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The Euler-Poincaré equations and semidirect products with applications to continuum theories. (English) Zbl 0951.37020
A Lagrangian analogue of the Hamiltonian semidirect product theory is studied. As a result of the reduction for a Lagrangian that depends on a parameter the problem reduces to Lie-Poisson type systems on the duals of semidirect products. The resulting equations generalize the basic Euler-Poincaré equations on a Lie algebra in that they depend on a parameter. A specific version of Noether’s theorem in an action principle formulation is proven and it leads to a Kelvin circulation type theorem for continuum mechanics. The authors suggest a number of applications of the Euler-Poincaré equations in ideal continuum dynamics which illustrate the power of the above approach.

MSC:
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
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