A direct approach to finding exact invariants for one-dimensional time-dependent classical Hamiltonians

For a classical Hamiltonian H=(1/2) p2+V(q,t) with an arbitrary time-dependent potential V(q,t), exact invariants that can be expressed as series in positive powers of p, I(q,p,t)=∑∞n=0pnfn(q,t), are examined. The method is based on direct use of the equation dI/dt=∂I/∂t +[I,H] =0. A recursion relation for the coefficients fn(q,t) is obtained. All potentials that admit an invariant quadratic in p are found and, for those potentials, all invariants quadratic in p are determined. The feasibility of extending the analysis to find invariants that are polynomials in p of higher degree than quadratic is discussed. The systems for which invariants quadratic in p have been found are transformed to autonomous systems by a canonical transformation.