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Abstract mechanical connection and Abelian reconstruction for almost Kähler manifolds. (English) Zbl 0998.53055

Let (P,ω) be a finite-dimensional symplectic manifold, G a Lie group, acting canonically on P, admitting an equivariant momentum map J, H a Hamiltonian on P. Under appropriate assumptions the space of group orbits P/G is a smooth manifold inheriting a Poisson structure from that of P. The Hamiltonian H on P drops to a Hamiltonian h on P/G, the Hamiltonian vector fields X H and X h as well as their solutions x t and y t are related by the projection π:PP/G. Let y t be periodic with period T. Then for any initial condition x 0 π -1 (y 0 ) there exists a unique gG such that x T =g·x 0 . The reconstruction problem consists in the computation of the associated “reconstruction phase” g.

Under certain conditions the Marsden-Weinstein reduction produces a principal bundle J -1 (μ 0 ) P μ 0 with μ 0 =J(x 0 ), P μ 0 being a symplectic leaf in P/G, containing the solution y t . The computation now uses an arbitrary principal connection on this principal bundle.

If P is the cotangent bundle of a Riemannian manifold, there is a canonical choice for such a connection, the so-called mechanical connection, but in general this is not the case. The authors show that there still exists a canonical connection, under the assumption that (P,ω) is an almost Kähler manifold, the symplectic form ω being given by the almost complex structure and the Kähler metric. This so-called abstract mechanical connection is defined by declaring horizontal spaces at each point to be metric orthogonal to the tangent to the group orbit. Explicit formulas for the corresponding connection form are given, which, in the Abelian case, simplify the computation considerably.

MSC:
53D20Momentum maps; symplectic reduction
37J15Symmetries, invariants, invariant manifolds, momentum maps, reduction
53D17Poisson manifolds; Poisson groupoids and algebroids
70H33Symmetries and conservation laws, reverse symmetries, invariant manifolds, etc.
70G45Differential-geometric methods for dynamical systems