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Geometric theory of alynawical systems, development and application

Research Project

Project/Area Number 13640209
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionKyoto University

Principal Investigator

IWAI Toshihiro  KYOTO UNIVERSITY Graduate School of Informatics, Prof., 情報学研究科, 教授 (10021635)

Co-Investigator(Kenkyū-buntansha) UWANO Yoshio  KYOTO UNIVERSITY Graduate School of Informatics, Associate Prof., 情報学研究科, 助教授 (80201953)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2002: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2001: ¥1,400,000 (Direct Cost: ¥1,400,000)
Keywordsmany-body system / reduction / stratification / 変換群論 / 力学系の層化簡約化 / 線形分子・非線形分子 / 量子力学系の層化・簡約化 / 多体系 / 力学系の対称性 / ピータ・ワイルの定理
Research Abstract

Geometric theory of dynamical systems in the title of this project is mainly concerned with many-body systems. If the collinear configurations are gotten rid of, the center-of-mass system for many bodies is made into a principal fiber bundle. This allows us to set up reduction theory for many-body dynamical system in terms of connection. However, we showed that the restriction can be removed. In quantum mechanics, a key to a geometric reduction theory is Peter-Weyl's theorem on unitary irreducible representations of compact Lie groups. By paying more attention to the rotation group and by applying the Peter-Weyl theorem to wave functins on the center-of-mass system, we were able to develop a quantum theory for many-body systems. The application of the Peter-Weyl theorem is interpreted as a process of reduction by symmetry. In the course of this project, we have found that the theory of connections can be extended to be set up even if the structure group acts non-freely. Thus we have established a stratified reduction theory for quantum many-body systems with rotational symmetry by stratifying the center-of-mass system into stratum and by applying the Peter-Weyl theorem to wave functions on each stratum. Actually, for quantum systems both for non-singular configurtions and for collinear configurations, we have performed the reduction procedure by using rotational symmetry. We have also shown that the singulality of the kinetic energy operator at boundary of the main stratum does not cause the divergence of the energy integral.

Report

(4 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] T.Iwai: "The Poincare compactification of the MIC-Kepler problem"J.Phys.A : Math.Gen.. 34. 1713-1723 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Iwai: "Geometric mechanics of many-body systems"J.Comp.Appl.Math.. 140. 403-422 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Iwai, T.Hirose: "The reduction of a quantum system of three identical particles"J.Math.Phys.. 43. 2907-2926 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Iwai, T.Hirose: "Reduction of quantum systems with symmetry"J.Math.Phys.. 43. 2927-2947 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Iwai, H.Yamaoka: "Stratified reduction of many-body kinetic energy operator"J.Math.Phys.. 44. 4411-4435 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Iwai, T.Hirose: "Boundary conditions on wave functions for three bodies"J.Phys.A : Math.Gen.. 37. 701-718 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Iwai: "The Poincare compactification of the MIC-Kepler problem with positive energies"J.Phys.A : Math.Gen.. vol.34. 1713-1723 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Y.Yamaguchi, T.Iwai: "Geometric approach to Lyapunov analysis in Hamiltonian dynamics"Phys.Rev.E. vol.64. 0066206-1-16 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Iwai: "Geometric mechanics of many-body systems"J.Comp.Appl.Math.. vol.140. 403-422 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Iwai, T.Hirose: "The reduction of a quantum system of three identical particles on a plane"J.Math.Phys.. vol.43. 2907-2926 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Iwai, T.Hirose: "Reduction of quantum systems with symmetry, continuous and discrete"J.Math.Phys.. vol.43. 2927-2947 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Iwai, H.Yamaoka: "Stratified reduction, of many-body kinetic energy operators"J.Math.Phys.. vol.44. 4411-4435 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Iwai, T.Hirose: "Boundary conditions on wavefunctions for three bodies at singular configurations"J.Phys.A : Math.Gen.. vol.37. 701-718 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Toshihiro Iwai, Hidetaka Yamaoka: "Stratified reduction of many-body kinetic energy operators"Journal of Mathematical Physics. 44巻10号. 4411-4435 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Gusev, Ukolov, Chekanov, Rostovtsev, Uwano, Vinitsky: "The program LINA for the normalization of polynomial Hamiltonians"Proceedings of the sixth workshop on computer algebra in scientific computing. 187-197 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Toshihiro Iwai, Toru Hirose: "Boundary conditions on wavefunctions for three bodies at singular configurations"Journal of Physics A : Mathematical and General. 37巻1号. 701-718 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Toshihiro Iwai: "Geometric mechanics of many-body systems"Journal of Computational and Applied Mathematics. 140巻1-2号. 403-422 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Toshihiro Iwai, Toru Hirose: "The reduction of a quantum system of three identical particles on a plane"Journal of Mathematical Physics. 43巻6号. 2907-2926 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Toshihiro Iwai, Toru Hirose: "Reduction of quantum systems with symmetry, continuous and discrete"Journal of Mathematical Physics. 43巻6号. 2927-2947 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Gusev, Chekanov, Rostovtsev, Uwano, Vinitsky: "The programs for normalization and quantization of polynomial Hamiltonians"Proceedings of the 5^<th> Workshop on Computer Algebra in Scientific Computing. 147-158 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Iwai, T.Hirose: "The reduction of a quantum system of three identical particles on a plane"Journal of Mathematical Physics. (掲載予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Iwai, T.Hirose: "Reduction of quantum systems with symmetry continuous and discrete"Journal of Mathematical Physics. (掲載予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Y.Yamaguchi, T.Iwai: "Geometric approach to Lyapunov analysis in Hamiltonian dynamics"Physical Review E. vol.64 no.6-1. 066206-1-066206-16 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Uwano: "A use of computer algebra for the separability of the purturbed harmonic oscillator with homogeneous polynomial potentials"Proceedings of the 7th international conference on applications of computer algebra. (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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