基于李群理论的几类自治系统的可积性研究
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摘要
本文基于李(Lie)群理论研究了二阶、三阶和n阶自治系统的可积性。主要内容和结果包括以下几个方面：
(1)首先给出了二阶恰当自治系统所接受的单参数李群的生成元的一般形式,然后讨论了该系统的一些性质；给出了两类特殊形式的二阶自治系统所接受的单参数李群的生成元的一般形式,并由此构造了系统的一个积分因子。
(2)基于李群理论,给出并证明了利用三阶自治系统所接受的两个单参数李群的生成元求该系统的积分因子的方法与结论。
(3)对于n阶自治系统,当系统为保守系统时,讨论了其积分因子和首次积分的关系；当系统为非保守系统时,给出并证明了系统积分因子存在的充要条件。
Based on Lie group theory, the integrability of second order, third order and n-th order autonomous system is studied. The main contents and results of this dissertation are included as following aspects:
(1) The general form of generators of one-parameter Lie groups accepted by second order exact automatic system is given. Then, some properties of the system are discussed. At the same time, the general form of generators of one-parameter Lie group which is accepted by two particular classes of second order automatic systems is given, thus the integrating factor of the system is constructed.
(2) Base on Lie group theory, the method to obtain the integrating factors of the third order autonomous system by using of two one-parameter Lie groups accepted by the system is given and shown.
(3) For the n-th order system, the relation between integrating factors and first integrals of the system is discussed when it is conservative system; At the same time, the sufficient and necessary condition for the existence of integrating factors of the system is given and proved when it is non-conservative system.
(1)首先给出了二阶恰当自治系统所接受的单参数李群的生成元的一般形式,然后讨论了该系统的一些性质；给出了两类特殊形式的二阶自治系统所接受的单参数李群的生成元的一般形式,并由此构造了系统的一个积分因子。
(2)基于李群理论,给出并证明了利用三阶自治系统所接受的两个单参数李群的生成元求该系统的积分因子的方法与结论。
(3)对于n阶自治系统,当系统为保守系统时,讨论了其积分因子和首次积分的关系；当系统为非保守系统时,给出并证明了系统积分因子存在的充要条件。
Based on Lie group theory, the integrability of second order, third order and n-th order autonomous system is studied. The main contents and results of this dissertation are included as following aspects:
(1) The general form of generators of one-parameter Lie groups accepted by second order exact automatic system is given. Then, some properties of the system are discussed. At the same time, the general form of generators of one-parameter Lie group which is accepted by two particular classes of second order automatic systems is given, thus the integrating factor of the system is constructed.
(2) Base on Lie group theory, the method to obtain the integrating factors of the third order autonomous system by using of two one-parameter Lie groups accepted by the system is given and shown.
(3) For the n-th order system, the relation between integrating factors and first integrals of the system is discussed when it is conservative system; At the same time, the sufficient and necessary condition for the existence of integrating factors of the system is given and proved when it is non-conservative system.
引文
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[2]V. I. Aronld. Geometrical Methods of the Theory of Ordinary Differential Equations. Springs-Verlag,1983.
[3]张芷芬,丁同仁,黄文灶,董镇喜.微分方程定性理论[M].北京：科学出版社,1985.
[4]黄琳.稳定性理论[M].北京：北京大学出版社,1992.
[5]Antoni Ferragut. Polynomial Inverse Integrating Factor of Quadratic Differential Systems and Other Result, thesis, July,2006.
[6]Jaume Llibre, Gerardo Rodriguez. The Inverse Integrating Factor and Limit Cycle, Eqiadiff 2003-International Conference on Differential Equations, July 22-26,2003, LUC Diepenbeel Hasselt, Belgium.
[7]丁同仁,李承治.常微分方程[M].北京：高等教育出版社,2004,第二版.
[8]Morris W. Hirsch, Stephen Smale, Robert L. Devancy. Differertial Equations, Dynamical Systems, and Introduction to Chaos [M].甘少波 译.北京：人民邮电出版社,2008,第二版.
[9]Garcia I.A. On the Integrability of Quasihomogeneous and Related Planer Vector Fields [J]. International Journal of Bifurcation and Choas.2003,13(4):995-1002
[10]Hu Yanxia. On the Integrability of Quasihomogeneous and Quasidegenerate Infinity System [J]. Advances in Difference Equations, Volume 2007, Article ID 98427,10 pages.
