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- W2146161082 abstract "As is well-known, there is a variational principle for theEuler–Poincare equations on a Lie algebra g of a Lie group G obtained by reducing Hamilton’s principle on G by the action of G by, say, left multiplication. The purpose of this paper is to give a variational principlefor the Lie–Poisson equations on g*, the dual of g, and also togeneralize this construction.The more general situation is that in which the original configurationspace is not a Lie group, but rather a configuration manifold Qon which a Lie group G acts freely and properly, so that Q → Q/Gbecomes a principal bundle. Starting with a Lagrangian system on TQinvariant under the tangent lifted action of G, the reduced equations on(TQ)/G, appropriately identified, are the Lagrange–Poincare equations.Similarly, if we start with a Hamiltonian system on T*Q, invariant underthe cotangent lifted action of G, the resulting reduced equations on(T*Q)/G are called the Hamilton–Poincare equations.Amongst our new results, we derive a variational structure for theHamilton–Poincare 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 associatedsymplectic leaves. We illustrate the formalism with a simple, butinteresting example, that of a rigid body with internal rotors." @default.
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- W2146161082 date "2003-01-01" @default.
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- W2146161082 title "Variational Principles for Lie—Poisson and Hamilton—Poincare Equations" @default.
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- W2146161082 doi "https://doi.org/10.17323/1609-4514-2003-3-3-833-867" @default.
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