#### DOCUMENTA MATHEMATICA, Vol. 7 (2002), 561-604

Anthony D. Blaom

Reconstruction Phases for Hamiltonian Systems on Cotangent Bundles

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 ${\mathrm T}^*Q$ 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.

2000 Mathematics Subject Classification: 70H33, 53D20.

Keywords and Phrases: mechanical system with symmetry, geometric phase, dynamic phase, reconstruction phase, Berry phase, cotangent bundle.

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