Semidirect products and reduction in mechanics

Marsden, Jerrold E.; Raţiu, Tudor; Weinstein, Alan

doi:10.1090/S0002-9947-1984-0719663-1

Semidirect products and reduction in mechanics
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by Jerrold E. Marsden, Tudor Raţiu and Alan Weinstein

Trans. Amer. Math. Soc. 281 (1984), 147-177

DOI: https://doi.org/10.1090/S0002-9947-1984-0719663-1

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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.

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