2013 Fiscal Year Final Research Report
Analysis of non-integrable system by the eigenvalue problem of the Liouvillian in classical mechanics
| Project/Area Number |
23654136
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Multi-year Fund |
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| Research Field |
Mathematical physics/Fundamental condensed matter physics
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| Research Institution | Osaka Prefecture University |
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Principal Investigator |
KENICHI Noba 大阪府立大学, 工学(系)研究科(研究院), 助教 (30316012)
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| Research Collaborator |
PETROSKY Tomio テキサス大学(アメリカ合衆国), 複雑量子系研究所, 上級研究員
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| Project Period (FY) |
2011 – 2013
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| Keywords | リウビル演算子の固有値問題 / 非可積分系 / 共鳴特異性 / 縮退がある場合の摂動論 / 小惑星の分布 / 制限三体問題 |
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| Research Abstract |
We have analyzed a non-integrable system in terms of the eigenvalue problem of the Liouville operator (the Liouvillian) that is the generator of motion in classical mechanics. We focus on the restricted three-body problem that consists of an asteroid, Sun, and Jupiter as one of the typical non-integrable systems. By using the degenerate perturbation theory which has been developed in quantum mechanics, we can analyze the resonance effect in the distribution of asteroids. We have obtained the approximate solutions of the eigenvalue problem of the perturbed Liouvillian at one of the resonance points where the ratios of the period of Jupiter to that of an asteroid are expressed by simple integers. These solutions indicate that the time scale of the formation of the striped pattern in the asteroid distribution is rather short in the order of thousand years.
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