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Relativistic Ermakov–Milne–Pinney Systems and First Integrals

Physics Institute, Federal University of Rio Grande do Sul, Avenida Bento Gonçalves 9500, Porto Alegre 91501-970, Rio Grande do Sul, Brazil
Received: 6 January 2021 / Revised: 2 February 2021 / Accepted: 4 February 2021 / Published: 12 February 2021
(This article belongs to the Section Classical Physics)
The Ermakov–Milne–Pinney equation is ubiquitous in many areas of physics that have an explicit time-dependence, including quantum systems with time-dependent Hamiltonian, cosmology, time-dependent harmonic oscillators, accelerator dynamics, etc. The Eliezer and Gray physical interpretation of the Ermakov–Lewis invariant is applied as a guiding principle for the derivation of the special relativistic analog of the Ermakov–Milne–Pinney equation and associated first integral. The special relativistic extension of the Ray–Reid system and invariant is obtained. General properties of the relativistic Ermakov–Milne–Pinney are analyzed. The conservative case of the relativistic Ermakov–Milne–Pinney equation is described in terms of a pseudo-potential, reducing the problem to an effective Newtonian form. The non-relativistic limit is considered to be well. A relativistic nonlinear superposition law for relativistic Ermakov systems is identified. The generalized Ermakov–Milne–Pinney equation has additional nonlinearities, due to the relativistic effects. View Full-Text
Keywords: Ermakov system; Ermakov–Milne–Pinney equation; relativistic Ermakov–Lewis invariant; relativistic Ray–Reid system; nonlinear superposition law Ermakov system; Ermakov–Milne–Pinney equation; relativistic Ermakov–Lewis invariant; relativistic Ray–Reid system; nonlinear superposition law
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MDPI and ACS Style

Haas, F. Relativistic Ermakov–Milne–Pinney Systems and First Integrals. Physics 2021, 3, 59-70. https://0-doi-org.brum.beds.ac.uk/10.3390/physics3010006

AMA Style

Haas F. Relativistic Ermakov–Milne–Pinney Systems and First Integrals. Physics. 2021; 3(1):59-70. https://0-doi-org.brum.beds.ac.uk/10.3390/physics3010006

Chicago/Turabian Style

Haas, Fernando. 2021. "Relativistic Ermakov–Milne–Pinney Systems and First Integrals" Physics 3, no. 1: 59-70. https://0-doi-org.brum.beds.ac.uk/10.3390/physics3010006

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