2003 Fiscal Year Final Research Report Summary
Geometric theory of alynawical systems, development and application
| Project/Area Number |
13640209
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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 |
Global analysis
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| Research Institution | Kyoto University |
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Principal Investigator |
IWAI Toshihiro KYOTO UNIVERSITY Graduate School of Informatics, Prof., 情報学研究科, 教授 (10021635)
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| Co-Investigator(Kenkyū-buntansha) |
UWANO Yoshio KYOTO UNIVERSITY Graduate School of Informatics, Associate Prof., 情報学研究科, 助教授 (80201953)
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| Project Period (FY) |
2001 – 2003
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| Keywords | many-body system / reduction / stratification |
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| Research Abstract |
Geometric theory of dynamical systems in the title of this project is mainly concerned with many-body systems. If the collinear configurations are gotten rid of, the center-of-mass system for many bodies is made into a principal fiber bundle. This allows us to set up reduction theory for many-body dynamical system in terms of connection. However, we showed that the restriction can be removed. In quantum mechanics, a key to a geometric reduction theory is Peter-Weyl's theorem on unitary irreducible representations of compact Lie groups. By paying more attention to the rotation group and by applying the Peter-Weyl theorem to wave functins on the center-of-mass system, we were able to develop a quantum theory for many-body systems. The application of the Peter-Weyl theorem is interpreted as a process of reduction by symmetry. In the course of this project, we have found that the theory of connections can be extended to be set up even if the structure group acts non-freely. Thus we have established a stratified reduction theory for quantum many-body systems with rotational symmetry by stratifying the center-of-mass system into stratum and by applying the Peter-Weyl theorem to wave functions on each stratum. Actually, for quantum systems both for non-singular configurtions and for collinear configurations, we have performed the reduction procedure by using rotational symmetry. We have also shown that the singulality of the kinetic energy operator at boundary of the main stratum does not cause the divergence of the energy integral.
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Research Products
(13 results)
