Variational approach to the n-body problem

• Project/Area Number: 22740104
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Global analysis
• Research Institution: Osaka University (2011-2013); Kyoto University (2010)
• Principal Investigator: SHIBAYAMA Mitsuru 大阪大学, 基礎工学研究科, 講師 (40467444)
• Project Period (FY): 2010-04-01 – 2014-03-31
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000); Fiscal Year 2013: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2012: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2011: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 多体問題 / 変分法 / 周期解 / 天体力学 / 力学系 / カオス / 分岐 / 対称性 / 特異点 / スケーリング / ブローアップ / N体問題 / ハミルトン力学系 / 衝突 / 摂動論 / 勾配法 / 峠の定理 / 特異点解析 / 可積分性

Research Abstract

I proved the existence of periodic solutions with regularizable collisions in the n-body problem. This includes the proof of the existence of Schubart orbit with arbitrary masses in the collinear three-body problem, Broucke orbit in the isosceles three-body problem and Sekiguchi orbit in the symmetric collinear four-body problem. I also proved the existence of super-eight solution by using the variational method.We investigated the bifurcation phenomena on periodic solutions in the isosceles three-body problem.
I proved the non-integrability of the collinear three-body problem by using the collision manifold theory.

Research Products

• [Journal Article] Variational proof of the existence of the super-eight orbit in the four-body problem (2014)
  • Author(s): M. Shibayama
  • Journal Title: Archive for Rational Mechanics and Analysis Volume: (In press) Issue: 1 Pages: 77-98
  • DOI: 10.1007/s00205-014-0753-x
  • Year and Date: 2014-04-23
  • Peer Reviewed
• [Journal Article] Variational proof of the existence of the super-eight orbit in the four-body problem (2014)
  • Author(s): Mitsuru Shibayama
  • Journal Title: Archive for Rational Mechanics and Analysis Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Minimax approach to the n-body problem (2014)
  • Author(s): Mitsuru Shibayama
  • Journal Title: ASPM, Nonlinear Dynamics in Partial Differential Equations Volume: 64
  • Peer Reviewed
• [Journal Article] Minimax approach to the n-body problem, ASPM (2013)
  • Author(s): M. Shibayama
  • Journal Title: Nonlinear Dynamics in Partial Differential Equations Volume: vol.64(In press)
  • Peer Reviewed
• [Journal Article] 柴山允瑠 (2013)
  • Author(s): ハミルトン系の非可積分性の証明
  • Journal Title: 数理解析研究所講究録 Volume: 1827 Pages: 1-17
• [Journal Article] Morse index of periodic solutions in the n-body problem (2012)
  • Author(s): M. Shibayama
  • Journal Title: 天体力学N体力学研究会2011集録
• [Journal Article] Action minimizing periodic solutions in the N-body problem (2012)
  • Author(s): M. Shibayama
  • Journal Title: proceedings of Sino-Japan conference 2011 Pages: 169-182
• [Journal Article] Action minimizing periodic solutions in the N-body problem (2012)
  • Author(s): Mitsuru Shibayama
  • Journal Title: proceedings of Sino-Japan conference Volume: 1 Pages: 169-182
• [Journal Article] Mitsuru Shibayama (2012)
  • Author(s): Morse index of periodic solutions in the n-body problem
  • Journal Title: 天体力学N 体力学研究会2011 集録 Volume: 1 Pages: 156-166
• [Journal Article] Families of symmetric relative periodic orbits originating from the circular Euler solution in the isosceles three body problem (2011)
  • Author(s): M. Shibayama and K. Yagasaki
  • Journal Title: Celestial Mechanics and Dynamical Astronomy Volume: Vol.110 Issue: 1 Pages: 53-70
  • DOI: 10.1007/s10569-011-9338-2
  • Peer Reviewed
• [Journal Article] Non-integrability of the collinear three-body problem (2011)
  • Author(s): M. Shibayama
