| Co-Investigator(Kenkyū-buntansha) |
YAMAGUCHI Yoshiyaki Kyoto University, Graduate School of Informatics, Research Associate, 情報学研究科, 助手 (40314257)
UWANO Yoshio Kyoto University, Graduate School of Informatics, Associate Professor, 情報学研究科, 助教授 (80201953)
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| Budget Amount *help |
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2000: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 1999: ¥1,800,000 (Direct Cost: ¥1,800,000)
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| Research Abstract |
This study, entitled "Geometric theory of dynamical systems and its applicatiopn, " has an aim to study many-body systems in a geometric manner. So far, in studying many-body systems, configurations that all particles are collinear have been excluded for a geometric reason ; for those configurations, isotropy subgroups of the rotaion group are not trivial, so that the center-of-mass system is not made into a principal fiber bundle. The present study has been made without such a restriction. As a by-product, the connection theory, which is set up usually on a principal fiber bundle, is generalized to such a case that the structure group action is not necessarily free. With this generalization, a theory of reduction for quantum many-body systems has been established. Further, the theory for many-particle systems is extended to systems of many rigid bodies. As an example, a system of two identical axially symmetric cylinders jointed together with a special kind of joint is treated. It is shown that this system serves as a model cat which can make a somersault with a suitable torque as an input. The somersault is indeed realized as a solution to the equations of motion with the constraint that the total angular momentum vanishes. In addition, it is shown that systems of identical particles admits the action of a permutation group and, in particular, that the action is represented in the matrix form for planar three particles.
As another topic of the geometric study of dynamical systems, this report contains the study of linearized equations of geodesic flows on the cotangent bundle of a Riemannian manifold. This has been made with Y.Yamaguchi. In this topic, we have treated Lyapunov exponents and Lyapunov vectors to show that the geometric method we have set up can provide Lyapunov vectors more precisely than those in the usual method, which has been confirmed by numerical calculations.
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