Reducibility of parametrized systems

In a previous paper [Phys. Rev. D 34, 1040 (1986)] we showed that the reduction method is often not applicable to parametrized systems because of topological obstructions. These are (1) the nonexistence of a cross section of the reparametrization-group orbits (Gribov effect) and/or (2) the existence of compact orbits of the group. We show that the mere existence of compact orbits is a consequence of an improper choice of variables for the description of the system and we give a general method of how the variables are to be transformed so that these difficulties disappear. The new variables are the so-called ‘‘embedding variables’’ of Kuchař. We study some general properties of the systems that can be made reducible by this method. For example, we find the symmetry which is necessary and sufficient for the Hamiltonian of the reduced theory to be independent of time. We demonstrate that the system can be reduced explicitly if such a symmetry exists. The explicit form of the reduced Hamiltonian also enables us to study the conditions under which it will be a polynomial of the momenta.