2005 Fiscal Year Final Research Report Summary
CRITERIA FOR INTEGRABILITY OF HAMILTONIAN SYSTEMS AND LIST OF INTEGRABLE SYSTEMS BASED ON SINGULARITY ANALYSIS
| Project/Area Number |
15540224
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Global analysis
|
|---|
| Research Institution | NATIONAL ASTRONOMICAL OBSERVATORY (NAO) |
|---|
Principal Investigator |
YOSHIDA Haruo NAO, DIVISION OF THEORETICAL ASTRONOMY, PROFESSOR, 理論研究部, 教授 (70220663)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
TANIKAWA Kiyotaka NAO, DIVISION OF THEORETICAL ASTRONOMY, ASSOCIATE PROFESSOR, 理論研究部, 助教授 (80125210)
NAKAI Hiroshi NAO, PUBLIC RELATIONS CENTERS, ASSOCIATE PROFESSOR, 天文情報センター, 助教授 (60155653)
|
|---|
| Project Period (FY) |
2003 – 2005
|
| Keywords | Hamiltonian dynamical system / Integrability / Criteria |
|---|
| Research Abstract |
The most important result obtained during the period, 2000-2002, is that the complete list of all integrable homogeneous polynomial potentials was obtained which has polynomial first integral of order four or less with respect to momenta. At the end of 19th century, Darboux obtained the condition such that there exist a first integral that is quadratic or less with respect to momenta. Three special potentials with fourth order integrals were found around 1980. The present result confirms that there are no more integrable potentials rigorously.
When the homogenous potential is cubic or quartic, there exists am example which has genuinely fourth order polynomial integral. However, when the degree of the potential is 5 or more, it is shown that there cannot exist genuinely fourth order polynomial integral. In order to prove this non-existence theorem, it is crucial to use the resultant which tells us when two algebraic equations have a common root.
|
