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

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 $\operatorname {SL}(2,\mathbb R)$ is investigated. In the considered reduction the constraints similar to those in the Hamiltonian reduction of the Wess–Zumino–Novikov–Witten model to Toda systems are used. It is shown that the reduced phase space is diffeomorphic either to the union of two two-dimensional planes, or to the cylinder $S^1 \times\mathbb R$. Canonical coordinates are constructed in both cases. In the first case the reduced phase space is sympectomorphic to the union of two cotangent bundles $T^*(\mathbb R)$ 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 sympectic structure.