Inflation of Hamiltonian System: The Spinning Top in Projective Space

Abstract

We present a method to enlarge the phase space of a canonical Hamiltonian System in order to remove coordinate singularities arising from a nontrivial topology of the configuration space. This ``inflation'' preserves the canonical structure of the system and generates new constants of motion that realize the constraints. As a first illustrative example the spherical pendulum is inflated by embedding the sphere $S^2$ in the three dimensional Euclidean space. The main application which motivated this work is the derivation of a canonical singularity free Hamiltonian for the general spinning top. The configuration space $SO(3)$ is diffeomorphic to the real projective space $\RP^3$ which is embedded in four dimensions using homogenous coordinates. The procedure can be generalized to $SO(n)$.