TimeDependentHarmonicOscillators 2009-05-29 13:46:17 2009-05-29 13:55:36 Topic Ermakov systems time-dependent harmonic oscillators \section{Time-dependent harmonic oscillators} Nonlinear equations are of increasing interest in Physics; Riccati and Ermakov systems enter the formalism of quantum theory in the study of cases where exact analytic Gaussian wave packet (WP) solutions of the time-dependent Schr\"odinger equation (SE) do exist, and in particular, in the harmonic oscillator (HO) and the free motion cases. One of the simplest examples of such nonlinear equations is the Milne--Pinney equation: $$d^2x/dt^2 = − {\omega}^2(t)x +k x^3,$$ (1) where $k$ is a real constant with values depending on the field in which the equation is to be applied. \subsection{Ermakov systems} This equation was introduced in the nineteenth century by V.P. Ermakov, as a way of looking for a first integral for the time-dependent harmonic oscillator. He employed some of Lie's ideas for dealing with ordinary differential equations with the tools of classical geometry. Lie had previously obtained a characterization of non-autonomous systems of first-order differential equations admitting a superposition rule: $$dx_i/dt = Y i(t, x), i = 1, . . . , n, $$ (2) Ermakov systems have also been broadly studied in Physics since its introduction in the nineteenth century until now. They appear in the study of the Bose–Einstein condensates, cosmological models, and the solution of time-dependent harmonic or anharmonic oscillators. Several recent reports are concerned with the use Hamiltonian or Lagrangian structures in the study of such a system, and many generalisations or new insights from the mathematical point of view have ben thus obtained. \textbf{[More to come]}