Semidirect Products and Reduction in Mechanics
Marsden, J. E., T. S. Ratiu, and A. Weinstein
Trans. Amer. Math. Soc., 281, 147-177
Abstract:
This paper shows how to reduce a Hamiltonian system on the cotangent
bundle of a Lie group to a Hamiltonian system in the dual of the Lie
algebra of a semidirect product. The procedure simplifies, unifies,
and extends work of Greene, Guillemin, Holm, Holmes, Kupershmidt, Marsden,
Morrison, Ratiu, Sternberg and others. The heavy top, compressible fluids,
magnetohydrodynamics, elasticity, the Maxwell-Vlasov equations and multifluid
plasmas are presented as examples. Starting with Lagrangian variables, our
method explains in a direct way why semidirect products occur so frequently
in examples. It also provides a framework for the systematic introduction
of Clebsch, or canonical, variables.