Relativistic Ermakov–Milne–Pinney Systems and First Integrals

Abstract

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.

Relativistic formulation of curl force, relativistic Kapitza equation and trapping

“…This version of the relativistic harmonic oscillator model has recently been probed experimentally also [28]. Recently, Haas [29] generalized Ermakov systems towards the special relativity domain. In fact relativistic nonlinear dynamics is an open area of research and deserves more attention and intense studies.…”

Destructive relativity

“…The study of the dynamics of classical systems in relativistic regimes is currently in a great activity of the scientific community. Let us recall the relativistic Kapitza system [1]; the relativistic hydrogen-like atom in a mag-netic field [2,3]; the relativistic two-dimensional harmonic and anharmonic oscillators in a uniform gravitational field [4][5][6]; the relativistic Lienard-type oscillators [7] or the relativistic time-dependent Ermakov-Milne-Pinney systems [8] to mention by name just a few.…”