| Project/Area Number |
17K05146
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Computational science
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| Research Institution | Kitasato University |
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Principal Investigator |
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| Project Period (FY) |
2017-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2017: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
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| Keywords | 8の字解 / 分岐 / 群論的分岐理論 / Lennard-Jones型ポテンシャル / 作用汎関数 / 3体問題 / モースインデックス / コレオグラフィー / オイラー標数 / Morse Index |
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| Outline of Final Research Achievements |
For equal mass three-body system in classical mechanics interacting through Lennard-Jones-type inhomogeneous potential, a model potential between atoms, there exist a lot of figure-eight choreographic solutions in which three bodies move in the same eight shaped orbit periodically. The solutions change shapes of the orbits by period T. We found that the solution bifurcate, that the necessary and almost sufficient condition for the bifurcation is the change of Morse index of the action functional, and that the bifurcated solution can be determined by the variation functions responsible to the change of Morse index. Further, we developed a bifurcation theory for periodic solution in n body system by action functional and the group theoretic bifurcation theory.
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| Academic Significance and Societal Importance of the Research Achievements |
天体の運動は，力が距離の2乗に反比例する，万有引力の法則に従う。本研究では，その力を，原子や分子の間に働く，距離が離れると距離の7乗に反比例するようになるLennard-Jones型の力に置き換えて，3つの同じ質量の天体が，8の字型の軌道上をおいかけっこする8の字運動の研究をした。この8の字運動には様々な形のものがあり，運動の周期Tを変化させると，8の字の形が変化し，解が2種類以上に分裂する分岐現象が起こる。我々は，特に分岐現象についてコンピュータで詳しく数値計算して，そのメカニズムを解明した。そして，一般にn個の天体の運動に適応できる分岐理論を構築した。
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