A Review in Ermakov Systems and Their Symmetries

Abstract

A review of the mathematical and physical aspects of the Ermakov systems is presented. The main properties of Lie algebra invariants are extensively used. The two and tridimensional Ermakov systems are fully analyzed and the correspondent invariants found. Then, we go over quantization with special emphasis in the two dimensional case. An application to Nonlinear Optics is hereby developed. We also treat the so-called “one dimensional” case, which is easily solved in the classical case but offers special interest in the quantum realm, where one can find exactly the Feynman propagator. We finish with the stationary phase approximation which contains also some interesting features when compared with the exact solution. Some prospects for future research are also discussed.

Dynamical Invariants for Generalized Coherent States via Complex Quantum Hydrodynamics

“…Even if the work of Lewis on the dynamical invariant associated with the Ermakov equation dates back to 1967, the importance of this invariant is still of present interest. Recently its mathematical properties are still studied [3], as well as its relation to quantum mechanical dynamics. Indeed, "it turns out that the evolution of the quantum states can be reduced to the solutions of the classical Ermakov-Milne-Pinney (EMP) equation" [4], especially for time-dependent quadratic potentials [5,6] and coupled oscillators [7].…”