2020 Fiscal Year Final Research Report
A new approach to construct planar equal-mass three-body choreographic solutions
Project/Area Number |
17K05588
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Research Category |
Grant-in-Aid for Scientific Research (C)
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Allocation Type | Multi-year Fund |
Section | 一般 |
Research Field |
Mathematical physics/Fundamental condensed matter physics
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Research Institution | Tokai University |
Principal Investigator |
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Project Period (FY) |
2017-04-01 – 2021-03-31
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Keywords | 三体問題 / 等質量3体8の字解 / 運動量保存則 / 角運動量保存則 / エネルギー保存則 / モース指数 / 分岐解 |
Outline of Final Research Achievements |
(1) We constructed an approximated choreographic solution of the planar equal masses three-body problem under the strong potential with Jacobian elliptic function. (2) We found a relation between three-body positions and the orbit function of the intersection point of three tangent lines from the three bodies with zero total angular momentum. (3) We found that change of the Morse index is not only necessary but also sufficient condition for bifurcations from the choreographic motion of equal mass three bodies under homogeneous potential, and Lenard-Jones type potential. The bifurcated periodic solutions are constructed with eigenfunctions belonging to zero eigenvalues under the eigenvalue problem of the Hessian, which is the second variation of the action integral.
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Free Research Field |
数理物理
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Academic Significance and Societal Importance of the Research Achievements |
三つの恒星が互いの重力で引き合いながら運動している恒星系が最近見つかりました.「三体の初期位置と初速度を与えたとき,三体のニュートンの運動方程式を解析的に解きなさい」という問題を「三体問題」と言います.三体問題は正攻法では解けないのですが,特解として直線解,正三角形解,8の字解の3つが知られています.3番目だけは,三体の質量がすべて等しい,という仮定が必要ですが,それでも手で解くことはできていません.高精度数値解がわかっているだけです.この研究では,謎の8の字解析解に手計算で挑みました.解くことはできませんでしたが,手で解くためのヒントをいくつか得ることができました.
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