First integrals and symmetries of time-dependent Hamiltonian systems.

In this paper, first integrals of one-dimensional time-dependent Hamiltonians, H(q,p,t), are studied. First of all, it is shown that the second first integral, J(q,H,t)=t-∫dq/Hp, valid for autonomous Hamiltonians H(q,p), can be extended to the case of nonautonomous Hamiltonians H(q,p,t), provided Hp (resp. t) is replaced with Ip [where I(q,p,t) is a first integral of H(q,p,t)] (resp. a new time T): J(q,I,T)=T-∫dq/Ip. In a second part, we derive a technique leading to a first integral, I(q,p,t), from a Lie symmetry. As an example, this technique, together with the expression of the second first integral, J(q,I,t), is used to completely integrate a special case of the linear harmonic oscillator with a time-dependent frequency. As a result, Lewis’ first integral is recovered from a Lie symmetry in a way that generalizes the recent work of Shivamoggi and Muilenberg [Phys. Lett. A 154, 1 (1991)]. [ABSTRACT FROM AUTHOR]