Topology of quadratic super-Hamiltonians

Abstract

Dynamical systems with finite number of degrees of freedom and one constraint are considered; this constraint, called a super-Hamiltonian, is assumed to be a quadratic function of all canonical coordinates and momenta. The authors examine those topological properties of the orbits generated by the super-Hamiltonians that are relevant to the existence of cross-sections (Gribov problem). The well known results on the so-called real normal form of quadratic Hamiltonian functions are extended to super-Hamiltonians and briefly summarised. They are used to separate the classical dynamical equation. The dynamics in the separation subspaces helps to find the topology of the orbits and to give its complete description. One of the results is that the number of generic super-Hamiltonian families that do not admit cross-sections grows with the number of degrees of freedom much more rapidly than the number of families that do. Thus, orbit topology that does not admit cross sections is not just a marginal, pathological effect.