REPRESENTATIONS OF ONE-DIMENSIONAL HAMILTONIANS IN TERMS OF THEIR INVARIANTS

A general formalism for representing the Hamiltonian of a system with one degree of freedom in terms of its invariants is developed. Those Hamiltonians H(q,p,t) are derived for which any particular function I(q,p,t) is an invariant. For each of those Hamiltonians, a function canonically conjugate to I(q,p,t) is derived which is also an invariant of H(q,p,t). The formalism is also presented for the case in which I(q,p,t) is expressed as a function of two canonically conjugate functions. The formalism is illustrated by applying it to the case of a particle moving in a time-dependent potential. Some earlier results are recovered and an invariant is found for a new potential. Lines for further study are outlined that may be fruitful for finding more examples of integrable systems.