TY - JOUR

T1 - Poisson geometry in constrained systems

AU - Bojowald, Martin

AU - Strobl, Thomas

N1 - Funding Information:
We thank A. Alekseev, M. Bordemann, P. Bressler, S. Lyakhovich, D. Sternheimer and in particular A. Weinstein for interesting discussions and suggestions, and L. Dittmann for help with drawing the diagrams. M. B. is grateful for support from NSF grant PHY00-90091 and the Eberly research funds of Penn State, and to A. Wipf and the TPI in Jena for hospitality during an essential part of the completion of this work. T. S. thanks the Erwin Schrödinger Institute in Vienna for hospitality during an inspiring workshop on Poisson geometry.

PY - 2003/9

Y1 - 2003/9

N2 - Associated to a constrained system with closed constraint algebra there are two Poisson manifolds P and Q forming a symplectic dual pair with respect to the original, unconstrained phase space: P is the image of the constraint map (equipped with the algebra of constraints) and Q the Poisson quotient with respect to the orbits generated by the constraints (the orbit space is assumed to be a manifold). We provide sufficient conditions so that the reduced phase space of the constrained system may be identified with a symplectic leaf of Q. By these methods, a second class constrained system with closed algebra is reformulated as an abelian first class system in an extended phase space. While any Poisson manifold (P, Π) has a symplectic realization (Karasev, Weinstein 87), it does not always permit a leafwise symplectic embedding into a symplectic manifold (M, ω). For regular P, it is seen that such an embedding exists, iff the characteristic form-class of Π, a certain element of the third relative cohomology of P, vanishes. A tubular neighborhood of the constraint surface of a general second class constrained system equipped with the Dirac bracket provides a physical example for such an embedding into the original symplectic manifold. In contrast, a leafwise symplectic embedding of e.g. (the maximal regular part of) a Poisson Lie manifold associated to a compact, semisimple Lie algebra does not exist.

AB - Associated to a constrained system with closed constraint algebra there are two Poisson manifolds P and Q forming a symplectic dual pair with respect to the original, unconstrained phase space: P is the image of the constraint map (equipped with the algebra of constraints) and Q the Poisson quotient with respect to the orbits generated by the constraints (the orbit space is assumed to be a manifold). We provide sufficient conditions so that the reduced phase space of the constrained system may be identified with a symplectic leaf of Q. By these methods, a second class constrained system with closed algebra is reformulated as an abelian first class system in an extended phase space. While any Poisson manifold (P, Π) has a symplectic realization (Karasev, Weinstein 87), it does not always permit a leafwise symplectic embedding into a symplectic manifold (M, ω). For regular P, it is seen that such an embedding exists, iff the characteristic form-class of Π, a certain element of the third relative cohomology of P, vanishes. A tubular neighborhood of the constraint surface of a general second class constrained system equipped with the Dirac bracket provides a physical example for such an embedding into the original symplectic manifold. In contrast, a leafwise symplectic embedding of e.g. (the maximal regular part of) a Poisson Lie manifold associated to a compact, semisimple Lie algebra does not exist.

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U2 - 10.1142/S0129055X0300176X

DO - 10.1142/S0129055X0300176X

M3 - Article

AN - SCOPUS:0348162351

SN - 0129-055X

VL - 15

SP - 663

EP - 703

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

IS - 7

ER -