A new approach to construct planar equal-mass three-body choreographic solutions

• Project/Area Number: 17K05588
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical physics/Fundamental condensed matter physics
• Research Institution: Tokai University
• Principal Investigator: Ozaki Hiroshi 東海大学, 理学部, 教授 (00407991)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2019: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000); Fiscal Year 2018: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000); Fiscal Year 2017: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000)
• 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 / 等質量3体8の字解析解 / 重心 / 全角運動量ゼロ / 分岐 / 等質量3体非平面解 / 3体問題 / 重心保存 / 三接線交点の軌跡 / 座標比関数 / 等質量3体8の字舞踏解 / 数理物理 / 等質量3体平面舞踏解 / 保存系 / 3接線定理

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.

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．

Research Products

• [Journal Article] Morse index and bifurcation for figure-eight choreographies of the equal mass three-body problem (2019)
  • 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
  • Peer Reviewed
• [Presentation] 等質量3体8の字解から分岐する非平面解 (2020)
  • Author(s): 福田 宏,藤原 俊朗,尾崎浩司
  • Organizer: 日本応用数理学会
• [Presentation] Bifurcations from the Figure-eight solution/bifurcation (2020)
  • Author(s): Fujiwara T, Fukuda H, Ozaki H
  • Organizer: Celestial Mechanics and Beyond
  • Int'l Joint Research / Invited
• [Presentation] Bifurcation of Simo H solution bifurcated from figure-eight choreographies of the equal mass three-body problem (2019)
  • Author(s): Fukuda H, Fujiwara T, Ozaki H
  • Organizer: 天体力学N体力学研究会
• [Presentation] Variational principle for bifurcation in Lagrangian mechanics (2019)
  • Author(s): Fujiwara T, Fukuda H, Ozaki H
  • Organizer: 天体力学N体力学研究会
• [Presentation] 変分原理の周期解分岐への応用 (2019)
  • Author(s): 藤原俊朗,福田宏，尾崎浩司
  • Organizer: 日本応用数理学会
• [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
• [Presentation] Periodic solutions around figure-eight choreography for the equal mass three-body problem (2018)
  • Author(s): Fukuda H, Fujiwara T, Ozaki H
  • Organizer: The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications AIMS 2018
  • Int'l Joint Research
• [Presentation] Linear Stability and Morse Index for the Figure-Eight and K=5 Slalom Solutions under Homogeneous Potential (2018)
  • Author(s): Fujiwara T, Fukuda H, Ozaki H
  • Organizer: The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications AIMS 2018
  • Int'l Joint Research