of the rigid body flow and in this paper we extend this representation to the geodesic flow on the ellipsoid and the more general Stiefel manifold case. The metric we choose on the Stiefel manifolds is the same as that used in the symmetric representation of the rigid body flow and that used by Moser and Veselov. In the extreme cases of the ellipsoid and the rigid body, the geodesic flows are known to be integrable. We obtain the extremal flows using both variational and optimal control approaches and elucidate the structure of the flows on general Stiefel manifolds.",
author = "Bloch, {Anthony M.} and Crouch, {Peter E.} and Amit Sanyal",
year = "2006",
month = "10",
day = "1",
doi = "10.1088/0951-7715/19/10/002",
language = "English (US)",
volume = "19",
pages = "2247--2276",
journal = "Nonlinearity",
issn = "0951-7715",
publisher = "IOP Publishing Ltd.",
number = "10",

}

TY - JOUR

T1 - A variational problem on Stiefel manifolds

AU - Bloch, Anthony M.

AU - Crouch, Peter E.

AU - Sanyal, Amit

PY - 2006/10/1

Y1 - 2006/10/1

N2 - In their paper on discrete analogues of some classical systems such as the rigid body and the geodesic flow on an ellipsoid, Moser and Veselov introduced their analysis in the general context of flows on Stiefel manifolds. We consider here a general class of continuous time, quadratic cost, optimal control problems on Stiefel manifolds, which in the extreme dimensions again yield these classical physical geodesic flows. We have already shown that this optimal control setting gives a new symmetric representation of the rigid body flow and in this paper we extend this representation to the geodesic flow on the ellipsoid and the more general Stiefel manifold case. The metric we choose on the Stiefel manifolds is the same as that used in the symmetric representation of the rigid body flow and that used by Moser and Veselov. In the extreme cases of the ellipsoid and the rigid body, the geodesic flows are known to be integrable. We obtain the extremal flows using both variational and optimal control approaches and elucidate the structure of the flows on general Stiefel manifolds.

AB - In their paper on discrete analogues of some classical systems such as the rigid body and the geodesic flow on an ellipsoid, Moser and Veselov introduced their analysis in the general context of flows on Stiefel manifolds. We consider here a general class of continuous time, quadratic cost, optimal control problems on Stiefel manifolds, which in the extreme dimensions again yield these classical physical geodesic flows. We have already shown that this optimal control setting gives a new symmetric representation of the rigid body flow and in this paper we extend this representation to the geodesic flow on the ellipsoid and the more general Stiefel manifold case. The metric we choose on the Stiefel manifolds is the same as that used in the symmetric representation of the rigid body flow and that used by Moser and Veselov. In the extreme cases of the ellipsoid and the rigid body, the geodesic flows are known to be integrable. We obtain the extremal flows using both variational and optimal control approaches and elucidate the structure of the flows on general Stiefel manifolds.

UR - http://www.scopus.com/inward/record.url?scp=33748861140&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748861140&partnerID=8YFLogxK

U2 - 10.1088/0951-7715/19/10/002

DO - 10.1088/0951-7715/19/10/002

M3 - Article

AN - SCOPUS:33748861140

VL - 19

SP - 2247

EP - 2276

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 10

M1 - 002

ER -