Computer-algebra approach to the inverse problem of the Birkhoff-Gustavson normalization and application to problems in science and engineering
| Project/Area Number |
16560050
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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 |
Engineering fundamentals
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| Research Institution | Future University-Hakodate (2005-2006)
Kyoto University (2004) |
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Principal Investigator |
UWANO Yoshio Future University-Hakodate, School of Systems Information Science, Professor, システム情報化学科, 教授 (80201953)
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| Co-Investigator(Kenkyū-buntansha) |
IWAI Toshihiro Kyoto University, Graduate School of Informatics, Professor, 情報学研究科, 教授 (10021635)
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| Project Period (FY) |
2004 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥3,700,000 (Direct Cost: ¥3,700,000)
Fiscal Year 2006: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2004: ¥1,800,000 (Direct Cost: ¥1,800,000)
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| Keywords | inverse problem / Birkhoff-Gustavson normalization / computer algebra / separation of variables / the Schrodinger equation / quantum computing / 保存力学系 / 標準化 / ジョセフソン接合 / シュレディンガー方程式 / 探索アルゴリズム / プログラム開発 / アルゴリズム開発 / 非線形ハミルトン系 |
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| Research Abstract |
1.The rediscovery of the Bertrand-Darboux condition from the Birkhoff-Gustavson (BG) normalization view point is refined and extended to the perturbed oscillators (PHOs) with homogeneous polynomials with any odd degree (Note : The primery result on the BD condition was obtained in my previous subject No.13650066 on the inverse problem of the BG normalization). The result envolves a new algebraic expression to the separability of the PHOs with homogeneous polynomial potentials, which at the same time is looked upon as a new algebraic expression of the BD condition to those oscillators.
2.A new symbolic computing program LINA is made, that works faster, in the direct and inverse problems of the BG normalization, than GITA^<-1> and ANFER developed by the head investigator with Prof. Vinitsky's group (foreign cooperated member). The effectiveness of LINA coming from the Hori-Deprit transformation adopted in the algorithm is confirmed on several computer algebra languages, REDUCE, MAPLE and MATHEMATICA.
3.As a extention of target systems for the direct and the inverse problems of the BG normalization, time-dependent Schrodinger equations are taken as time-dependent Hamiltonian systems (with infinitely many degree of freedom). Our established method and algorithms turn out to be very effective on the preparatory stages in the propose numerical programs for time-dependent Schrodinger equations.
4.Quantum computation is taken as a target area for finding an application of the inverse problem to science and engineering. Through our research process, an integrable system is discovered on a quantum information space that admits a similar dynamical structure often found in classical system sciences. Further, a Josephson junction circuit model is shown to share a BG normal form with the Henon-Heiles systems.
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Report
(4 results)
Research Products
(26 results)
