Analytic central orbits and their transformation group
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Bibliographic record
Abstract
A useful crude approximation for Abelian functions is developed and applied to orbits. The bound orbits in the power-law potentials A r−α take the simple form (ℓ/r)k= 1 +e cos (mφ), where k= 2 −α > 0 and ℓ and e are generalizations of the semi-latus-rectum and the eccentricity. m is given as a function of ‘eccentricity’. For nearly circular orbits m is , while the above orbit becomes exact at the energy of escape where e is 1 and m is k. Orbits in the logarithmic potential that gives rise to a constant circular velocity are derived via the limit α→ 0. For such orbits, r2 vibrates almost harmonically whatever the ‘eccentricity’. Unbound orbits in power-law potentials are given in an appendix. The transformation of orbits in one potential to give orbits in a different potential is used to determine orbits in potentials that are positive powers of r. These transformations are extended to form a group which associates orbits in sets of six potentials, e.g. there are corresponding orbits in the potentials proportional to r, r−2/3, r−3, r−6, r−4/3 and r4. A degeneracy reduces this to three, which are r−1, r2 and r−4 for the Keplerian case. A generalization of this group includes the isochrone with the Kepler set.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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