Velocity Syzygies and Bounding Syzygy Moments in the Planar Three-Body Problem
We consider the Newtonian planar three-body problem, defining a syzygy (velocity syzygy) as a configuration where the positions (velocities) of the three bodies become collinear. We demonstrate that if the total energy is negative, every collision-free solution has an infinite number of velocity syz...
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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.683/ |
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| author | Tsygvintsev, Alexei |
| author_facet | Tsygvintsev, Alexei |
| author_sort | Tsygvintsev, Alexei |
| collection | DOAJ |
| description | We consider the Newtonian planar three-body problem, defining a syzygy (velocity syzygy) as a configuration where the positions (velocities) of the three bodies become collinear. We demonstrate that if the total energy is negative, every collision-free solution has an infinite number of velocity syzygies. Specifically, the velocities of the three bodies become parallel within every interval of time containing three consecutive syzygies. Using comparison theory for matrix Riccati equations, we derive new upper and lower bounds on the moments when syzygies occur. |
| format | Article |
| id | doaj-art-5215a0c5a7a84cbc84f9ec109eadb09e |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-5215a0c5a7a84cbc84f9ec109eadb09e2025-02-07T11:26:37ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G121785179110.5802/crmath.68310.5802/crmath.683Velocity Syzygies and Bounding Syzygy Moments in the Planar Three-Body ProblemTsygvintsev, Alexei0https://orcid.org/0000-0002-8744-4100ENS de Lyon, UMPA, 46 allée d’Italie, 69364 Lyon Cedex 07, FranceWe consider the Newtonian planar three-body problem, defining a syzygy (velocity syzygy) as a configuration where the positions (velocities) of the three bodies become collinear. We demonstrate that if the total energy is negative, every collision-free solution has an infinite number of velocity syzygies. Specifically, the velocities of the three bodies become parallel within every interval of time containing three consecutive syzygies. Using comparison theory for matrix Riccati equations, we derive new upper and lower bounds on the moments when syzygies occur.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.683/dynamical systemscelestial mechanicsthree-body problemsyzygies |
| spellingShingle | Tsygvintsev, Alexei Velocity Syzygies and Bounding Syzygy Moments in the Planar Three-Body Problem Comptes Rendus. Mathématique dynamical systems celestial mechanics three-body problem syzygies |
| title | Velocity Syzygies and Bounding Syzygy Moments in the Planar Three-Body Problem |
| title_full | Velocity Syzygies and Bounding Syzygy Moments in the Planar Three-Body Problem |
| title_fullStr | Velocity Syzygies and Bounding Syzygy Moments in the Planar Three-Body Problem |
| title_full_unstemmed | Velocity Syzygies and Bounding Syzygy Moments in the Planar Three-Body Problem |
| title_short | Velocity Syzygies and Bounding Syzygy Moments in the Planar Three-Body Problem |
| title_sort | velocity syzygies and bounding syzygy moments in the planar three body problem |
| topic | dynamical systems celestial mechanics three-body problem syzygies |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.683/ |
| work_keys_str_mv | AT tsygvintsevalexei velocitysyzygiesandboundingsyzygymomentsintheplanarthreebodyproblem |