Reconstruction phases for Hamiltonian systems on cotangent bundles

Abstract

Reconstruction phases describe the motions experienced by dynamical systems whose symmetry-reduced variables are undergoing periodic motion. A well known example is the non-trivial rotation experienced by a free rigid body after one period of oscillation of the body angular momentum vector. Here reconstruction phases are derived for a general class of Hamiltonians on a cotangent bundle ${\mathrm{T}}^{\ast }Q$ possessing a group of symmetries $G$, and in particular for mechanical systems. These results are presented as a synthesis of the known special cases $Q=G$ and $G$ Abelian, which are reviewed in detail.