  • Journal Title: Discrete and Continuous Dynamical Systems-A Volume: 30 Issue: 1 Pages: 299-312
  • DOI: 10.3934/dcds.2011.30.299
  • Peer Reviewed
• [Journal Article] Minimizing periodic orbits with regularizable collisions in the n body problem (2011)
  • Author(s): M. Shibayama
  • Journal Title: Archive for Rational Mechanics and Analysis Volume: Vol.199 Issue: 3 Pages: 821-841
  • DOI: 10.1007/s00205-010-0334-6
  • NAID: 120003001325
  • Peer Reviewed
• [Journal Article] 舞踏解に関する第二変分の数値計算 (2011)
  • Author(s): 柴山允瑠
  • Journal Title: 天体力学N体力学研究会・ちばN体2010集録 Pages: 198-206
• [Journal Article] 舞踏解に関する第二変分の数値計算 (2011)
  • Author(s): 柴山允瑠
  • Journal Title: 天体力学N体力学研究会集録 Pages: 198-206
• [Journal Article] 二等辺三体問題の三体衝突間のヘテロクリニック軌道と相対周期軌道 (2010)
  • Author(s): 柴山允瑠, 矢ヶ崎一幸
  • Journal Title: 数理解析研究所講究録 Volume: 1692 Pages: 39-45
• [Presentation] Variational proof of the existence of the super-eight orbit in the four-body problem (2014)
  • Author(s): Mitsuru Shibayama
  • Organizer: the 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: The Universidad Autonoma, Madrid, Spain
  • Invited
• [Presentation] Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities (2014)
  • Author(s): Mitsuru Shibayama
  • Organizer: the 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: The Universidad Autonoma, Madrid, Spain
  • Invited
• [Presentation] Variational Proof of the Existence of the Super-Eight Orbit in the Four-Body Problem (2013)
  • Author(s): M. Shibayama
  • Organizer: Taiwan-Japan Symposium on Celestial Mechanics and N-Body Dynamics, NCTS
  • Place of Presentation: Taiwan
  • Year and Date: 2013-12-06
• [Presentation] 変分法による軌道の探索 (2013)
  • Author(s): 柴山允瑠
  • Organizer: 第56回自動制御連合講演会
  • Place of Presentation: 新潟大学
  • Year and Date: 2013-11-16
• [Presentation] V ariational proof of the existence of the super-eight orbit in the four-body problem (2013)
  • Author(s): M. Shibayama
  • Organizer: The Asian Mathematical Conference 2013
  • Place of Presentation: Busan, Korea
  • Year and Date: 2013-07-01
  • Invited
• [Presentation] Variational proof of the existence of the super-eight orbit in the four-body problem (2013)
  • Author(s): 柴山允瑠
  • Organizer: KAM理論とその周辺
  • Place of Presentation: 金沢大学サテライトプラザ
  • Year and Date: 2013-06-19
• [Presentation] Variational proof of the existence of the super-eight orbit in the four-body problem (2013)
  • Author(s): M. Shibayama
  • Organizer: The Sixth International Meeting on Celestial Mechanics
  • Place of Presentation: Viterbo, Itary
• [Presentation] Variational proof of the existence of the super-eight orbit in the four-body problem (2013)
  • Author(s): Mitsuru Shibayama
  • Organizer: The Sixth International Meeting on Celestial Mechanics
  • Place of Presentation: The Belletti Park Hotel San Martino al Cimino, Viterbo, Itary
• [Presentation] Minimax approach to the n-body problem (2012)
  • Author(s): 柴山允瑠
  • Organizer: 微分方程式セミナー
  • Place of Presentation: 大阪大学理学部数学教室
  • Year and Date: 2012-10-19
• [Presentation] 天体の周期運動 (2012)
  • Author(s): 柴山允瑠
  • Organizer: 研究会「非線形現象の解明と制御」
  • Place of Presentation: 大阪大学基礎工学研究科
  • Year and Date: 2012-10-04
• [Presentation] 天体力学における周期解 (2012)
  • Author(s): 柴山允瑠
  • Organizer: 非線形テクノサイエンス講演会
  • Place of Presentation: 大阪大学
  • Year and Date: 2012-02-22
• [Presentation] On the super-eight orbit in the four-body problem (2012)
  • Author(s): 柴山允瑠
  • Organizer: 冬の力学系研究集会
  • Place of Presentation: 日大軽井沢研修所
  • Year and Date: 2012-01-07
• [Presentation] On the super-eight orbit in the four-body problem (2012)
  • Author(s): 柴山允瑠
  • Organizer: 2011年度冬の力学系研究集会
  • Place of Presentation: 日本大学軽井沢研修所、長野県
  • Year and Date: 2012-01-07
