The Relativistic Anharmonic Oscillator within a Double-Well Potential

Abstract

[EN] Non-harmonic oscillations described by a double-well potential allow to model various physical phenomena and provide insight into their properties without overly simplifying the problem, thereby extending the more elementary harmonic case. Important applications of the anharmonic oscillator range from classical instantons and field theory to quantum theory and particle physics [1¿3]. This presentation focuses on the treatment of the anharmonic oscillator in connection with the double-well potential including the non-negligible relativistic effects of a strong and uniform gravitational field. It extends previous work of the relativistic harmonic oscillator to the anharmonic model [4]. Our analysis parts from a relativistic action principle, where the dynamics is described as usual by the corresponding Euler-Lagrange equations. However, complications arise on how to handle the relativistic potential of the strong gravitational field. We proceed by deriving an integral equation for this potential following an approach¿a method introduced by Goldstein & Bender for the brachistochrone problem [5] and more recently applied to the pendulum [6, 7]. Finally, a numerical simulation of the relativistic results is carried out to examine the dynamics of the model and compare it with the purely classical calculations. A detailed study of the phase space for both models will enable us to detect very distinct features among them, and thereby confirms the necessity for including relativistic corrections to yield reliable predictions for the strong-gravity case.