Nonlinearity from linearity: The Ermakov–Pinney equation revisited

In this short note, we revisit the so-called Ermakov–Pinney (EP) equation viewing its properties from a physically motivated perspective. We discuss its ties with the Schrödinger equation from such a perspective, demonstrating how the Ermakov–Pinney equation arises essentially due to the conservation of angular momentum. One of the main findings of the present work is how to use this conservation law to give a simple geometric proof of the nonlinear superposition principle applicable to the solutions of the EP equation. We also present ways in which the EP equation can be generalized and discuss their connections to earlier work. The other main novelty of this work consists of the generalization of the EP equation to higher dimensions.