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arXiv · 1510.08992

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

Richard Michael Morris·Peter Gavin Lawrence Leach
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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.

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Richard Michael Morris, Peter Gavin Lawrence Leach. 2015-10-30. The Ermakov-Pinney Equation: its varied origins and the effects of the introduction of symmetry-breaking functions. https://arxiv.org/abs/1510.08992

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