Perturbartions of integrable Hamiltonian

可积哈密顿量的扰动

• 批准号: 09640239
• 负责人: KAMETANI Makoto
• 金额: $ 0.64万
• 依托单位: Maizuru National College of Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640239/
• 关键词: integrable systems; perturbation problem; KAM theory; 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

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.

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 investment 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环上的扭曲图t由t（r，s）=（r，r + s）定义，其中（r，s）代表环的极坐标。现在，我们用产品空间g x d替换环，并在t（a，z）=（a，a（z））上引入twist映射t。由于该映射保留了所有a中的每个{a} x d，并且由于它们的结合涵盖了整个空间g x d，因此由t确定的动态系统是g xD上的一个可以集成的系统。我们的结果如下：如果a是g的椭圆形元素，并且每个a的旋转角度，并且每个diopophantine n is，diophantine n e e e e e e e e e e e e e e e El persiant persiant persiant persiant pers the persiant e}在G中类似于旋转Z→EXP（IK）Z，我们称此实数k为a的旋转角度。

项目成果

亀谷 睦: "複素領域における積分可能系とその摂動について"岩崎敷久教授記念偏微分方程式研究集会講演記録集. 31-40 (2001)

Mutsumi Kameya：“论可积系统及其在复数域中的扰动”纪念岩崎志久教授的偏微分方程研究会议记录 31-40 (2001)。

Kametani, M.: "A KAM theory on unit disk, to appear in RIMS kokyuroku, Partial differential equations and time-frequency analysis, (in Japanese)"(2004)

Kametani, M.：“单位圆盘上的 KAM 理论，出现在 RIMS kokyuroku，偏微分方程和时频分析，（日语）”(2004)

亀谷 睦: "単位円板上でのKAM理論"京都大学数理解析研究所講究録. (印刷中). (2004)

Mutsumi Kametani：“单位圆盘上的 KAM 理论”京都大学数学科学研究所 Kokyuroku（出版中）。

Kametani, M.: "On Integrable systems in complex domains and their perturbations, Professor N. Iwasaki memorial symposium on partial differential equations"Kyoto University (in Japanese). 31-40 (2001)

Kametani, M.：“关于复杂域中的可积系统及其扰动，N. Iwasaki 教授偏微分方程纪念研讨会”京都大学（日语）。