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提出构造二阶微分方程的Lagrange函数和Hamilton函数的新路径. 将二阶方程写成一阶方程组并构造出对应的一阶Lagrange函数后,直接从一阶Lagrange函数导出二阶Lagrange函数和Hamilton函数. 利用上述方法得到若干耗散和类耗散系统的一阶和二阶Lagrange函数以及Hamilton函数;讨论了这种方法的优点. 举例说明所得结果的应用.
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关键词:
- 逆问题 /
- 耗散系统 /
- Lagrange函数 /
- Hamilton函数
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.-
Keywords:
- inverse problem /
- dissipative system /
- Lagrangian /
- Hamiltonian
[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 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 56 5041]
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[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 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 56 5041]
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