Recent Applications of the Theory of Lie Systems in Ermakov Systems
Jos\'e F. Cari\~nena, Javier De Lucas, Manuel F. Ra\~nada
TL;DR
This paper reviews recent advances in Lie systems theory applied to Ermakov systems, revealing fundamental properties and deriving new superposition rules for related equations.
Contribution
It introduces new superposition rules for the Pinney equation using Lie systems, expanding understanding of Ermakov systems from this perspective.
Findings
Superposition rules for Ermakov systems are derived.
New superposition rule for the Pinney equation involving Riccati solutions.
Fundamental properties like invariants are characterized via Lie systems.
Abstract
We review some recent results of the theory of Lie systems in order to apply such results to study Ermakov systems. The fundamental properties of Ermakov systems, i.e. their superposition rules, the Lewis-Ermakov invariants, etc., are found from this new perspective. We also obtain new results, such as a new superposition rule for the Pinney equation in terms of three solutions of a related Riccati equation.
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