2013/02/28 by Vivek Narayanan, P. Morrison, Narayanan, Vivek +2
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1302.7267
arxiv created 2013/02/28 · openalex publication_date 2013/02/28 · arxiv updated 2013/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is shown in this paper how a connection may be made between the symmetry generators of the Hamiltonian (or potential) invariant under a symmetry group G, and the subcasimirs that come about when the rank of the Poisson structure of a dynamical system drops by an even integer. This \em kinematics-dynamics connection is made by using the algebraic geometry of the orbit space in the vicinity of rank change, and the extra null eigenvectors of the mass matrix (Hessian with respect to symmetry generators) of the Hamiltonian (or potential). Some physical interpretations of this point of view include a control-theoretic prescription to study stability on various symplectic leaves of the Poisson structure. Methods of Invariant Theory are utilized to provide parametrization for the leaves of a Poisson dynamical system for the case where a compact Lie group acts properly on the phase space, which is assumed to be modeled by Poisson geometry.