Abstract
Milne–Pinney equation \(\ddot x=-\omega^2(t)x+ k/{x^3}\) is usually studied together with the time-dependent harmonic oscillator \(\ddot y+\omega^2(t) y=0\) and the system is called Ermakov system, and actually Pinney showed in a short paper that the general solution of the first equation can be written as a superposition of two solutions of the associated harmonic oscillator. A recent generalization of the concept of Lie systems for second order differential equations and the usual techniques of Lie systems will be used to study the Ermakov system. Several applications of Ermakov systems in Quantum Mechanics as the relation between Schroedinger and Milne equations or the use of Lewis–Riesenfeld invariant will be analysed from this geometric viewpoint.
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