Hamiltonian reduction of free particle motion on the group SL(2, ℝ)

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 SL(2, 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 × 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 * (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.

Coadjoint Orbits of the Virasoro Algebra and the Global Liouville Equation

“…For other groups, the reduced WZNW systems [3] contain the standard Toda field theories in their trivial topological sector. Some nontrivial aspects of these reduced systems have been investigated in the point particle case [5,18,19,20]. In the field theoretic case the SL(3, R) model was studied [21] using the known description of the symplectic leaves of the W 3 -algebra [22] that replaces the Virasoro algebra as one goes from SL(2, R) to SL(3, R).…”

Physical Properties of 0.24Pb(Zn1/3Nb2/3)O3·0.384PbZrO3·0.376PbTiO3Thin Films Crystallized by Hot Isostatic Pressing

“…The authors thank L Fehér for reading the manuscript and for drawing our attention to [8]. ZB was supported by OTKA F019477, D25517, T029802.…”

“…to the Lagrangian, so it describes a physically different system as we will see at the quantum level. It was shown in [8], that the reduced phase space is, in fact, symplectomorphic to T * S 1 , but the Hamiltonian is very complicated: H = sin −2 ϕ (cos 2 ϕ − exp(p ϕ sin 2 ϕ)), so this description is not useful in quantizing the system. The Hamiltonian in our language, however, is very simple:…”

W-algebras for non-abelian Toda systems

“…It is worth to note here that majority of the results on W -algebras for Toda systems was obtained by the method of Hamiltonian reduction that is based on the fact that Toda systems can be obtained if one starts with a WZNW model based on a Lie group G and then imposes relevant constraints on the conserved currents forming with respect to the Poisson bracket two copies of loop algebras associated with the Lie algebra g [17,18,19]. Here the Toda 'field space' arises as a factor in the generalized Gauss decomposition of the Lie group G, which is valid only for a dense subset of G. This results in that the true reduced system is different from a Toda system; see, in this respect, papers [20,21,22,23,24,25,26]. Such our conclusion is justified at least by the fact that the Toda systems have singular solutions corresponding to some nonsingular initial conditions, and that is impossible for a system being a reduction of a WZNW model which does not have such solutions.…”