Topic
Kepler problem
About: Kepler problem is a research topic. Over the lifetime, 1103 publications have been published within this topic receiving 19681 citations.
Papers published on a yearly basis
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Papers
TL;DR: In this article, the authors apply to the planetary problem the theorems proved by Lochak about stability for near integrable Hamiltonian systems over timescales which are exponentially long with respect to the inverse of the size of the perturbation.
Abstract: In this article, we apply to the planetary problem the theorems proved by Lochak about stability for near integrable Hamiltonian systems over timescales which are exponentially long with respect to the inverse of the size of the perturbation. The Lochak’s theorems allow for important improvements of the original results obtained by Nekhorochev. Thus, we can give analytic results of stability in the problem with n planets (n arbitrary), whose eccentricities and inclinations are comparable with those met in our solar system. The threshold of application of these theorems on the masses of the planets are still low but can be improved and, perhaps, be pushed close to realistic values by making one (or more) numerical averaging on the system associated with the planetary problem before applying our theorems.
TL;DR: In this article, the classical hypervirial and Hellmann-Feynman theorems are used to formulate a perturbation theory without Fourier series that can be used to generate canonical series expansions for the energies of perturbed periodic orbits for separable classical Hamiltonians.
Abstract: The classical hypervirial and Hellmann–Feynman theorems are used to formulate a ‘‘perturbation theory without Fourier series’’ that can be used to generate canonical series expansions for the energies of perturbed periodic orbits for separable classical Hamiltonians. As in the case where these theorems are used to generate quantum mechanical Rayleigh–Schrodinger perturbation series, the method is very efficient and may be used to generate expansions to large order either numerically or in algebraic form. Here, the method is applied to one‐dimensional anharmonic oscillators and radial Kepler problems. In all cases, the classical series for energies and expectation values are seen to correspond to the expansions associated with their quantum mechanical counterparts through an appropriate action preserving classical limit as discussed by Turchetti, Graffi, and Paul. This ‘‘action fixing’’ is inherent in the classical Hellmann–Feynman theorem applied to periodic orbits.
TL;DR: In this article, it was shown that the solution is related to the Kepler problem; the trajectories in velocity space are conic sections and the force experienced by roller coaster riders is the normal force, so a natural question to ask is: what shape of the track gives a normal force of constant magnitude?
Abstract: A particle that moves along a smooth track in a vertical plane is influenced by two forces: gravity and normal force. The force experienced by roller coaster riders is the normal force, so a natural question to ask is: what shape of the track gives a normal force of constant magnitude? Here we solve this problem. It turns out that the solution is related to the Kepler problem; the trajectories in velocity space are conic sections.
TL;DR: In this article, the WKB series for the angular momentum and the non-relativistic three-dimesional Kepler problem is calculated and the torus quantization of the leading WKB term is shown to be exact.
Abstract: We calculate the WKB series for the angular momentum and the non-relativistic three-dimesional Kepler problem. This is the first semiclassical treatment of the angular momentum for terms beyond the leading WKB approximation. We explain why the torus quantization (the leading WKB term) of the full problem is exact, even if the individual torus quantization of the angular momentum and of the radial Kepler problem separately is not exact. In this way we derive Langer's rule, calculate the first correction to the leading Langer's term and conjecture the form of all higher terms.
TL;DR: In this article, the generalization of the motion of a particle in a central field to the case of a constant curvature space is investigated and the integrability of the generalized two-centre problem remains even if elastic forces are added.
Abstract: In this article the generalization of the motion of a particle in a central field to the case of a constant curvature space is investigated. We found out that orbits on a constant curvature surface are closed in two cases: when the potential satisfies Iaplace-Beltrami equation and can be regarded as an analogue of the potential of the gravitational interaction, and in the case when the potential is the generalization of the potential of an elastic spring. Also the full integrability of the generalized two-centre problem on a constant curvature surface is discovered and it is shown that integrability remains even if elastic “forces” are added.
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 12 |
| 2024 | 9 |
| 2023 | 23 |
| 2022 | 51 |
| 2021 | 30 |
| 2020 | 29 |