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Numerical study on periodic solution of three-body problem under inhomogeneous interaction potential

Research Project

Project/Area Number 17K05146
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Computational science
Research InstitutionKitasato University

Principal Investigator

Fukuda Hiroshi  北里大学, 一般教育部, 教授 (70238484)

Project Period (FY) 2017-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
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)
Keywords8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
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.

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。

Report

(4 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (16 results)

All 2020 2019 2018 2017 Other

All Journal Article (4 results) (of which Open Access: 2 results,  Peer Reviewed: 2 results) Presentation (10 results) (of which Int'l Joint Research: 3 results) Book (1 results) Remarks (1 results)

  • [Journal Article] Variational principle of action and group theory for bifurcation of figure-eight solutions2020

    • Author(s)
      Fujiwara T, Fukuda H, Ozaki H
    • Journal Title

      arXiv

      Volume: arXiv:2002.03496

    • Related Report
      2019 Annual Research Report
    • Open Access
  • [Journal Article] Variational principle for bifurcation in Lagrangian mechanics2019

    • Author(s)
      Fujiwara T, Fukuda H, Ozaki H
    • Journal Title

      arXiv

      Volume: arXiv:1905.11073

    • Related Report
      2019 Annual Research Report
    • Open Access
  • [Journal Article] Morse index and bifurcation for figure-eight choreographies of the equal mass three-body problem2019

    • Author(s)
      Fukuda Hiroshi、Fujiwara Toshiaki、Ozaki Hiroshi
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 52 Issue: 18 Pages: 185201-185201

    • DOI

      10.1088/1751-8121/ab1270

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Morse index for figure-eight choreographies of the planar equal mass three-body problem2018

    • Author(s)
      Fukuda Hiroshi、Fujiwara Toshiaki、Ozaki Hiroshi
    • Journal Title

      J. Phys. A.

      Volume: 51 Issue: 14 Pages: 145201-145201

    • DOI

      10.1088/1751-8121/aab06d

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Presentation] Variational principle of action and group theory for bifurcation of figure-eight solutions2020

    • Author(s)
      Fujiwara T, Fukuda H, Ozaki
    • Organizer
      Celestial Mechanics and Beyond
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Bifurcation of Simo H solution bifurcated from figure-eight choreographies of the equal mass three-body problem2019

    • Author(s)
      Fukuda H, Fujiwara T, Ozaki
    • Organizer
      天体力学N体力学研究会2019
    • Related Report
      2019 Annual Research Report
  • [Presentation] 変分原理の周期解の分岐への応用2019

    • Author(s)
      藤原 俊朗,福田 宏,尾崎 浩司
    • Organizer
      応用数理学会2019年度年会
    • Related Report
      2019 Annual Research Report
  • [Presentation] 三体8の字解の分岐 -作用と線形安定性の2つの視点から-2019

    • Author(s)
      藤原 俊朗,福田 宏,尾崎 浩司
    • Organizer
      2018年度冬の力学系研究集会
    • Related Report
      2018 Research-status Report
  • [Presentation] Morse index and periodic solutions bifurcated from the figure-eight choreography for the equal mass three-body problem2018

    • Author(s)
      Fukuda H, Fujiwara T, Ozaki H
    • Organizer
      HAMSYS 2018
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] 三体8の字解と、それから分岐する解の線形安定性2018

    • Author(s)
      藤原 俊朗,福田 宏,尾崎 浩司
    • Organizer
      応用数理学会2018年度年会
    • Related Report
      2018 Research-status Report
  • [Presentation] Periodic solutions around figure-eight choreography for the equal mass three-body problem2018

    • Author(s)
      Fukuda H, Fujiwara T, Ozaki H
    • Organizer
      AIMS 2018
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] Linear Stability and Morse Index for the Figure-Eight and K = 5 Slalom Solutions Under Homogeneous Potential2018

    • Author(s)
      Fujiwara T, Fukuda H, Ozaki H
    • Organizer
      AIMS 2018
    • Related Report
      2018 Research-status Report
  • [Presentation] 等質量3 体8 の字コレオグラフィーの近くに存在する周期解2018

    • Author(s)
      福田宏
    • Organizer
      天体力学N体力学研究会2018
    • Related Report
      2017 Research-status Report
  • [Presentation] 等質量3体8の字解のモースインデックス2017

    • Author(s)
      福田宏,藤原俊朗,尾崎浩司
    • Organizer
      応用数理学会2017年度年会
    • Related Report
      2017 Research-status Report
  • [Book] 数学セミナー2020年1月号通巻699号2020

    • Author(s)
      藤原 俊朗,福田 宏, 尾崎 浩司
    • Total Pages
      6
    • Publisher
      日本評論社
    • Related Report
      2019 Annual Research Report
  • [Remarks]

    • URL

      https://kilin.clas.kitasato-u.ac.jp/3body/bifurcation/

    • Related Report
      2018 Research-status Report

URL: 

Published: 2017-04-28   Modified: 2021-02-19  

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