Hamiltonian reduction of free particle motion on group SL(2, ${\Bbb R}$)

TL;DR

This paper analyzes the Hamiltonian reduction of a free particle on the group SL(2, R), revealing the structure of the reduced phase space and constructing canonical coordinates for different cases.

Contribution

It characterizes the reduced phase space for the particle on SL(2, R) and shows its diffeomorphism to familiar symplectic manifolds, with explicit canonical coordinates.

Findings

Reduced phase space is either two planes or a cylinder.

Reduced phase space is symplectomorphic to cotangent bundles of R or S^1.

Canonical coordinates are explicitly constructed for both cases.

Abstract

The structure of the reduced phase space arising in the Hamiltonian reduction of the phase space corresponding to a free particle motion on the group ${\rm SL}(2, {\Bbb R})$ is investigated. The considered reduction is based on the constraints similar to those used in the Hamiltonian reduction of the Wess--Zumino--Novikov--Witten model to Toda systems. It is shown that the reduced phase space is diffeomorphic either to the union of two two--dimensional planes, or to the cylinder $S^1 \times {\Bbb R}$. Canonical coordinates are constructed for the both cases, and it is shown that in the first case the reduced phase space is symplectomorphic to the union of two cotangent bundles $T^*({\Bbb R})$ endowed with the canonical symplectic structure, while in the second case it is symplectomorphic to the cotangent bundle $T^*(S^1)$ also endowed with the canonical symplectic structure.