The Euler-Poincaré equations and semidirect products with applications to continuum theories (Q1269471)

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.