Momentum Maps and Hamiltonian Reduction

Details Momentum Maps and Hamiltonian Reduction Momentum Maps and Hamiltonian Reduction

Dec 1998

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1998tx19.confe.108f&link_type=abstract

Abstracts of the 19th Texas Symposium on Relativistic Astrophysics and Cosmology, held in Paris, France, Dec. 14-18, 1998. Eds.:

Astronomy and Astrophysics

Astrophysics

Scientific paper

We discuss the geometry underlying the symplectic reduction of an infinite-dimensional symplectic manifold (P,omega) with an infinite-dimensional symmetry group G and a momentum map Psi:P --> {G}^*. Our main tool is a generalized adjoint formulation which asserts that Z_A(p) = (omega^sharp_pcirc dΨ(p)^star)cdot A; where Z_A:P --> TP is the infinitesimal generator corresponding to A in {G} and dΨ(p)^ star:{G} --> T^*_pP is the natural adjoint of dΨ(p):T_pP --> {G} ^*. We show how the generalized adjoint formulation plays a fundamental role in symplectic reduction and discuss various applications of this formulation to (1) the construction of a universal symplectic splitting for almost Hermitian G-manifolds, (2) the geometry of symplectic reduction and dynamical reconstruction, (3) linearization stability and bifurcations of momentum maps, (4) the structure of the constraint space, and (5) Hamiltonian reduction of Einstein's equations of general relativity.

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