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Hamiltonian reduction of free particle motion on the group \(SL(2,\mathbb{R})\). (English. Russian original) Zbl 0984.37086
Theor. Math. Phys. 110, No. 1, 119-128 (1997); translation from Teor. Mat. Fiz. 110, No. 1, 149-161 (1997).
Summary: We analyze the structure of the reduced phase space that arises in the Hamiltonian reduction of the phase space of free particle motion over the group \(\text{SL}(2,\mathbb{R})\). The reduction considered is based on introducing constraints that are analogous to those used in the reduction of the Wess-Zumino-Novikov-Witten model to Toda systems. It is shown that the reduced phase space is diffeomorphic either to a union of two two-dimensional planes or to a cylinder \(S^1\times \mathbb{R}\). We construct canonical coordinates for both cases and show that in the first case, the reduced phase space is symplectomorphic to the union of two cotangent bundles \(T^*(\mathbb{R})\) endowed with a canonical symplectic structure, while in the second case, it is symplectomorphic to the cotangent bundle \(T^*(S^1)\), which is also endowed with a canonical symplectic structure.

MSC:
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
70H05 Hamilton’s equations
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