2003 Fiscal Year Final Research Report Summary
The inverse problem of the Birkhoff-Gustavson normalization and its application
| Project/Area Number |
13650066
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Engineering fundamentals
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| Research Institution | Kyoto University |
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Principal Investigator |
UWANO Yoshio Kyoto University, Graduate School of Informatics, Associate Professor, 情報学研究科, 助教授 (80201953)
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| Co-Investigator(Kenkyū-buntansha) |
IWAI Toshihiro Kyoto University, Graduate School of Informatics, Professor, 情報学研究科, 教授 (10021635)
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| Project Period (FY) |
2001 – 2003
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| Keywords | Birkhoff-Gustavson normalization / inverse problem / computer algebra / separation of variables / integrability of systems / algorithm |
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| Research Abstract |
Application to the integrabilty of certain perturbed harmonic oscillators:
The separability the perturbed harmonic oscillators with homogeneous polynomial potentials of odd-degree, say δ, (the δ-PHOs for short) within rotations of the Cartesian coordinates is characterized as a necessary and sufficient condition for any δ-PHO to share its BG normal form with a (2δ -2)-PHO.
A necessary and sufficient condition for any 3-PHO to share its BG normal form with a 4-PHO in a linear magnetic field is obtained. A half of the 3-PHOs subject to the condition thus found are shown to be non-integrable through the Ziglin analysis. In the process of finding both of the results above, computer-algebra program ANFER developed by the head investigator are utilized very effectively.
Computer algebraic approach to the inverse problem:
The programs named ANFER and GITA^<-1> by the head investigator are unified into the new program GITAN. A faster program, named LINA, is written up by applying the Hori-Deprit transformation method to the inverse problem. This result is a part of the joint work with Dr.S.I. Vinitsky, the foreign collaborator of this project.
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