Symplectic Manifolds, Dynamical Groups, and Hamiltonian Mechanics

Summary

Let (M, ω) be a symplectic manifold and G a Lie group acting on M by symplectic diffeomorphisms (i.e. g*ω = ω for all g ∈ G). Souriau has defined the moment of this group action as a suitable map J : M → G* (dual of the Lie algebra G of G). Some well-known results are first briefly outlined in Part I: there exists an action of G on G* for which J is equivariant, whose orbits ϑ_ξ are symplectic manifolds (Kirillov-Souriau-Kostant’s theorem); if ξ is a regular value of J, J ⁻¹(ξ) quotiented by the action of the isotropy subgroup of ξ is, under suitable assumptions, a symplectic manifold (Meyer’s theorem).

Part II is devoted to a synthesis of these two theorems. It is shown that J ⁻¹(ϑ_ξ), quotiented by a suitable equivalence relation, is a symplectic manifold, which has a local product structure (Theorem 3). Examples of Hamiltonian systems which may be reduced by this theorem are given (Theorem 4). Then Part III deals with groups of symplectic similarities, for which the property of equivariance of J is extended (Theorem 5). Finally, Part IV shows how these results may be applied to mechanical systems with time-dependent Hamiltonians: Lie groups of canonical transformations are defined and related to dynamical groups of a product symplectic manifold. A way for constructing groups of canonical transformations from groups of isometries of the base manifold is indicated; in particular, it explains how the Galilean group arises in mechanics from the group of space isometries.