A Novel Potential Featuring Off-Center Circular Orbits
In Book 1, Proposition 7, Problem 2 of his 1687 Philosophiae Naturalis Principia Mathematica, Isaac Newton poses and answers the following question: Let the orbit of a particle moving in a central force field be an off-center circle. How does the magnitude of the force depend on the position of the...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2023 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2023
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211883 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | A Novel Potential Featuring Off-Center Circular Orbits, Maxim Olshanii, SIGMA 19 (2023), 001, 8 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862669577191686144 |
|---|---|
| author | Olshanii, Maxim |
| author_facet | Olshanii, Maxim |
| citation_txt | A Novel Potential Featuring Off-Center Circular Orbits, Maxim Olshanii, SIGMA 19 (2023), 001, 8 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In Book 1, Proposition 7, Problem 2 of his 1687 Philosophiae Naturalis Principia Mathematica, Isaac Newton poses and answers the following question: Let the orbit of a particle moving in a central force field be an off-center circle. How does the magnitude of the force depend on the position of the particle onthat circle? In this article, we identify a potential that can produce such a force, only at zero energy. We further map the zero-energy orbits in this potential to finite-energy free motion orbits on a sphere; such a duality is a particular instance of a general result by Goursat, from 1887. The map itself is an inverse stereographic projection, and this fact explains the circularity of the zero-energy orbits in the system of interest. Finally, we identify an additional integral of motion—an analogue of the Runge-Lenz vector in the Coulomb problem—that is responsible for the closeness of the zero-energy orbits in our problem.
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| first_indexed | 2026-03-16T13:28:44Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211883 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-16T13:28:44Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Olshanii, Maxim 2026-01-14T09:57:13Z 2023 A Novel Potential Featuring Off-Center Circular Orbits, Maxim Olshanii, SIGMA 19 (2023), 001, 8 pages 1815-0659 2020 Mathematics Subject Classification: 37J35; 68-03 arXiv:2207.09606 https://nasplib.isofts.kiev.ua/handle/123456789/211883 https://doi.org/10.3842/SIGMA.2023.001 In Book 1, Proposition 7, Problem 2 of his 1687 Philosophiae Naturalis Principia Mathematica, Isaac Newton poses and answers the following question: Let the orbit of a particle moving in a central force field be an off-center circle. How does the magnitude of the force depend on the position of the particle onthat circle? In this article, we identify a potential that can produce such a force, only at zero energy. We further map the zero-energy orbits in this potential to finite-energy free motion orbits on a sphere; such a duality is a particular instance of a general result by Goursat, from 1887. The map itself is an inverse stereographic projection, and this fact explains the circularity of the zero-energy orbits in the system of interest. Finally, we identify an additional integral of motion—an analogue of the Runge-Lenz vector in the Coulomb problem—that is responsible for the closeness of the zero-energy orbits in our problem. We thank Francisco J. Jauffred and Bala Sundaram for inspiring this project, and Vanja Dunjko and Steven Jackson for the helpful discussions. We are indebted to the anonymous referee who inspired us to reorganize the original draft drastically and provided the absolutely crucial references [8] and [6]. This work was supported by NSF grant PHY-1912542. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Novel Potential Featuring Off-Center Circular Orbits Article published earlier |
| spellingShingle | A Novel Potential Featuring Off-Center Circular Orbits Olshanii, Maxim |
| title | A Novel Potential Featuring Off-Center Circular Orbits |
| title_full | A Novel Potential Featuring Off-Center Circular Orbits |
| title_fullStr | A Novel Potential Featuring Off-Center Circular Orbits |
| title_full_unstemmed | A Novel Potential Featuring Off-Center Circular Orbits |
| title_short | A Novel Potential Featuring Off-Center Circular Orbits |
| title_sort | novel potential featuring off-center circular orbits |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211883 |
| work_keys_str_mv | AT olshaniimaxim anovelpotentialfeaturingoffcentercircularorbits AT olshaniimaxim novelpotentialfeaturingoffcentercircularorbits |