构造Birkhoff函数(组)的参数调节法

宋端; 刘畅; 郭永新

doi:10.3879/j.issn.1000-0887.2013.09.013

构造Birkhoff函数(组)的参数调节法

doi: 10.3879/j.issn.1000-0887.2013.09.013

宋端¹,
刘畅²,
郭永新^2,3

¹辽东学院影像物理教研室,辽宁丹东118001;²辽宁大学物理学院,沈阳110036;³辽东学院机电工程系,辽宁丹东118001

基金项目: 国家自然科学基金资助项目（11172120;11202090;10932002）

详细信息

作者简介:
宋端（1962—），女，辽宁丹东人，副教授;郭永新（1963—），男，辽宁东港人，教授(通讯作者. E-mail:yxguo@lnu.edu.cn).

中图分类号: O316
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出版历程
- 收稿日期: 2013-07-22
- 修回日期: 2013-09-09
- 刊出日期: 2013-09-15

Parameter-Adjusting Method of Constructing Birkhoffian Functions

¹Department of Imaging Physics, Eastern Liaoning University, Dandong, Liaoning 118001, P.R.China;²College of Physics, Liaoning University, Shenyang 110036, P.R.China;³School of Mechanical and Electrical Engineering, Eastern Liaoning University, Dandong, Liaoning 118001, P.R.China

Funds: The National Natural Science Foundation of China（11172120;11202090;10932002）

摘要

摘要: 根据偏微分方程的Cauchy-Kovalevski可积性定理，将欠定的Birkhoff方程组转化为以Birkhoff函数组为未知变量的完备的偏微分方程组，提出了构造Birkhoff动力学函数的参数调节法．通过调节补偿方程中的两类可调的函数参数就能得到不同的Birkhoff函数组．并把构造Birkhoff函数组的参数调节法与Santilli构造方法进行了比较，例如研究了利用动力学系统独立的第一积分构造Birkhoff函数组的Hojman方法与参数调节法之间的关系．最后，给出应用实例验证了参数调节法的实用性及其与Santilli 3种构造方法的关系
- Birkhoff方程 /
- Cauchy-Kovalevski定理 /
- 自伴随微分方程 /
- 参数调节法
Abstract: The parameter-adjusting method to construct dynamical functions of Birkhoff's equations is put forward based on realizing the completeness of Birkhoff’s equations, which are under-determinate, by means of Cauchy-Kovalevski integrability theorem for partial differential equations. The two kinds of parameters in the compensatory equation were capable of adjusting to get different sets of Birkhoffian functions. The existing methods, such as Hojman’s method using 2n-first integrals for dynamical systems with symmetry, were compared with the parameter-adjusting method. Finally, The compensatory equation for the Birkhoff's equations can be simplified by means of some limitations on the two kinds of parameters, where the relation between the Birkhoffian functions and parameters become more evident.
- Birkhoff’s equations /
- Cauchy-Kovalevski theorem /
- self-adjoint differential equations /
- parameter-adjusting method

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参考文献(15)

[1]	Birkhoff G D.Dynamical Systems[M]. New York: AMS College Publishers Providence, RI, Vol9, 1927.
[2]	Santilli R M.Foundations of Theoretical Mechanics II[M]. New York: SpringerVerlag, 1983.
[3]	梅凤翔, 史荣昌, 张永发, 吴惠彬. Birkhoff系统动力学[M]. 北京：北京理工大学出版社, 1996.（MEI Feng-xiang, SHI Rong-chang, ZHANG Yong-fa, WU Hui-bin.Dynamics of Birkhoffian Systems[M]. Beijing: Beijing Institute of Technology Press, 1996.(in Chinese)）
[4]	Hojman S. Construction of genotypic transformations for first order systems of differential equations[J].Hadronic J,1981,5: 174-184.
[5]	Ionescu D. A geometric Birkhoffian formalism for nonlinear RLC networks[J].J Geom Phys,2006,56(12): 2545-2572.
[6]	Ionescu D, Scheurle J. Birkhoffian formulation of the dynamics of LC circuits[J]. Z angew Math Phys,2007,58(2): 175-208.
[7]	GUO Yong-xin, LIU Chang, LIU Shi-xing. Generalized Birkhoffian realization of nonholonomic systems[J].Communications in Mathematics,2010,18(1): 21-35.
[8]	LIU Shi-xing, LIU Chang, GUO Yong-xin. Geometric formulations and variational integrators of discrete autonomous Birkhoff systems[J].Chin Phys B,2011,20(3): 034501.
[9]	Sun Y J, Shang Z J. Structure-preserving algorithms for Birkhoffian systems[J].Phys Lett A,2005,336(4/5): 358-369.
[10]	Van der Schaft A J, Maschke B M. On the Hamiltonian formulation of nonholonomic mechanical systems[J].Rep Math Phys,1994,34(2): 225-233.
[11]	Bloch A M, Fernandez O E, Mestdag T. Hamiltonization of nonholonomic systems and the inverse problem of the calculus of variations[J].Rep Math Phys,2009,63(2): 225-249.
[12]	LIU Chang, LIU Shi-xing, GUO Yong-xin. Inverse problem for Chaplygin’s nonholonomic systems[J].Sci China Tech Sci,2011,54(8): 2100-2106.
[13]	刘畅, 宋端, 刘世兴, 郭永新. 非齐次Hamilton系统的Birkhoff表示[J]. 中国科学：物理学力学天文学, 2013,43(4): 541-548. （LIU Chang, SONG Duan, LIU Shi-xing, GUO Yong-xin. Birkhoffian representation of non-homogenous Hamiltonian systems[J].Sci Sin Phys Mech Astron,2013,43(4): 541-548.(in Chinese)）
[14]	Guo Y X, Luo S K, Shang M, Mei F X. Birkhoffian formulation of nonholonomic constrained systems[J].Rep Math Phys,2001,47(3): 313-322.
[15]	Courant R, Hilbert D.Methods of Mathematical Physics II[M]. New York: Interscience Publishers, 1966.