Original manuscript: 2007/06/26

Reduction theory for systems with symmetry deals with the problem of understanding dynamics on a manifold with an action of a Lie group. In geometric mechanics, this problem can be formulated in the Lagrangian, Hamiltonian or affine connection frameworks. While the Lagrangian and Hamiltonian formulations have been well developed, the results obtained in these setups are based on variational principles and symplectic geometry. These methods cannot be used directly in the affine connection formulation unless additional structure is available.

In this thesis, a manifold with an arbitrary affine connection is considered, and the geodesic spray associated with the connection is studied in the presence of a Lie group action. In particular, results are obtained that provide insight into the structure of the reduced dynamics associated with the given invariant affine connection. The geometry of the frame bundle of the given manifold is used to provide an intrinsic description of the geodesic spray. A fundamental relationship between the geodesic spray, the tangent lift and the vertical lift of the symmetric product is obtained, which provides a key to understanding reduction in this formulation.

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