A Note on the Time-Dependent Harmonic Oscillator

Abstract

H. R. Lewis has shown that in the discussion of the equation $\ddot x + \Omega ^2 (t)x = 0$ which gives the motion in a straight line of a harmonic oscillator, there could be of use an invariant $I = \frac{1}{2}[ {{h^2 x^2 \rho + ( {\rho p - \dot \rho x} )^2 }} ]$, where $\rho (t)$ is a function introduced into the discussion and which satisfies $\ddot \rho + \Omega ^2 \rho = {h^2 / \rho ^3 }$. A physical meaning to the origin of the invariant is presented here. The motion is viewed as the projection of an auxiliary two-dimensional motion. The relationship between the solutions of the two equations is pursued, and some particular cases when $\Omega $ has certain values are discussed.