Completely integrable one-dimensional classical and relativistic time-dependent hamiltonians

In this paper, we look for first integrals $I(q;p;t)$ of time-dependent one-dimensional Hamiltonians $H(q;p;t)$. We first present a formalism based on the use of canonical transformations, and it is seen that $I(q;p;t)$ can always be written in terms of two variables $I=P(u;v)$, whereu andv are functions of $q$, $p$ andt, without loss of generality. Moreover, it is shown that any Hamiltonian with first integral $I(q;p;t)$ can be made autonomous in the space $(u,v,T)$, where $T$ is a new time. On the other hand, the cases of a particle moving classically and relativistically in a time-dependent potential $V(q;t)$ are studied. In both cases, completely integrable potentials, together with the corresponding first integrals, are derived.