Artículo
Variational Principles for Lie-Poisson and Hamilton-Poincaré Equations
Fecha de publicación:
07/2003
Editorial:
Independent Univ Moscow
Revista:
Moscow Mathematical Journal
ISSN:
1609-3321
e-ISSN:
1609-4514
Idioma:
Inglés
Tipo de recurso:
Artículo publicado
Clasificación temática:
Resumen
As is well-known, there is a variational principle for theEuler–Poincar ́e equations on a Lie algebragof a Lie groupGobtainedby reducing Hamilton’s principle onGby the action ofGby, say, leftmultiplication. The purpose of this paper is to give a variational prin-ciple for the Lie–Poisson equations ong∗, the dual ofg, and also togeneralize this construction.The more general situation is that in which the original configura-tion space is not a Lie group, but rather a configuration manifoldQon which a Lie groupGacts freely and properly, so thatQ→Q/Gbecomes a principal bundle. Starting with a Lagrangian system onTQinvariant under the tangent lifted action ofG, the reduced equations on(TQ)/G, appropriately identified, are the Lagrange–Poincar ́e equations.Similarly, if we start with a Hamiltonian system onT∗Q, invariant un-der the cotangent lifted action ofG, the resulting reduced equations on(T∗Q)/Gare called the Hamilton–Poincar ́e equations.Amongst our new results, we derive a variational structure for theHamilton–Poincar ́e equations, give a formula for the Poisson structureon these reduced spaces that simplifies previous formulas of Montgomery,and give a new representation for the symplectic structure on the asso-ciated symplectic leaves. We illustrate the formalism with a simple, butinteresting example, that of a rigid body with internal rotors.
Palabras clave:
VARIATIONAL PRINCIPLES
,
LIEPOISSON EQUATIONS
,
HAMILTONPOINCARE EQUATIONS
Archivos asociados
Licencia
Identificadores
Colecciones
Articulos(INMABB)
Articulos de INST.DE MATEMATICA BAHIA BLANCA (I)
Articulos de INST.DE MATEMATICA BAHIA BLANCA (I)
Citación
Cendra, Hernan; Marsden, Jerrold E.; Pekarsky, Sergey; Ratiu, Tudor S.; Variational Principles for Lie-Poisson and Hamilton-Poincaré Equations; Independent Univ Moscow; Moscow Mathematical Journal; 3; 3; 7-2003; 833-867
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