TITLE:
On the Relativistic Harmonic Oscillator
AUTHORS:
Yair Zarmi
KEYWORDS:
Relativistic Harmonic Oscillator, Weak-Relativistic Limit, Extreme-Relativistic Limit
JOURNAL NAME:
Applied Mathematics,
Vol.14 No.1,
January
10,
2023
ABSTRACT: The relativistic harmonic oscillator represents a unique energy-conserving oscillatory system. The detailed characteristics of the solution of this oscillator are displayed in both weak- and extreme-relativistic limits using different expansion procedures, for each limit. In the weak-relativistic limit, a Normal Form expansion is developed, which yields an approximation to the solution that is significantly better than in traditional asymptotic expansion procedures. In the extreme-relativistic limit, an expansion of the solution in terms of a small parameter that measures the proximity to the limit (v/c) → 1 yields an excellent approximation for the solution throughout the whole period of oscillations. The variation of the coefficients of the Fourier expansion of the solution from the weak- to the extreme-relativistic limits is displayed.