Dynamical invariants for time-dependent real and complex Hamiltonian systems

Abstract

The Struckmeier and Riedel (SR) approach is extended in real space to isolate dynamical invariants for one- and two-dimensional time-dependent Hamiltonian systems. We further develop the SR-formalism in $z\bar{z}$ complex phase space characterized by z = x + iy and $\bar{z}=x-iy$ and construct invariants for some physical systems. The obtained quadratic invariants contain a function f₂(t), which is a solution of a linear third-order differential equation. We further explore this approach into extended complex phase space defined by x = x₁ + ip₂ and p = p₁ + ix₂ to construct a quadratic invariant for a time-dependent quadratic potential. The derived invariants may be of interest in the realm of numerical simulations of explicitly time-dependent Hamiltonian systems.