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On the Damped Pinney Equation from Noether Symmetry Principles
Version 1
: Received: 13 September 2024 / Approved: 14 September 2024 / Online: 15 September 2024 (07:32:51 CEST)
How to cite: Haas, F. On the Damped Pinney Equation from Noether Symmetry Principles. Preprints 2024, 2024091121. https://doi.org/10.20944/preprints202409.1121.v1 Haas, F. On the Damped Pinney Equation from Noether Symmetry Principles. Preprints 2024, 2024091121. https://doi.org/10.20944/preprints202409.1121.v1
Abstract
There are several versions of the damped form of the celebrated Pinney equation, which is the natural partner of the undamped time-dependent harmonic oscillator. In this work these dissipative versions of the Pinney equation are briefly reviewed. We show that Noether’s theorem for the usual time-dependent harmonic oscillator as a guiding principle for derivation of the Pinney equation also works in the damped case, selecting a Noether symmetry based damped Pinney equation. The results are extended to general nonlinear damped Ermakov systems. A certain time-rescaling always allows to remove the damping from the final equations.
Keywords
Pinney equation; Noether theorem; Ermakov systems; dissipative systems
Subject
Physical Sciences, Mathematical Physics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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