On the Sundman-Sperling estimates for the restricted one-center-two-body problem
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| Format: | Preprint |
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2024
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| _version_ | 1866913323375656960 |
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| author | Hsu, Ku-Jung Liu, Lei |
| author_facet | Hsu, Ku-Jung Liu, Lei |
| contents | In the past two decades, since the discovery of the figure-8 orbit by Chenciner and Montgomery, the variational method has became one of the most popular tools for constructing new solutions of the $N$-body problem and its extended problems. However, finding solutions to the restricted three-body problem, in particular, the two primaries form a collision Kepler system, remains a great difficulty. One of the major reasons is the essential differences between two-body collisions and three-body collisions.
In this paper, we consider a similar three-body system with less difficulty, i.e. the restricted one-center-two-body system, that is involving a massless particle and a collision Kepler system with one body fixed. It is an intermediate system between the restricted three-body problem and the two-center problem. By an in-depth analysis of the asymptotic behavior of the minimizer, and an argument of critical and infliction points, we prove the Sundman-Sperling estimates near the three-body collision for the minimizers. With these estimates, we provide a class of collision-free solutions with prescribed boundary angles. Finally, under the extended collision Kepler system from Gordon, we constructed a family of periodic and quasi-periodic solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_13572 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Sundman-Sperling estimates for the restricted one-center-two-body problem Hsu, Ku-Jung Liu, Lei Dynamical Systems Analysis of PDEs 70F07, 70F15, 70F16, 70G75, 70M20 In the past two decades, since the discovery of the figure-8 orbit by Chenciner and Montgomery, the variational method has became one of the most popular tools for constructing new solutions of the $N$-body problem and its extended problems. However, finding solutions to the restricted three-body problem, in particular, the two primaries form a collision Kepler system, remains a great difficulty. One of the major reasons is the essential differences between two-body collisions and three-body collisions. In this paper, we consider a similar three-body system with less difficulty, i.e. the restricted one-center-two-body system, that is involving a massless particle and a collision Kepler system with one body fixed. It is an intermediate system between the restricted three-body problem and the two-center problem. By an in-depth analysis of the asymptotic behavior of the minimizer, and an argument of critical and infliction points, we prove the Sundman-Sperling estimates near the three-body collision for the minimizers. With these estimates, we provide a class of collision-free solutions with prescribed boundary angles. Finally, under the extended collision Kepler system from Gordon, we constructed a family of periodic and quasi-periodic solutions. |
| title | On the Sundman-Sperling estimates for the restricted one-center-two-body problem |
| topic | Dynamical Systems Analysis of PDEs 70F07, 70F15, 70F16, 70G75, 70M20 |
| url | https://arxiv.org/abs/2404.13572 |