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Symmetric Collinear Four Body Problem via Symbolic Dynamics
https://ir.soken.ac.jp/records/404
https://ir.soken.ac.jp/records/404c18c3950-0998-4d85-9634-d301eeff3fc3
| 名前 / ファイル | ライセンス | アクション |
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要旨・審査要旨 / Abstract, Screening Result (340.0 kB)
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本文 / Thesis (3.0 MB)
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| Item type | 学位論文 / Thesis or Dissertation(1) | |||||
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| 公開日 | 2010-02-22 | |||||
| タイトル | ||||||
| タイトル | Symmetric Collinear Four Body Problem via Symbolic Dynamics | |||||
| タイトル | ||||||
| タイトル | Symmetric Collinear Four Body Problem via Symbolic Dynamics | |||||
| 言語 | en | |||||
| 言語 | ||||||
| 言語 | eng | |||||
| 資源タイプ | ||||||
| 資源タイプ識別子 | http://purl.org/coar/resource_type/c_46ec | |||||
| 資源タイプ | thesis | |||||
| 著者名 |
関口, 昌由
× 関口, 昌由 |
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| フリガナ |
セキグチ, マサヨシ
× セキグチ, マサヨシ |
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| 著者 |
SEKIGUCHI, Masayoshi
× SEKIGUCHI, Masayoshi |
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| 学位授与機関 | ||||||
| 学位授与機関名 | 総合研究大学院大学 | |||||
| 学位名 | ||||||
| 学位名 | 博士（理学） | |||||
| 学位記番号 | ||||||
| 内容記述タイプ | Other | |||||
| 内容記述 | 総研大乙第91号 | |||||
| 研究科 | ||||||
| 値 | 数物科学研究科 | |||||
| 専攻 | ||||||
| 値 | 09 天文科学専攻 | |||||
| 学位授与年月日 | ||||||
| 学位授与年月日 | 2001-09-28 | |||||
| 学位授与年度 | ||||||
| 値 | 2001 | |||||
| 要旨 | ||||||
| 内容記述タイプ | Other | |||||
| 内容記述 | Symmetric collinear four-body problem (hereafter, referred to as SC4BP) is a special case of the general Newtonian four-body problem in which the bodies are distributed symmetrically about the center of masses on a fixed common line. SC4BP is a Hamiltonian system of two degree of freedom. We analytically study SC4BP independently of the values of mass ratio and energy. We numerically study the case of equal-mass and negative energy. The present work is the first systematic study of SC4BP, and provides rich qualitative features in SC4BP with the aid of symbolic dynamics. The first study of SC4BP was made by Simo and Lacomba (1982, Celes. Mech., 28, 49-62). They studied the flow on the quadruple collision manifold, especially the invariant manifolds associated with the critical points. They obtained some special values of mass ratio for which the invariant manifolds change their behavior qualitatively. Roy and Steaves (1998, Planet Sp. Sci., 46(11/12), 1475-1486) obtained homothetic solutions for some special four-body systems including SC4BP. Sweatman (2001, Celes. Mech., to appear) numerically found some interesting behavior of orbits in SC4BP, and pointed out similarity of the phase structure to the one in the collinear three-body problem. It is easily understood that there are four states of motion in SC4BP. They are quadruple collision, 2-2 ejection (distant two binaries), 1-2-1 ejection (one central binary and two isolated bodies), and interplay (repeating different binary collisions). In general, orbits travel among these four states. It is natural to ask how they travel among the states. This question requires a qualitative study of the phase structure in SC4BP. Our purpose is to clarify qualitatively the structure of phase space in SC4BP, namely (1) classification of motion, (2) search of the classified orbits, and (3) their distribution in the phase space. The suitable method for (1) is symbolic dynamics. In SC4BP, any orbit (except for the homothetic solution) experiences an infinity of binary collisions. There are two types of binary collision. One is a single binary collision. The other one is a simultaneous binary collision. We assign different symbols to them (“0” to the former, “2” to the latter). Thus, orbits are replaced by symbol sequences. For instance, orbits in 2-2 (resp. 1-2-1) ejection state are replaced by sequences repeating 2 (resp. 0). Replacement of orbits by symbol sequences ignores quantatively small differences among orbits, but it keeps their qualitative differences. We introduce a Poincaré mapping on a surface of section. This is one of basic tools for the study of dynamical systems. The Poincaré mapping reflects the flow in the phase space. Our surface of section is the set of phase points in central configuration. The surface of section is the global surface of section, i.e., all orbits in SC4BP intersect the surface of section at least once. In addition, the surface of section is a compact, two dimensional disk. Our Poincaré mapping and surface of section, therefore, are very convenient to study the totality of motion. Our purpose is achieved by studying the mapping on the surface of section, and by both analytically and numerically investigating the distribution of symbols sequences on the surface of section. In order to remove collision singularities, we define new variables after McGehee (1974, Inventiones Mathematicae, 27, 191-227), and we blow-up the phase point of quadruple collision to the quadruple collision manifold (QCM for short). This technique enables us to understand totally the mapping on the surface of section and the flow on QCM. Our main theoretical results are summarized into five issues. First, the surface of section is proved to be a global surface of section as we mentioned above. Second, the immediate future symbols after crossing the surface of section and the immediate past symbols before crossing the surface of section depend on where the orbit crosses the surface of section. Any orbit except for the homothetic solution experiences two different types of binary collision before and after crossing the surface of section. This allows us to find un-realizable words of symbols. Third, the set of points leading to quadruple collision forms arcs. We call them quadruple collision curves (QCC for short). The end-points of a QCC are on QCM, and never meet each other. Fourth, QCCs form boundaries between regions of different symbol sequences. Fifth, we obtain escape criteria by simple two-body consideration. We express these escape criteria on the surface of section the surface of section. In numerical analyses, we regard the surface of section as the set of initial values as well, and carry out numerical integrations for points on the surface of section with fine mesh, and replace them with words. Thus, we have a distribution of words on the surface of section, and derive the following results. The region is thought to contain the invariant region. This is an evidence for the existence of orbits staying permanently in the interplay state. Second, some families of unstable periodic orbits are confirmed. This indicates the existence of orbits from ejection to interplay state permanently (or from interplay to ejection). We can classify various periodic according to the length of symbols. Third, we find some un-realizable words. There never exists any orbit whose symbol sequence contains an un-realizable word. In order to list up all un-realizable words, we divide the surface of section into some sub-regions and obtain mapping-rules among the sub-regions. Then, we have some sequences of un-realizable words. Fourth, escape regions on the surface of section are numerically determined. These contain the escape regions obtained analytically. Fifth, it is found that the distribution of words has the stratified structure in the sea of chaos on the surface of section. This is consistent with our theoretical results. Finally, initial points leading to quadruple collision are numerically discovered to form curves on the surface of section. Thus, a method searching for such points is established. This work has been the first detailed description of the phase structure in SC4BP. Our results are expected to be applied for and are extended to the more general case of four-body problem. |
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| 所蔵 | ||||||
| 値 | 有 | |||||
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| 内容記述タイプ | Other | |||||
| 内容記述 | application/pdf | |||||
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| 出版タイプ | AM | |||||
| 出版タイプResource | http://purl.org/coar/version/c_ab4af688f83e57aa | |||||
