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2006 Fiscal Year Final Research Report Summary

Development of innovative numerical integrators which preserving all of the conserved quantities

Research Project

Project/Area Number 15340030
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKyoto University

Principal Investigator

NAKAMURA Yoshimasa  Kyoto University, Graduate School of Informatics, Professor, 情報学研究科, 教授 (50172458)

Co-Investigator(Kenkyū-buntansha) TSUJIMOTO Satoshi  Kyoto University, Graduate School of Informatics, Lecturer, 情報学研究科, 講師 (60287977)
MINESAKI Yokitaka  Kyoto University, Graduate School of Informatics, Research Associate, 情報学研究科, 助手 (70378834)
OHTA Yasuhiro  Kobe University, Department of Mathematics, Associate Professor, 大学院自然科学研究科, 助教授 (10213745)
Project Period (FY) 2003 – 2006
Keywordsgravitational 3・body motion / totally conservative integrator / Kepler motion / Langrage's equilateral triangle / energy preserving difference solution / numerical integrator / symplectic integrator method
Research Abstract

Symplectic integrators and energy preserving difference methods have been widely used for Hamiltonian dynamical systems. However, since such numerical integrators do not preserve all of the conserved quantities, their behavior may be rather different from the trajectory of Hamiltonian systems. Indeed, these integrators can not describe a long-time behavior of the gravitational 2-body motion of Kepler. The purpose of the research project is to develop a new numerical integrator named TCI, totally conservative integrator, which preserving all of the conserved quantities of Hamiltonian systems. It was shown in a paper by Minesaki and Nakamura that a long-time behavior of the Kepler motion is completely preserved by the TCI.
In 2006, the last year of the project, Minesaki and Inoue developed a new integrator which preserving all the conserved quantities of the gravitational 3-body motion. Since the general 3-body motion is a chaotic dynamical system, a candidate of such an integrator is formulated for the case of Langrage's equilateral triangle solution on a plain. The basic design of the integrator is to divide the equations of motion into 2-body parts and interaction parts of 3-body and then discretize them, individually. A key idea is a use of a gap in a time variable which appears in the discrete 2-body motion to keep the sum of relative coordinates zero. Consequently, it is shown that the resulting numerical integrator preserves Langrage's equilateral triangle solution exactly. Such a remarkable property is viewed in numerical simulation except for a round-off error in computer. The new integrator is shown to be superior than Stormer-Verlet's symplectic integrator for 3-body motion. It is also verified that the new integrator behaves well for the letter "8" solution of the 3-body motion and the corresponding energy is kept constant for a long period.

  • Research Products

    (13 results)

All 2007 2006

All Journal Article (12 results) Patent(Industrial Property Rights) (1 results)

  • [Journal Article] New type of soliton equations2007

    • Author(s)
      Y.Ohta, R.Hirota
    • Journal Title

      J. Phys. Soc. Japan 76巻2号

      Pages: 024005 (14)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Discretization of the new type of soliton equations2007

    • Author(s)
      R.Hirota, Y.Ohta
    • Journal Title

      J. Phys. Soc. Japan 76巻3号

      Pages: 034001 (9)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] New type of soliton equations2007

    • Author(s)
      Y.Ohta, R.Hirota
    • Journal Title

      J. Phys. Soc. Japan Vol.76,No.2

      Pages: 024005(14)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Discretization of the new type of soliton equations2007

    • Author(s)
      R.Hirota, Y.Ohta
    • Journal Title

      J. Phys. Soc. Japan Vol.76, No.3

      Pages: 034001(9)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A numerical integrator for the two-fixed-centres problem conserving all constants of motion2006

    • Author(s)
      T.Inoue, Y.Minesaki
    • Journal Title

      J. Phys. A, Math. Gen. 39巻

      Pages: 9437-9452

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Determinant structure of non-autonomous Toda-type integrable systems2006

    • Author(s)
      A.Mukaihira, S.Tsujimoto
    • Journal Title

      J. Phys. A, Math. Gen. 39巻

      Pages: 779-788

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Accurate computation of singular values in terms of shifted integrable schemes2006

    • Author(s)
      M.Iwasaki, Y.Nakamura
    • Journal Title

      Japan J. Indust. Appl. Math. 23巻

      Pages: 239-259

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] New numerical integrator for Stackel system which conserves all constants of motion2006

    • Author(s)
      Y.Minesaki, Y.Nakamura
    • Journal Title

      J. Phys. A, Math. Gen. 39巻

      Pages: 9453-9476

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A numerical integrator for the two-fixed-centres problem conserving all constants of motion2006

    • Author(s)
      T.Inoue, Y.Minesaki
    • Journal Title

      J. Phys. A, Math. Gen. Vol.39

      Pages: 9437-9452

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Determinant structure of non-autonomous Toda-type integrable systems2006

    • Author(s)
      A.Mukaihira, S.Tsujimoto
    • Journal Title

      J. Phys. A, Math. Gen. Vol.39

      Pages: 779-788

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Accurate computation of singular values in terms of shifted integrable schemes2006

    • Author(s)
      M.Iwasaki, Y.Nakamura
    • Journal Title

      Japan J. Indust. Appl. Math. Vol.23

      Pages: 239-259

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] New numerical integrator for Stackel system which conserves all constants of motion2006

    • Author(s)
      Y.Minesaki, Y.Nakamura
    • Journal Title

      J. Phys. A, Math. Gen. Vol.39

      Pages: 9453-9476

    • Description
      「研究成果報告書概要(欧文)」より
  • [Patent(Industrial Property Rights)] 特異値分解装置,及び特異値分解方法2006

    • Inventor(s)
      中村佳正ほか4名
    • Industrial Property Rights Holder
      国立大学法人京都大学100%
    • Industrial Property Number
      PCT出願(出願番号:PCT/JP2006/318713)
    • Filing Date
      2006-09-21
    • Description
      「研究成果報告書概要(和文)」より

URL: 

Published: 2008-05-27  

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