2013 Fiscal Year Final Research Report
Normal forms for superintegrable systems at singular points and their perturbation problems
| Project/Area Number |
22540180
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Basic analysis
|
|---|
| Research Institution | Kanazawa University |
|---|
Principal Investigator |
ITO Hidekazu 金沢大学, 数物科学系, 教授 (90159905)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
YAGASAKI Kazuyuki 広島大学, 理学研究科, 教授 (40200472)
|
|---|
| Co-Investigator(Renkei-kenkyūsha) |
SHIBAYAMA Mitsuru 大阪大学, 基礎工学研究科, 講師 (40467444)
|
|---|
| Project Period (FY) |
2010-04-01 – 2013-03-31
|
| Keywords | 超可積分系 / ハミルトン系 / バーコフ標準形 / 共鳴条件 / 標準形理論 |
|---|
| Research Abstract |
The purpose of this research is to understand the global structure of solutions for the so-called superintegrable systems admitting integrals the number of which is greater than the degrees of freedom. We generalized Liouville-Arnold theorem which is the fundamental theorem for integrable systems. Namely, under some additional conditions on resonances, we showed the existence of special coordinates in a neighbourhood of singularities of the map defined by those integrals so that the system can be solved explicitly in those coordinates.
Moreover, we defined superintegrability for general vector fields and showed that such a superintegrable vector field can be solved explicitly in a neighbourhood of some type of equilibrium point by obtaining its convergent normal form.
|
