Reduction and Hamiltonian structures on duals of semidirect product Lie algebras

摘要

With the heavy top and compressible flow as guiding examples, this paper discusses the Hamiltonian structure of systems on duals of semidirect product Lie algebras by reduction from Lagrangian to Eulerian coordinates. Special emphasis is placed on the left-right duality which brings out the dual role of the spatial and body (i.e. Eulerian and convective) descriptions. For example, the heavy top in spatial coordinates has a Lie-Poisson structure on the dual of a semidirect product Lie algebra in which the moment of inertia is a dynamic variable. For compressible fluids in the convective picture, the metric tensor similarly becomes a dynamic variable. Relationships to the existing literature are given.