The Ermakov-Pinney Equation: its varied origins and the effects of the introduction of symmetry-breaking functions

Abstract

The Ermakov-Pinney Equation, $$\ddot{x}+{\omega }^{2}x=\frac{h^2}{x^3},$$ has a varied provenance which we briefly delineate. We introduce time-dependent functions in place of the ${\omega }^2$ and $h^2$. The former has no effect upon the algebra of the Lie point symmetries of the equation. The latter destroys the $sl(2,\Re )$ symmetry and a single symmetry persists only when there is a specific relationship between the two time-dependent functions introduced. We calculate the form of the corresponding autonomous equation for these cases.

References in corpus (1)

• Variations on the adiabatic invariance: the Lorentz pendulum