Research on particular solutions and dynamics around them for Hamiltonian systems

• Project/Area Number: 26800059
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Kyoto University (2015-2018); Osaka University (2014)
• Principal Investigator: Shibayama Mitsuru 京都大学, 情報学研究科, 准教授 (40467444)
• Project Period (FY): 2014-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000); Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 力学系 / 非可積分性 / 変分法 / 微分ガロア理論 / 記号力学系 / 3体問題 / Sitnikov問題 / 周期解 / 変分問題 / 可積分性 / 制限n体問題 / n中心問題 / ハミルトン力学系 / 天体力学

Outline of Final Research Achievements

We estimate the number of singularities which weak solutions can have.
We proved the non-integrability of the restricted n-body problem.
We show the existence of solutions realizing given symbolic sequences using the variational methods. Obtained periodic solutions are shown to be unstable by applying results by Ortega and Offin.

Academic Significance and Societal Importance of the Research Achievements

ハミルトン系において与えられたエネルギー面に周期解が存在するかとういう問題は活発に研究されてきた．特にエネルギー面がコンパクトでない場合として，ポテンシャル関数が特異点を持つポテンシャル系の解の持つ特異点の個数の最適な評価を与えた．制限n体問題の非可積分性の証明は，ポアンカレが非常に強い仮定のもとでなされて以来，新たな証明はなかった．本研究により，微分ガロア理論を用いた新たな証明が与えられた．特に，パラメータは任意に固定して良いという意味で，仮定を弱められている．変分法による特殊解の存在証明は8の字解の発見以来大きく発展している分野で，その中で記号列の変分法による実現は新たな展開を示している．

