The Relativistic Harmonic Oscillator in a Uniform Gravitational Field

Michael M. Tung
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Michael M. Tung: Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de Valencia, Camino de Vera, s/n, 46022 Valencia, Spain

Mathematics, 2021, vol. 9, issue 4, 1-12

Abstract: We present the relativistic generalization of the classical harmonic oscillator suspended within a uniform gravitational field measured by an observer in a laboratory in which the suspension point of the spring is fixed. The starting point of this analysis is a variational approach based on the Euler–Lagrange formalism. Due to the conceptual differences of mass in the framework of special relativity compared with the classical model, the correct treatment of the relativistic gravitational potential requires special attention. It is proved that the corresponding relativistic equation of motion has unique periodic solutions. Some approximate analytical results including the next-to-leading-order term in the non-relativistic limit are also examined. The discussion is rounded up with a numerical simulation of the full relativistic results in the case of a strong gravity field. Finally, the dynamics of the model is further explored by investigating phase space and its quantitative relativistic features.

Keywords: relativistic harmonic oscillator; kinematics of a particle; special relativity; nonlinear problems in mechanics; equations of motion in gravitational theory (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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