Complete sets of invariants for classical and quantum systems Willard Miller (miller@ima.umn.edu)
Abstract We consider the general problem of determining exactly when a classical Hamiltonian H in n dimensions admits a constant of the motion that is polynomial in the momenta. If the associated Hamilton-Jacobi equation admits an orthogonal separation of variables, then it is possible to generate algorithmically a canonical basis Q, P where P_1=H, P_2, ..., P_n are the other 2nd-order constants of the motion associated with the separable coordinates, and {Q_i,Q_j}={P_i,P_j}=0, {Q_i,P_j}=\delta_{ij}. The 2n-1 functions Q_2,... Q_n,P_1,...,P_n form a basis for the invariants. We show how to determine for exactly which spaces and potentials the invariant Q_j is a polynomial in the original momenta. |