Research Products

• [Int'l Joint Research] 南開大学(中国)
• [Int'l Joint Research] メリーランド大学(米国)
• [Int'l Joint Research] ベオグラード大学(セルビア)
• [Journal Article] Variational proof of the existence of brake orbits in the planar 2-center problem (2019)
  • Author(s): Yuika Kajihara and Mitsuru Shibayama
  • Journal Title: Discrete and Continuous Dynamical Systems-A Volume: 印刷中
  • NAID: 120006712787
  • Peer Reviewed
• [Journal Article] Variational Construction of Orbits Realizing Symbolic Sequences in the Planar Sitnikov Problem (2019)
  • Author(s): Shibayama Mitsuru
  • Journal Title: Regular and Chaotic Dynamics Volume: 24 Issue: 2 Pages: 202-211
  • DOI: 10.1134/s1560354719020060
  • NAID: 120007173782
  • Peer Reviewed
• [Journal Article] Non-integrability of the spacial n-center problem (2018)
  • Author(s): Mitsuru Shibayama
  • Journal Title: Journal of Differential Equations Volume: 印刷中 Issue: 6 Pages: 2461-2469
  • DOI: 10.1016/j.jde.2018.04.037
  • NAID: 120006601517
  • Peer Reviewed
• [Journal Article] Linear stability of periodic three-body orbits with zero angular momentum and topological dependence of Kepler’s third law: a numerical test (2018)
  • Author(s): Dmitrasinovic V、Hudomal Ana、Shibayama Mitsuru、Sugita Ayumu
  • Journal Title: Journal of Physics A: Mathematical and Theoretical Volume: 51 Issue: 31 Pages: 315101-315101
  • DOI: 10.1088/1751-8121/aaca41
  • NAID: 120006601519
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Periodic solutions of a perscribed-energy problem for a singular Hamiltonian system (2017)
  • Author(s): M. Shibayama
  • Journal Title: Discrete and Continuous Dynamical Systems A Volume: 37 Issue: 5 Pages: 2705-2715
  • DOI: 10.3934/dcds.2017116
  • Peer Reviewed
• [Journal Article] Three topologically nontrivial choreographic motions of three bodies (2016)
  • Author(s): Milovan Suvakov, Mitsuru Shibayama
  • Journal Title: Celestial Mechanics and Dynamical Astronomy Volume: 124 Issue: 2 Pages: 155-162
  • DOI: 10.1007/s10569-015-9657-9
  • NAID: 120006601520
  • Peer Reviewed / Int'l Joint Research
• [Presentation] 一般的な相互作用力による平面三体問題の非可積分性について (2019)
  • Author(s): 山田淳二，柴山允瑠
  • Organizer: 日本数学会2019年度年会
• [Presentation] 変分法による平面2中心問題におけるbrake軌道の存在証明 (2019)
  • Author(s): 梶原唯加，柴山允瑠
  • Organizer: 日本数学会2019年度年会
• [Presentation] Variational construction of periodic and heteroclinic orbits in the planar Sitnikov problem (2018)
  • Author(s): Mitsuru Shibayama
  • Organizer: HAMSYS 2018
• [Presentation] Variational construction of periodic and heteroclinic orbits in the planar Sitnikov problem (2018)
  • Author(s): Mitsuru Shibayama
  • Organizer: :The 12th AIMS Conference on Dynamical Systems
• [Presentation] Variational construction of periodic and heteroclinic orbits in the planar Sitnikov problem (2018)
  • Author(s): Mitsuru Shibayama
  • Organizer: BOSTON UNIVERSITY/KEIO UNIVERSITY WORKSHOP 2018- Dynamical Systems
• [Presentation] Non-integrability of the spatial n-center and restricted n+1-body problem (2018)
  • Author(s): Mitsuru Shibayama
  • Organizer: Hamiltonian systems, from topology to applications through analysis II
• [Presentation] 平面 Sitnikov 問題における周期解とヘテロクリニック軌道について (2018)
  • Author(s): 柴山允瑠
  • Organizer: RIMS 研究集会「力学系 -理論と応用の融合-」
• [Presentation] Variational construction of periodic and connecting orbits in the planar Sitnikov problem (2018)
  • Author(s): Mitsuru Shibayama
  • Organizer: The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications
  • Int'l Joint Research
• [Presentation] Variational construction of periodic and connecting orbits in the planar Sitnikov problem (2018)
  • Author(s): Mitsuru Shibayama
  • Organizer: RIMS共同研究「力学系-理論と応用の融合-」
• [Presentation] Variational construction of periodic and connecting orbits in the planar Sitnikov problem (2018)
  • Author(s): Mitsuru Shibayama
  • Organizer: Dynamics Seminar, Department of Mathematics, University of Maryland, USA
• [Presentation] 平面Sitnikov問題における記号力学系と変分構造 (2018)
