Perturbartions of integrable Hamiltonian
Project/Area Number  09640239 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
解析学

Research Institution  Maizuru National College of Technology 
Principal Investigator 
KAMETANI Makoto Maizuzu National College of Technology, Division of natural science, Associate professor, 自然化学科, 助教授 (80270297)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 1998 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1997 : ¥500,000 (Direct Cost : ¥500,000)

Keywords  integrable systems / perturbation problem / KAM theory / 可積分系 / 摂動問題 / KAM理論 / 1階偏微分方程式 / 非線型偏微分方程式 / 分岐解 / ハミルトン系 / 可積分系の摂動 / 非線型 / 摂動 
Research Abstract 
There is a famous problem, namely, the perturbation problem of the twist map defined on a twodimensional annulus, which is treated in "Lectures on Celestial Mechanics" written by SiegelMoser 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.

Report
(4results)
Research Output
(4results)