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A new approach to the construction of Lagrangians and Hamiltonians for one-dimensional dissipative systems with variable coefficients

Ding Guang-Tao

A new approach to the construction of Lagrangians and Hamiltonians for one-dimensional dissipative systems with variable coefficients

Ding Guang-Tao
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  • A new approach to the construction of Lagrangian and Hamiltonian for a second-order differential equation is presented. By writing the second-order equation in the first-order form and constructing first-order Lagranian corresponding to the set of the first-order equations, the second-order Lagrangian and Hamiltionian are deduced from the first-order Lagrangian directly. By using the above method the first-order ane the second-order Lagrangians and the Hamiltonians for some of dissipative and dissipative-like systems are obtained. The advantage of the approach is discussed. Four examples are given to illustrate the applications of the results.
    [1]

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

    [2]

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

    [3]

    Mei F X 1988 Special Problems of Analytical Mechanics (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔 1988 分析力学专题 北京:北京工业学院出版社)]

    [4]

    Currie D F, Saletan E J 1966 J. Math. Phys. 7 967

    [5]

    Hojman S, Urrutia L F 1981 J. Math. Phys. 22 1896

    [6]

    Lopez G 1996 Ann. Phys. 251 363

    [7]

    Lopez G 1996 Ann. Phys. 251 372

    [8]

    Ding G T 1996 J. Anhui Normal Univ. 19 382 (in Chinese)[丁光涛 1996 安徽师范大学学报 19 382]

    [9]

    Pen H W 1980 Acta Phys. Sin. 29 1084 (in Chinese) [彭恒武 1980 物理学报 29 1084] 〖10] Lopez G, Lopez P 2006 Int. J. Theor. Phys. 45 734

    [10]

    Musielak Z E 2008 J. Phys. A: Math. Theor. 41 055205

    [11]

    Cieslinski J L, Nikiciuk T 2010 J. Phys. A: Math. Theor. 43 175205

    [12]

    Ding G T, Tao S T 2008 Chin. Sci. Bull. 53 872 (in Chinese) [丁光涛、 陶松涛 2008 科学通报 53 872]

    [13]

    Ding G T 2009 Science in China G 39 813 (in Chinese) [丁光涛 2009 中国科学 G 辑 39 813]

    [14]

    Ding G T 2009 China. Sci. Bull. 54 337 (in Chinese) [丁光涛 2009 科学通报 54 337]

    [15]

    Mei F X, Xie J F, Gang T Q 2007 Acta Phys. Sin. 56 5041 (in Chinese)[梅凤翔、 解加芳、 江铁强 2007 物理学报 56 5041]

  • [1]

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

    [2]

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

    [3]

    Mei F X 1988 Special Problems of Analytical Mechanics (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔 1988 分析力学专题 北京:北京工业学院出版社)]

    [4]

    Currie D F, Saletan E J 1966 J. Math. Phys. 7 967

    [5]

    Hojman S, Urrutia L F 1981 J. Math. Phys. 22 1896

    [6]

    Lopez G 1996 Ann. Phys. 251 363

    [7]

    Lopez G 1996 Ann. Phys. 251 372

    [8]

    Ding G T 1996 J. Anhui Normal Univ. 19 382 (in Chinese)[丁光涛 1996 安徽师范大学学报 19 382]

    [9]

    Pen H W 1980 Acta Phys. Sin. 29 1084 (in Chinese) [彭恒武 1980 物理学报 29 1084] 〖10] Lopez G, Lopez P 2006 Int. J. Theor. Phys. 45 734

    [10]

    Musielak Z E 2008 J. Phys. A: Math. Theor. 41 055205

    [11]

    Cieslinski J L, Nikiciuk T 2010 J. Phys. A: Math. Theor. 43 175205

    [12]

    Ding G T, Tao S T 2008 Chin. Sci. Bull. 53 872 (in Chinese) [丁光涛、 陶松涛 2008 科学通报 53 872]

    [13]

    Ding G T 2009 Science in China G 39 813 (in Chinese) [丁光涛 2009 中国科学 G 辑 39 813]

    [14]

    Ding G T 2009 China. Sci. Bull. 54 337 (in Chinese) [丁光涛 2009 科学通报 54 337]

    [15]

    Mei F X, Xie J F, Gang T Q 2007 Acta Phys. Sin. 56 5041 (in Chinese)[梅凤翔、 解加芳、 江铁强 2007 物理学报 56 5041]

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  • Received Date:  10 June 2010
  • Accepted Date:  05 July 2010
  • Published Online:  15 April 2011

A new approach to the construction of Lagrangians and Hamiltonians for one-dimensional dissipative systems with variable coefficients

  • 1. The Collega of Physics and Electronic Information, Anhui Normal University, Wuhu 241000, China

Abstract: A new approach to the construction of Lagrangian and Hamiltonian for a second-order differential equation is presented. By writing the second-order equation in the first-order form and constructing first-order Lagranian corresponding to the set of the first-order equations, the second-order Lagrangian and Hamiltionian are deduced from the first-order Lagrangian directly. By using the above method the first-order ane the second-order Lagrangians and the Hamiltonians for some of dissipative and dissipative-like systems are obtained. The advantage of the approach is discussed. Four examples are given to illustrate the applications of the results.

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