Details

Autor(en) / Beteiligte

• Jahanpanah, J.

Titel

Relativistic potential energy of a non-dissipative classical harmonic oscillator

Ist Teil von

• Communications in nonlinear science & numerical simulation, 2024-01, Vol.128, p.107614, Article 107614

Ort / Verlag

Elsevier B.V

Erscheinungsjahr

2024

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• •The Lorentz factor γ=(1−X˙2/c2)−12 transforms the state of the second-order differential equation of a classical harmonic oscillator X¨+ω02γ−3X=0 from linear into nonlinear.•The relativistic form of Newton's second law Frel=dprel/dt⌣=−kX is a semi relativistic equation due to the nonrelativistic Hook force −kX.•The relativistic relations of the Hook force and potential energy of a classical harmonic oscillator are introduced for the first time.•The relativistic harmonic oscillator obeys from the principles of special relativity. 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.