Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Relativistic Ermakov-Milne-Pinney Systems and First Integrals

Version 1 : Received: 6 January 2021 / Approved: 8 January 2021 / Online: 8 January 2021 (11:13:35 CET)

A peer-reviewed article of this Preprint also exists.

Haas, F. Relativistic Ermakov–Milne–Pinney Systems and First Integrals. Physics 2021, 3, 59-70. Haas, F. Relativistic Ermakov–Milne–Pinney Systems and First Integrals. Physics 2021, 3, 59-70.

Journal reference: Physics 2021, 3, 6
DOI: 10.3390/physics3010006


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 as 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.

Subject Areas

Ermakov system; Ermakov-Milne-Pinney equation; relativistic Ermakov-Lewis invariant; relativistic Ray-Reid system; nonlinear superposition law.

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