1998 Fiscal Year Final Research Report Summary
Perturbartions of integrable Hamiltonian
| Project/Area Number |
09640239
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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 |
解析学
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| Research Institution | Maizuru National College of Technology |
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Principal Investigator |
KAMETANI Makoto Maizuzu National College of Technology, Division of natural science, Associate professor, 自然化学科, 助教授 (80270297)
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| Project Period (FY) |
1997 – 1998
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| Keywords | integrable systems / perturbation problem / KAM theory |
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| Research Abstract |
There is a famous problem, namely, the perturbation problem of the twist map defined on a two-dimensional annulus, which is treated in "Lectures on Celestial Mechanics" written by Siegel-Moser in 1971. Our investigation is concerned with the perturbation of a twist map which is defined not on the annulus but on the product space of the unit disk D and the Lie group G of the fractional linear transformations acting on D. We recall that the twist map t on an annulus is defined by t(r, s)=(r, r + s), where (r, s) stands for the polar coordinate of the annulus. Now we replace the annulus with the product space G x D and introduce the twist map T on G x D by T(a, z)=(a, a(z)). Since this map preserves each {a} x D for all a in G and since their union covers the whole space G x D, the dynamical system determined by T is an integrable system on G x D. Our result is as follows : if a is an elliptic element of G, and the rotation angle of a is a, Diophantine number, then the invariant set {a} x D is persistent under small perturbations of T. Here, recall that every eliptic a in G is similar to an rotation z→exp(ik)z, and we call this real number k the rotation angle of a.
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Research Products
(4 results)
