2006 Fiscal Year Final Research Report Summary
Development of innovative numerical integrators which preserving all of the conserved quantities
| Project/Area Number |
15340030
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kyoto University |
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Principal Investigator |
NAKAMURA Yoshimasa Kyoto University, Graduate School of Informatics, Professor, 情報学研究科, 教授 (50172458)
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| 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)
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| Project Period (FY) |
2003 – 2006
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| Keywords | gravitational 3・body motion / totally conservative integrator / Kepler motion / Langrage's equilateral triangle / energy preserving difference solution / numerical integrator / symplectic integrator method |
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| 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.
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