ISSN 0253-2778

CN 34-1054/N

open
Free to Publish. Free to Read.
Open AccessOpen Access JUSTC Original Paper

Poisson bracket method for obtaining normal coordinates of quadratic Hamiltonian

Funds:  Supported by the National Natural Science Foundation of China (11775208), Education and Sci-Tech Projects for Young and Middle-aged Teachers in Fujian Province (JK2014053), Educational Research Projects for Young and Middle-aged Teachers in Fujian Province (JAT170582).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2018.08.007
More Information
  • Corresponding author: LIN Quan (corresponding author), male, born in 1980, master/ associate Prof. Research field:Theoretical mechanics and mechanical design .E-mail: 12959392@qq.com
  • Received Date: 29 March 2018
  • Accepted Date: 25 May 2018
  • Rev Recd Date: 25 May 2018
  • Publish Date: 31 August 2018
    Abstract

  • Abstract

    It was found that the problem of searching for normal coordinates W of quadratic Hamiltonian H can be ascribed to solving the newly established secular equation with two consecutive Poisson bracket operations. Solving W would simultaneously lead to the normal frequency. Some examples about quadratic Hamiltonian were presented to demonstrate the effectiveness of the presented method.

    Abstract

    It was found that the problem of searching for normal coordinates W of quadratic Hamiltonian H can be ascribed to solving the newly established secular equation with two consecutive Poisson bracket operations. Solving W would simultaneously lead to the normal frequency. Some examples about quadratic Hamiltonian were presented to demonstrate the effectiveness of the presented method.
    • FullText(HTML)
  • loading
    • References(12)
  • [1]
    HERBERT G, POOLE C P, SAFKO J L. Classical Mechanics[M]. Boston, MA: Addison Wesley, 2002.
    [2]
    TAYLOR J R. Classical Mechanics[M]. Herndon, VA: University Science Books, 2005.
    [3]
    STECHER J V, GREENE C H. Correlated Gaussian hyperspherical method for few-body systems[J]. Physical Review A, 2009, 80(2): 022504.
    [4]
    LACOURSIERE C, LINDE M. Spook: a variational time-stepping scheme for rigid multibody systems subject to dry frictional contacts[R]. Ume, Sweden: Ume University, 2014.
    [5]
    LOU Zhimei. Normal coordinates and normal vibration modals of multidimension free vibration[J]. Mechanics in Engineering, 2002, 24(6): 50-53. （in Chinese）
    [6]
    LOU Zhimei. The common method of finding normal coordinate for the multidimensional linear microvibration[J]. College Physics, 2004, 23(7): 3-7, 31.（in Chinese）
    [7]
    KONISHI K, PAFFUTI G. A theorem on cyclic harmonic oscillators[J]. International Journal of Modern Physics A, 2004, 21(15):3199-3211.
    [8]
    TOMIYA M, YOSHINAGA N. Numerical analysis of level statistical properties of two- and three-dimensional coupled quartic oscillators[J]. Computer Physics Communications, 2001, 142(1): 82-87.
    [9]
    JI Yinghua, LEI Minsheng. Diagonalization of Hamiltonian for three harmonically coupled non-identical oscillators[J]. College Physics, 2001, 20(10): 24-25. （in Chinese）
    [10]
    LINDENFELD Z, EISENBERG E, LIFSHITZ R. Phonon-mediated damping of mechanical vibrations in a finite atomic chain coupled to an outer environment[EB/OL]. [2018-03-01] https://arxiv.org/abs/1309.5772.
    [11]
    FAN H Y, LU H L, FAN Y. Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations[J]. Ann Phys, 2006, 321: 480-494.
    [12]
    FAN H Y, LI C. Invariant ‘eigen-operator’ of the square of Schrdinger operator for deriving energy-level gap[J]. Phys Lett A, 2004, 321: 75-78.)
    • Supplements(0)
    • Related Articles
    • Track Citations
  • 加载中

Catalog

    |
    Get Citation
    PDF
    XML
    [1]
    HERBERT G, POOLE C P, SAFKO J L. Classical Mechanics[M]. Boston, MA: Addison Wesley, 2002.
    [2]
    TAYLOR J R. Classical Mechanics[M]. Herndon, VA: University Science Books, 2005.
    [3]
    STECHER J V, GREENE C H. Correlated Gaussian hyperspherical method for few-body systems[J]. Physical Review A, 2009, 80(2): 022504.
    [4]
    LACOURSIERE C, LINDE M. Spook: a variational time-stepping scheme for rigid multibody systems subject to dry frictional contacts[R]. Ume, Sweden: Ume University, 2014.
    [5]
    LOU Zhimei. Normal coordinates and normal vibration modals of multidimension free vibration[J]. Mechanics in Engineering, 2002, 24(6): 50-53. （in Chinese）
    [6]
    LOU Zhimei. The common method of finding normal coordinate for the multidimensional linear microvibration[J]. College Physics, 2004, 23(7): 3-7, 31.（in Chinese）
    [7]
    KONISHI K, PAFFUTI G. A theorem on cyclic harmonic oscillators[J]. International Journal of Modern Physics A, 2004, 21(15):3199-3211.
    [8]
    TOMIYA M, YOSHINAGA N. Numerical analysis of level statistical properties of two- and three-dimensional coupled quartic oscillators[J]. Computer Physics Communications, 2001, 142(1): 82-87.
    [9]
    JI Yinghua, LEI Minsheng. Diagonalization of Hamiltonian for three harmonically coupled non-identical oscillators[J]. College Physics, 2001, 20(10): 24-25. （in Chinese）
    [10]
    LINDENFELD Z, EISENBERG E, LIFSHITZ R. Phonon-mediated damping of mechanical vibrations in a finite atomic chain coupled to an outer environment[EB/OL]. [2018-03-01] https://arxiv.org/abs/1309.5772.
    [11]
    FAN H Y, LU H L, FAN Y. Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations[J]. Ann Phys, 2006, 321: 480-494.
    [12]
    FAN H Y, LI C. Invariant ‘eigen-operator’ of the square of Schrdinger operator for deriving energy-level gap[J]. Phys Lett A, 2004, 321: 75-78.)
    Recommended articles
    TrendMD
    Volume 48 Issue 8 page: 643-648
    Cover

    Article Metrics

    Article views (43) PDF downloads(95)
    Proportional views

    Copyright © Editorial Office of JUSTC, All Rights Reserved. 皖ICP备05002528号

    Supported by: Beijing Renhe Information Technology Co. Ltd    

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return