Exact Partial Solution in a Form of Convex Tetragons Interacting According to the Arbitrary Law for Four Bodies
We prove the existence of exact partial solutions in the form of convex quadrilaterals in the general problem of four bodies mutually acting according to an arbitrary law ~1/rк , where k ≥ 2. For each fixed k ≥ 2 the distances between the bodies and their corresponding sets of four masses are found,...
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| Format: | Article |
| Language: | English |
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Peoples’ Friendship University of Russia (RUDN University)
2024-12-01
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| Series: | RUDN Journal of Engineering Research |
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| Online Access: | https://journals.rudn.ru/engineering-researches/article/viewFile/42384/24385 |
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| author | Yulianna V. Perepelkina |
| author_facet | Yulianna V. Perepelkina |
| author_sort | Yulianna V. Perepelkina |
| collection | DOAJ |
| description | We prove the existence of exact partial solutions in the form of convex quadrilaterals in the general problem of four bodies mutually acting according to an arbitrary law ~1/rк , where k ≥ 2. For each fixed k ≥ 2 the distances between the bodies and their corresponding sets of four masses are found, which determine private solutions in the form of square, rhombus, deltoid and trapezoid. On the basis of the methodology of classical works the equations of motion in Raus - Lyapunov variables in the general problem of four bodies interacting according to a completely arbitrary law are derived, as it took place when Laplace proved the existence of exact partial triangular solutions of the general problem of three bodies with arbitrary masses. An explanation of the problem of existence of this type of solutions is given, due, in particular, to the more complicated geometry of quadrangular solutions in comparison with triangular ones, the existence of which is proved in the general three-body problem by the classics of celestial mechanics. It is suggested that if the arbitrariness of the interaction law is somewhat restricted, it is possible to prove by numerical methods the existence of exact partial solutions at different fixed values k ≥ 2 and unequal values of the masses of the four bodies. |
| format | Article |
| id | doaj-art-c3c12a9ba71d45beac6c105cecead6c0 |
| institution | Kabale University |
| issn | 2312-8143 2312-8151 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Peoples’ Friendship University of Russia (RUDN University) |
| record_format | Article |
| series | RUDN Journal of Engineering Research |
| spelling | doaj-art-c3c12a9ba71d45beac6c105cecead6c02025-01-14T08:09:51ZengPeoples’ Friendship University of Russia (RUDN University)RUDN Journal of Engineering Research2312-81432312-81512024-12-0125328829510.22363/2312-8143-2024-25-3-288-29521131Exact Partial Solution in a Form of Convex Tetragons Interacting According to the Arbitrary Law for Four BodiesYulianna V. Perepelkina0https://orcid.org/0000-0001-8115-8253Russian Institute for Scientific and Technical Information of Russian Academy of SciencesWe prove the existence of exact partial solutions in the form of convex quadrilaterals in the general problem of four bodies mutually acting according to an arbitrary law ~1/rк , where k ≥ 2. For each fixed k ≥ 2 the distances between the bodies and their corresponding sets of four masses are found, which determine private solutions in the form of square, rhombus, deltoid and trapezoid. On the basis of the methodology of classical works the equations of motion in Raus - Lyapunov variables in the general problem of four bodies interacting according to a completely arbitrary law are derived, as it took place when Laplace proved the existence of exact partial triangular solutions of the general problem of three bodies with arbitrary masses. An explanation of the problem of existence of this type of solutions is given, due, in particular, to the more complicated geometry of quadrangular solutions in comparison with triangular ones, the existence of which is proved in the general three-body problem by the classics of celestial mechanics. It is suggested that if the arbitrariness of the interaction law is somewhat restricted, it is possible to prove by numerical methods the existence of exact partial solutions at different fixed values k ≥ 2 and unequal values of the masses of the four bodies.https://journals.rudn.ru/engineering-researches/article/viewFile/42384/24385celestial mechanicsfour-body problemraus - lyapunov variablesparticular solutionslaws of interactionbounded problems |
| spellingShingle | Yulianna V. Perepelkina Exact Partial Solution in a Form of Convex Tetragons Interacting According to the Arbitrary Law for Four Bodies RUDN Journal of Engineering Research celestial mechanics four-body problem raus - lyapunov variables particular solutions laws of interaction bounded problems |
| title | Exact Partial Solution in a Form of Convex Tetragons Interacting According to the Arbitrary Law for Four Bodies |
| title_full | Exact Partial Solution in a Form of Convex Tetragons Interacting According to the Arbitrary Law for Four Bodies |
| title_fullStr | Exact Partial Solution in a Form of Convex Tetragons Interacting According to the Arbitrary Law for Four Bodies |
| title_full_unstemmed | Exact Partial Solution in a Form of Convex Tetragons Interacting According to the Arbitrary Law for Four Bodies |
| title_short | Exact Partial Solution in a Form of Convex Tetragons Interacting According to the Arbitrary Law for Four Bodies |
| title_sort | exact partial solution in a form of convex tetragons interacting according to the arbitrary law for four bodies |
| topic | celestial mechanics four-body problem raus - lyapunov variables particular solutions laws of interaction bounded problems |
| url | https://journals.rudn.ru/engineering-researches/article/viewFile/42384/24385 |
| work_keys_str_mv | AT yuliannavperepelkina exactpartialsolutioninaformofconvextetragonsinteractingaccordingtothearbitrarylawforfourbodies |