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March 2016, 8(1): 13-34. doi: 10.3934/jgm.2016.8.13
Symplectic reduction at zero angular momentum
Joshua Cape 1, , Hans-Christian Herbig 2, and Christopher Seaton 3,
Department of Applied Mathematics and Statistics, Johns Hopkins University, 3400 N. Charles St, Baltimore, MD 21218, United States
Departamento de Matemática Aplicada, Av. Athos da Silveira Ramos 149, Centro de Tecnologia - Bloco C, CEP: 21941-909 - Rio de Janeiro, Brazil
Department of Mathematics and Computer Science, Rhodes College, 2000 N. Parkway, Memphis, TN 38112, United States
Received April 2015 Revised August 2015 Published February 2016
We study the symplectic reduction of the phase space describing $k$ particles in $\mathbb{R}^n$ with total angular momentum zero. This corresponds to the singular symplectic quotient associated to the diagonal action of $O_n$ on $k$ copies of $T^\ast\mathbb{R}^n$ at the zero value of the homogeneous quadratic moment map. We give a description of the ideal of relations of the ring of regular functions of the symplectic quotient. Using this description, we demonstrate $\mathbb{Z}^+$-graded regular symplectomorphisms among the $O_n$- and $SO_n$-symplectic quotients and determine which of these quotients are graded regularly symplectomorphic to linear symplectic orbifolds. We demonstrate that when $n \leq k$, the zero fibre of the moment map has rational singularities and hence is normal and Cohen-Macaulay. We also demonstrate that for small values of $k$, the ring of regular functions on the symplectic quotient is graded Gorenstein.
Keywords: moment map, rational singularities., $O_n$-representation, angular momentum, Symplectic reduction.
Mathematics Subject Classification: Primary: 53D20, 13A50; Secondary: 57S15, 37J15, 20G2.
Citation: Joshua Cape, Hans-Christian Herbig, Christopher Seaton. Symplectic reduction at zero angular momentum. Journal of Geometric Mechanics, 2016, 8 (1) : 13-34. doi: 10.3934/jgm.2016.8.13
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Joshua Cape Hans-Christian Herbig Christopher Seaton