Relativistic potential energy of a non-dissipative classical harmonic oscillator

Abstract

The well-known relation of Einstein's relativistic energy E=mc2 for a free particle is extended to cover the total relativistic energy of a classical harmonic oscillator (CHO) by calculating the relativistic potential energy. This study is essentially concerned with the relativistic mass m=γm0, where the Lorentz factor γ transforms the state of the second-order differential equation of a CHO from linear into nonlinear. Although the nonlinear solution still remains periodic, the amplitude A(β) and angular frequency ω(β) are determined only by the dimensionless factor β=X˙(0)/c (the ratio of the initial velocity X˙(0) to the light speed c). It is demonstrated that the time period T(β) and Hook force respectively tend to infinity and zero when β→1 so that the state of the bound particle approaches that of a free particle. By contrast, the relativistic behavior of a quantum harmonic oscillator (QHO) in subatomic scales, such as the relativistic ro-vibrational motion of an electron in circular Bohr orbits, is completely different. The relativistic model of Bohr atom is discussed in detail by using the relativistic relations of a CHO according to the Ehrenfest theorem. Finally, the results are confirmed by demonstrating energy conservation.