Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2001-12-10
Rev.Math.Phys. 15 (2003) 663-703
Physics
High Energy Physics
High Energy Physics - Theory
41 pages, more detailed abstract in paper; v2: minor corrections and an additional reference
Scientific paper
10.1142/S0129055X0300176X
Constrained Hamiltonian systems fall into the realm of presymplectic geometry. We show, however, that also Poisson geometry is of use in this context. For the case that the constraints form a closed algebra, there are two natural Poisson manifolds associated to the system, forming a symplectic dual pair with respect to the original, unconstrained phase space. We provide sufficient conditions so that the reduced phase space of the constrained system may be identified with a symplectic leaf in one of those. In the second class case the original constrained system may be reformulated equivalently as an abelian first class system in an extended phase space by these methods. Inspired by the relation of the Dirac bracket of a general second class constrained system to the original unconstrained phase space, we address the question of whether a regular Poisson manifold permits a leafwise symplectic embedding into a symplectic manifold. Necessary and sufficient for this is the vanishing of the characteristic form-class of the Poisson tensor, a certain element of the third relative cohomology.
Bojowald Martin
Strobl Thomas
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