Hamiltonian Symmetry Reduction via Localizations: Theory and Application to a Barbell System

We specialize a recently introduced variant of orbit space reduction for symmetric Hamiltonian systems. This variant works with suitable localizations of the algebra of polynomial invariants of the corresponding symmetry group action, and provides reduction to a variety that is embedded in a low-dimensional affine space, which makes efficient computations possible. As an example, we discuss the mechanical system of a “barbell” in a general central force field.