  • Author(s): 柴山允瑠
  • Organizer: 天体力学N体力学研究会
• [Presentation] シンプレクティック写像の不動点のモース指数と安定性 (2018)
  • Author(s): 柴山允瑠
  • Organizer: 天体力学N体力学研究会
• [Presentation] 制限n体問題の非可積分性 (2017)
  • Author(s): 柴山允瑠
  • Organizer: 日本数学会秋季総合分科会応用数学分科会
• [Presentation] 制限n体問題の非可積分性 (2017)
  • Author(s): 柴山允瑠
  • Organizer: 応用数理学会年会応用力学系
• [Presentation] ケプラー型ポテンシャル系のエネルギー固定問題における周期解の存在 (2017)
  • Author(s): 柴山允瑠
  • Organizer: 日本数学会秋季総合分科会函数方程式論分科会
• [Presentation] Non-integrability of the restricted n-body problem (2017)
  • Author(s): Mitsuru Shibayama
  • Organizer: RIMS研究集会「界面運動，力学系に現れる漸近問題への粘性解的手法とその周辺」
  • Invited
• [Presentation] Non-integrability of the restricted n-body problem (2017)
  • Author(s): Mitsuru Shibayama
  • Organizer: 天体力学N体力学研究会
  • Place of Presentation: 東京工業大学(東京都目黒区)
• [Presentation] Non-integrability of the restricted n-body problem, (2017)
  • Author(s): Mitsuru Shibayama
  • Organizer: 第６回「ハミルトン系とその周辺」研究集会
  • Place of Presentation: 京都大学(京都府京都市)
• [Presentation] 制限$n$体問題の非可積分性 (2017)
  • 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(長野県軽井沢町)
• [Presentation] Singularity analysis of the restricted three-body problem (2016)
  • Author(s): 柴山允瑠
  • Organizer: 天体力学N体力学研究会
  • Place of Presentation: 豊橋技術科学大学サテライトオフィス(愛知県豊橋市)
  • Year and Date: 2016-06-11
• [Presentation] n体問題の周期軌道の変分解析 (2016)
  • Author(s): 柴山允瑠
  • Organizer: 日本数学会2016年度年会函数方程式分科会特別講演
  • Place of Presentation: 筑波大学(茨城県，つくば市)
  • Year and Date: 2016-03-16
  • Invited
• [Presentation] Periodic solutions of a prescribed-energy problem for a singular Hamiltonian system (2016)
  • Author(s): Mitsuru Shibayama
  • Organizer: The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: Hyatt Regency Orlando, Florida, USA
  • Int'l Joint Research / Invited
• [Presentation] A perturbed Kepler problem in which there are no periodic solutions (2016)
  • Author(s): 柴山允瑠
  • Organizer: 天体力学N体力学研究会
  • Place of Presentation: 豊橋技術科学大学サテライトオフィス(愛知県豊橋市)
• [Presentation] 特異点のブローアップと標準形理論の応用による第一積分の個数の評価 (2015)
  • Author(s): 柴山允瑠
  • Organizer: 天体力学$N$体力学研究会
  • Place of Presentation: SMBCイベントスペース-千葉(千葉県，千葉市)
  • Year and Date: 2015-12-25
• [Presentation] 特異点をもつハミルトン系の周期軌道の変分解析 (2015)
  • Author(s): 柴山允瑠
  • Organizer: ダイナミクス研究会中野
  • Place of Presentation: 明治大学(東京都，中野区)
  • Year and Date: 2015-11-11
  • Invited
• [Presentation] Periodic solutions of a prescribed-energy problem for a singular Hamiltonian system (2015)
  • Author(s): 柴山允瑠
  • Organizer: RIMS研究集会「実領域における常微分方程式の定性的研究」
  • Place of Presentation: 京都大学数理解析研究所(京都府，京都市)
  • Year and Date: 2015-11-10
  • Invited
• [Presentation] 特異点をもつハミルトン系の周期解について (2015)
  • Author(s): 柴山允瑠
  • Organizer: 南大阪応用数学セミナー
  • Place of Presentation: 大阪市立大学(大阪府，大阪市)
  • Year and Date: 2015-05-16
  • Invited
• [Presentation] Perturbation Theory in Celestical Mechanics (2015)
  • Author(s): 柴山允瑠
  • Organizer: Dynamical Systems in Mathematical Physics
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2015-02-24
• [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
  • Place of Presentation: Universidad Autónoma de Madrid, Spain
  • Year and Date: 2014-07-09
  • Invited
• [Presentation] Non-integrability criterion for homogeneous Hamiltonian systems via blowingup technique of singularities (2014)
  • Author(s): Mitsuru Shibayama
  • Organizer: The 10th AIMS Conference
  • Place of Presentation: Universidad Autónoma de Madrid, Spain
  • Year and Date: 2014-07-09
  • Invited
• [Book] ハミルトン力学系-可積分系とKAM理論を中心に- (2016)
  • Author(s): 柴山允瑠
  • Total Pages: 184
  • Publisher: サイエンス社
• [Remarks] 柴山允瑠のホームページ
  • URL: http://yang.amp.i.kyoto-u.ac.jp/~shibayama/index.html
• [Remarks]
  • URL: http://yang.amp.i.kyoto-u.ac.jp/~shibayama/index.html
• [Remarks]
  • URL: https://kyouindb.iimc.kyoto-u.ac.jp/j/jS9lX
• [Remarks]
  • URL: http://yang.amp.i.kyoto-u.ac.jp/~shibayama/index.html
• [Remarks] 柴山允瑠のホームページ
  • URL: http://www.sigmath.es.osaka-u.ac.jp/~shibayama/