PlanetPhysics/Time Dependent Harmonic Oscillators

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

Time-dependent harmonic oscillators

Nonlinear equations are of increasing interest in Physics; Riccati equation and Ermakov systems enter the formalism of [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum theory]] in the study of cases where exact analytic Gaussian [[../CosmologicalConstant/|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 [[../CosmologicalConstant2/|motion]] cases.

One of the simplest examples of such nonlinear equations is the Milne--Pinney equation: $d^{2}x/d{t}^2=-{\omega }^2(t)x+k{x}^3,$ (1) where $k$ is a real constant with values depending on the [[../CosmologicalConstant/|field]] in which the equation is to be applied.

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 [[../DifferentialEquations/|ordinary differential equations]] with the tools of classical geometry. Lie had previously obtained a characterization of non-autonomous [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|systems]] of first-order [[../DifferentialEquations/|differential equations]] admitting a superposition rule: $d{x}_i/dt=Yi(t,x),i=1,...,n,$ (2).

This approach has been recently reformulated from a geometric perspective in which the role of the superposition function is played by an appropriate algebriac connection. This geometric approach allows one to consider a superposition of solutions of a given system in order to obtain solutions of another system as a kind of mathematical construction that might be generalized even further; such a superposition rule may be understood from a geometric viewpoint in some interesting cases, as in the Milne--Pinney equation (1), or in the Ermakov system and its generalisations. One recalls here that Ermakov systems are defined as systems of second-order differential equations composed by the Milne--Pinney differential equation (1) together with the corresponding time--dependent harmonic oscillator.

Ermakov systems have been also broadly studied in Physics since their introduction in the nineteenth century. They also 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 [[../Hamiltonian2/|Hamiltonian]] or [[../LagrangesEquations/|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. Ermakov--Lewis invariants naturally emerge as [[../Bijective/|functions]] defining the foliation associated to the superposition rule. It has been shown by Ermakov in 1880 that the [[../DifferentialEquations/|system of differential equations]] coupled via the possibly time-dependent frequency $\omega$, leads to a dynamical invariant that has been rediscovered by several authors in the 20th century: $I_L=0.5[(d\eta /dt)\alpha -\eta (d\alpha /dt){]}^2+(\eta \alpha {)}^2=const.22:34,25June2015(UTC)(3)$ (3)

It is straightforward to show that $d/dt(I_L)=0.$ The above Ermakov invariant $I_L$ depends not only on the classical variables $\eta (t)$ and its time derivative, but also on the [[../QuantumParticle/|quantum uncertainty]] related to $\alpha (t)$ and its time derivative. Additional interesting insight into the [[../Bijective/|relation]] between variables $\eta$ and $\alpha$ can be obtained by considering also the [[../RiccatiEquation2/|Riccati equation]].