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New kind of inverse problems of Noether’s theory for Hamiltonian systems

Ding Guang-Tao

New kind of inverse problems of Noether’s theory for Hamiltonian systems

Ding Guang-Tao
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  • In this paper, a new kind of inverse problems of Noether’s theory for Hamiltonian systems is studied. The general solution and the specific solutions of constructing the Hamiltonians and the symmetries from known first integrals by using Noether’s theory are obtained. Two corollaries according to which the conserved quantities can be deduced directly from the Hamiltonians are presented. Two examples are given to illustrate the application of the results.
    [1]

    [1]Mei F X, Liu D, Luo Y 1991 Advanced analytical mechanics (Beijing: Beijing Institute of Technology Press) (in Chinese)[梅凤翔、刘端、罗勇 1991 高等分析力学 (北京:北京理工大学出版社)]

    [2]

    [2]Галиуллин А С 1986 Методы решения обратных задач динамики (Москва: Наука)

    [3]

    [3]Mei F X 1991 Mechanics in Engineering 13 17 (in Chinese)[梅凤翔 1991 力学与实践 13 17]

    [4]

    [4]Luo S K, Zhang Y F 2008 Advances in the Study of Dynamics of Constrained Systems (Beijing: Science Press) (in Chinese)[罗绍凯、张永发等 2008 约束系统动力学研究进展 (北京:科学出版社)]

    [5]

    [5]Santilli R M 1978 Foundations of Theoretical Mechanics I (New York: Springer-Verlag)

    [6]

    [6]Santilli R M 1983 Foundations of Theoretical Mechanics II (New York: Springer-Verlag)

    [7]

    [7]Noether A E 1918 Nachr.Akad.Wiss.Gottingen.Math.Phys. K I II 235

    [8]

    [8]Zhao Y Y, Mei F X 1999 Symmetries and Invariants of Mechanical Systems (Beijing: Science Press) (in Chinese)[赵跃宇、梅凤翔 1999 力学系统的对称性与不变量 (北京:科学出版社)]

    [9]

    [9]Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) (in Chinese)[梅凤翔 1999 李群和李代数对约束力学系统的应用 (北京:科学出版社)]

    [10]

    [10]Li Z P 1981 Acta Phys.Sin. 30 1659 (in Chinese)[李子平 1981 物理学报 30 1659]

    [11]

    [11]Liu D 1990 Sci.China A (11) 1189 (in Chinese)[刘端 1990 中国科学 A辑 (11)1189]

    [12]

    [12]Li Z P 1991 Chin.Sci.Bull 36 958 (in Chinese)[李子平 1991 科学通报 36 958]

    [13]

    [13]Mei F X 1993 Sci.China A 36 1456

    [14]

    [14]Zhang Y, Shang M, Mei F X 2000 Chin.Phys. 9 401

    [15]

    [15]Luo S K 2003 Acta Phys.Sin. 52 2941 (in Chinese)[罗绍凯 2003 物理学报 52 2941]

    [16]

    [16]Goldstein H, Poole C, Safko J 2002 Classical Mechanics 3rd ed. (Redwood City: Addison-Wesley)

  • [1]

    [1]Mei F X, Liu D, Luo Y 1991 Advanced analytical mechanics (Beijing: Beijing Institute of Technology Press) (in Chinese)[梅凤翔、刘端、罗勇 1991 高等分析力学 (北京:北京理工大学出版社)]

    [2]

    [2]Галиуллин А С 1986 Методы решения обратных задач динамики (Москва: Наука)

    [3]

    [3]Mei F X 1991 Mechanics in Engineering 13 17 (in Chinese)[梅凤翔 1991 力学与实践 13 17]

    [4]

    [4]Luo S K, Zhang Y F 2008 Advances in the Study of Dynamics of Constrained Systems (Beijing: Science Press) (in Chinese)[罗绍凯、张永发等 2008 约束系统动力学研究进展 (北京:科学出版社)]

    [5]

    [5]Santilli R M 1978 Foundations of Theoretical Mechanics I (New York: Springer-Verlag)

    [6]

    [6]Santilli R M 1983 Foundations of Theoretical Mechanics II (New York: Springer-Verlag)

    [7]

    [7]Noether A E 1918 Nachr.Akad.Wiss.Gottingen.Math.Phys. K I II 235

    [8]

    [8]Zhao Y Y, Mei F X 1999 Symmetries and Invariants of Mechanical Systems (Beijing: Science Press) (in Chinese)[赵跃宇、梅凤翔 1999 力学系统的对称性与不变量 (北京:科学出版社)]

    [9]

    [9]Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) (in Chinese)[梅凤翔 1999 李群和李代数对约束力学系统的应用 (北京:科学出版社)]

    [10]

    [10]Li Z P 1981 Acta Phys.Sin. 30 1659 (in Chinese)[李子平 1981 物理学报 30 1659]

    [11]

    [11]Liu D 1990 Sci.China A (11) 1189 (in Chinese)[刘端 1990 中国科学 A辑 (11)1189]

    [12]

    [12]Li Z P 1991 Chin.Sci.Bull 36 958 (in Chinese)[李子平 1991 科学通报 36 958]

    [13]

    [13]Mei F X 1993 Sci.China A 36 1456

    [14]

    [14]Zhang Y, Shang M, Mei F X 2000 Chin.Phys. 9 401

    [15]

    [15]Luo S K 2003 Acta Phys.Sin. 52 2941 (in Chinese)[罗绍凯 2003 物理学报 52 2941]

    [16]

    [16]Goldstein H, Poole C, Safko J 2002 Classical Mechanics 3rd ed. (Redwood City: Addison-Wesley)

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  • Received Date:  24 May 2009
  • Accepted Date:  19 June 2009
  • Published Online:  15 March 2010

New kind of inverse problems of Noether’s theory for Hamiltonian systems

  • 1. 安徽师范大学物理与电子信息学院,芜湖 241000

Abstract: In this paper, a new kind of inverse problems of Noether’s theory for Hamiltonian systems is studied. The general solution and the specific solutions of constructing the Hamiltonians and the symmetries from known first integrals by using Noether’s theory are obtained. Two corollaries according to which the conserved quantities can be deduced directly from the Hamiltonians are presented. Two examples are given to illustrate the application of the results.

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