Exact invariants for time-dependent Hamiltonian systems with one degree-of-freedom

Abstract

Generalising the ideas of two previous papers (Lewis (1968), Kruskal (1962)), a method is devised for obtaining exact invariants for time-dependent Hamiltonian systems with one degree-of-freedom. It consists in firstly transforming to a new Hamiltonian which is linear in the momentum variable, and secondly in solving the related Hamilton-Jacobi equation. The Hamiltonian of an oscillator with a supplementary inverse quadratic potential is treated as an illustrative example. After that, a complete application is given to a class of polynomial Hamiltonians, including an interpretation and discussion of the possible extent of the results.