The Relativistic Harmonic Oscillator in a Uniform Gravitational Field
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 Eule...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/4/294 |
| _version_ | 1797416149614329856 |
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| author | Michael M. Tung |
| author_facet | Michael M. Tung |
| author_sort | Michael M. Tung |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-09T05:59:09Z |
| format | Article |
| id | doaj.art-7c5143f076f04042a6afc654439ff76c |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T05:59:09Z |
| publishDate | 2021-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-7c5143f076f04042a6afc654439ff76c2023-12-03T12:10:45ZengMDPI AGMathematics2227-73902021-02-019429410.3390/math9040294The Relativistic Harmonic Oscillator in a Uniform Gravitational FieldMichael M. Tung0Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de Valencia, Camino de Vera, s/n, 46022 Valencia, SpainWe 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.https://www.mdpi.com/2227-7390/9/4/294relativistic harmonic oscillatorkinematics of a particlespecial relativitynonlinear problems in mechanicsequations of motion in gravitational theory |
| spellingShingle | Michael M. Tung The Relativistic Harmonic Oscillator in a Uniform Gravitational Field Mathematics relativistic harmonic oscillator kinematics of a particle special relativity nonlinear problems in mechanics equations of motion in gravitational theory |
| title | The Relativistic Harmonic Oscillator in a Uniform Gravitational Field |
| title_full | The Relativistic Harmonic Oscillator in a Uniform Gravitational Field |
| title_fullStr | The Relativistic Harmonic Oscillator in a Uniform Gravitational Field |
| title_full_unstemmed | The Relativistic Harmonic Oscillator in a Uniform Gravitational Field |
| title_short | The Relativistic Harmonic Oscillator in a Uniform Gravitational Field |
| title_sort | relativistic harmonic oscillator in a uniform gravitational field |
| topic | relativistic harmonic oscillator kinematics of a particle special relativity nonlinear problems in mechanics equations of motion in gravitational theory |
| url | https://www.mdpi.com/2227-7390/9/4/294 |
| work_keys_str_mv | AT michaelmtung therelativisticharmonicoscillatorinauniformgravitationalfield AT michaelmtung relativisticharmonicoscillatorinauniformgravitationalfield |