The Euler-Poincare Equations and Semidirect Products with Applications to Continuum Theories

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dc.creatorHolm, D. D.
dc.creatorMarsden, J. E.
dc.creatorRatiu, T. S.
dc.date1998-01-13
dc.date.accessioned2026-07-07T02:35:20Z
dc.date.available2026-07-07T02:35:20Z
dc.descriptionWe study Euler-Poincare systems (i.e., the Lagrangian analogue of Lie-Poisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler-Poincare equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d'Alembert type. Then we derive an abstract Kelvin-Noether theorem for these equations. We also explore their relation with the theory of Lie-Poisson Hamiltonian systems defined on the dual of a semidirect product Lie algebra. The Legendre transformation in such cases is often not invertible; so it does not produce a corresponding Euler-Poincare system on that Lie algebra. We avoid this potential difficulty by developing the theory of Euler-Poincare systems entirely within the Lagrangian framework. We apply the general theory to a number of known examples, including the heavy top, ideal compressible fluids and MHD. We also use this framework to derive higher dimensional Camassa-Holm equations, which have many potentially interesting analytical properties. These equations are Euler-Poincare equations for geodesics on diffeomorphism groups (in the sense of the Arnold program) but where the metric is H^1 rather than L^2.
dc.description72 pages, LATeX, no figures
dc.identifierhttps://arxiv.org/abs/chao-dyn/9801015
dc.identifierhttp://arxiv.org/abs/chao-dyn/9801015
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/15574
dc.subjectChaotic Dynamics
dc.titleThe Euler-Poincare Equations and Semidirect Products with Applications to Continuum Theories
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