• [Presentation] Variational approach to the n-body problem (2011)
  • Author(s): M. Shibayama
  • Organizer: Emerging topic on differential equations and their applications
  • Place of Presentation: Nankai University
  • Year and Date: 2011-12-06
  • Invited
• [Presentation] Variational approach to the n-body problem (2011)
  • Author(s): 柴山允瑠
  • Organizer: Emerging Topics on Differential Equations and their Applications---Sino-Japan Conference of Young Mathematicians
  • Place of Presentation: Nankai University, China(招待講演)
  • Year and Date: 2011-12-06
• [Presentation] 多体問題への変分法的アプローチ (2011)
  • Author(s): 柴山允瑠
  • Organizer: ハミルトン系研究会「関西保存会」
  • Place of Presentation: 京都大学文学部、京都府
  • Year and Date: 2011-10-15
• [Presentation] Variational approach to the n-body problem (2011)
  • Author(s): M. Shibayama
  • Organizer: The 4th MSJ-SI conference on Nonlinear Dynamics in Partial Differential Equations
  • Place of Presentation: Kyushu University
  • Year and Date: 2011-09-13
• [Presentation] Non-integrability of the three-body problem via blowing-up of the collision singularity (2011)
  • Author(s): M. Shibayama
  • Organizer: Workshop on Symplectic Geometry and Topology
  • Place of Presentation: Kyoto University
  • Year and Date: 2011-02-18
• [Presentation] Heteroclinic connections between triple collisions and relative periodic orbits in the isosceles three-body problem, Differential Equations and Applications (2010)
  • Author(s): M. Shibayama
  • Organizer: The 8th AIMS Conference on Dynamical Systems
  • Place of Presentation: Dresden University of Technology, Germany
  • Year and Date: 2010-05-27
• [Presentation] Variational proof of the existence of the super-eight orbit in the four-body problem
  • Author(s): Mitsuru Shibayama
  • Organizer: The Asian Mathematical Conference 2013
  • Place of Presentation: BEXCO, Busan, Korea
  • Invited
• [Presentation] Variational Proof of the Existence of the Super-Eight Orbit in the Four-Body Problem
  • Author(s): Mitsuru Shibayama
  • Organizer: Taiwan-Japan Symposium on Celestial Mechanics and N-Body Dynamics,
  • Place of Presentation: The national Hua University, Hsinchu city, Taiwan
• [Presentation] 特異点のブローアップによるハミルトン系の可積分性の判定
  • Author(s): 柴山允瑠
  • Organizer: 日本数学会2014年度年会無限可積分系分科会
  • Place of Presentation: 学習院大学, 東京都豊島区
• [Presentation] 変分法による軌道の探索
  • Author(s): 柴山允瑠
  • Organizer: 第56回自動制御連合講演会
  • Place of Presentation: 新潟大学, 新潟県新潟市
• [Presentation] Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up theory of singularities
  • Author(s): Mitsuru Shibayama
  • Organizer: New Perspectives on the N-body Problem
  • Place of Presentation: BIRS, Banff, Canada
  • Invited
• [Presentation] Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up theory of sin- gularities
  • Author(s): Mitsuru Shibayama
  • Organizer: Workshop on Variational methods in N-body and Vortex Dynamics
  • Place of Presentation: Dipartimento di Matematica e Fisica ”Ennio De Giorgi”, Italy
  • Invited
• [Presentation] 斉次ハミルトン系の特異点のブローアッフと可積分性の判定
  • Author(s): 柴山允瑠
  • Organizer: 数理解析研究所共同研究集会『力学 系の作る集団タイナミクス-保存系・散逸系の枠組みを越えて- 』
  • Place of Presentation: 京都大学数理解析研究所
  • Invited
• [Presentation] 斉次ハミルトン系の特異点のブローアップによる非可積分性の証明
  • Author(s): 柴山允瑠
  • Organizer: 天体力学 N 体力学研究会
  • Place of Presentation: 千葉市美術館
• [Presentation] 勾配法を用いた制限三体問題における Lambert 問題の数値解法
  • Author(s): 柴山允瑠
  • Organizer: 天体力学 N 体力学研究会
  • Place of Presentation: 千葉市美術館
• [Remarks]
  • URL: http://www.sigmath.es.osaka-u.ac.jp/~shibayama/
• [Remarks] 柴山允瑠のホームページ
  • URL: http://www.sigmath.es.osaka-u.ac.jp/~shibayama/index.html
• [Remarks]
  • URL: http://www.sigmath.es.osaka-u.ac.jp~/shibayama
• [Remarks]
  • URL: http://www.kurims.kyoto-u.ac.jp/~sibayama