Reconstruction phases via Poisson reduction

Abstract

A new formula for reconstruction phases in Hamiltonian systems with symmetry expresses the phase in terms of the Poisson-reduced solution curve and certain derivatives transverse to the symplectic leaf containing the curve. Specifically, the “dynamic” part of the phase depends on transverse derivatives in the Poisson-reduced Hamiltonian, while the “geometric” part is determined by transverse derivatives in the leaf symplectic structures. Intermediate results include a decomposition theorem for Hamiltonian vector fields with symmetry, and a new expression for curvature in Marsden–Weinstein reduction bundles. Applications are made to mechanical systems and resonant three-wave interactions.