[11]刘许成.复合型积分因子的存在定理及其应用[J].数学的实践与认识,2004,34(2)：168-171.
[12]程惠东,孟新柱.积分因子存在定理的一般充要条件[J].数学的实践与认识,2006,36(8)：309-312.
[13]G. W. Bluman, S. Kumei. Symmetrics and Differential Equations [M],2ed. New York:Springer-Verlag,1989.
[14]GUnal. Algebraic integrability and generalized symmetries of dynamical systems [J]. Physics Letters A 260,1999.352-359.
[15]E. S. Cheb-Terrab, A. D. Roche. Integrating Factors for Second-order ODEs [J]. JGurnal of Symbolic Computation,Volume 27, Issue 5, May 1999, Pages 501-519.
[16]Sibusiso Moyo, P.GL.Leach. Symmetry Properties of Autonomous Integrating Factors [J]. Symmetry, Integrability and Geometry:Methods and Applications, Vol.1 (2005), Paper 024,12 pages.
[17]Sibusiso Moyo, P.GL.Leach. A Note on the Integrability of a Class of Nonlinear Ordinary Differential Equations [J]. Journal of Nonlinear Mathematical Physics Volume 15, Supplement 1 (2008),159-164.
[18]Olver P. J. Applications of Lie Group to Differential Equations [M],2ed. New York:Springer-Verlag,1989.
[19]Guan Keying, Liu Sheng, Lei Jinzhi. Lie Algebra Admitted by an Ordinary Equation System [J]. Ann. of Diff. Eqs.,1998,14(2):131-142.
[20]胡彦霞,管克英.常微分方程组接受Lie群的判定条件[J].北京交通大学学报,2004,28(6)：19-21.
[21]刘胜,管克英.二阶非自治系统首次积分的一种构造方法[J].内蒙古大学学报(自然科学版),1999,30(2)：135-139.
[22]刘洪伟,管克英.用两个单参数Lie群求3阶自治系统的首次积分[J].应用数学学报,2006,29(3),567-573.
[23]刘洪伟.用n-1个单参数李群求n阶自治系统的首次积分[J].应用数学学报,2006,29(3)：567-573.
[24]Yanxia Hu, Xiaozhong Yang. A Method for Obtaining First Integrals and Integrating Factors of Autonmous Systems and Application to Euler-Poisson Equations [J]. Reports On Mathematical Physics.2006,58(1),41-50.
[25]史少云.非线性系统有理首次积分的不存在性和部分存在性[J].数学物理学报,2008,28A(4)：603—611.
[26]赵红发,黎文磊,骆昱成.多项式系统的有理首次积分[J].吉林大学学报(理学版).2009年3月.第47卷第2期：77-280.
[27]焦佳,许志国.周期系统的形式首次积分[J].吉林大学学报(理学版),2009年3月,第47卷第2期：289—291.
[28]杨恩浩.微分方程可积性的群分析理论[M].广州：暨南大学出版社,1990.
[29]韩锋.二阶恰当自治系统所接受的Lie群研究[J].大庆师范学院学报,2006(02)：16-18.
[30]V. I. Arnold. Mathematical Methods of Classical mechanics [M]. New York: Springer-Verlag,1978.
[31]Cooke. R. The Mathematics of Sonya Kovalevskaya [M], New York: Springer-Verlag,1984.
[2]V. I. Aronld. Geometrical Methods of the Theory of Ordinary Differential Equations. Springs-Verlag,1983.
[3]张芷芬,丁同仁,黄文灶,董镇喜.微分方程定性理论[M].北京：科学出版社,1985.
[4]黄琳.稳定性理论[M].北京：北京大学出版社,1992.
[5]Antoni Ferragut. Polynomial Inverse Integrating Factor of Quadratic Differential Systems and Other Result, thesis, July,2006.
[6]Jaume Llibre, Gerardo Rodriguez. The Inverse Integrating Factor and Limit Cycle, Eqiadiff 2003-International Conference on Differential Equations, July 22-26,2003, LUC Diepenbeel Hasselt, Belgium.
[7]丁同仁,李承治.常微分方程[M].北京：高等教育出版社,2004,第二版.
[8]Morris W. Hirsch, Stephen Smale, Robert L. Devancy. Differertial Equations, Dynamical Systems, and Introduction to Chaos [M].甘少波 译.北京：人民邮电出版社,2008,第二版.
[9]Garcia I.A. On the Integrability of Quasihomogeneous and Related Planer Vector Fields [J]. International Journal of Bifurcation and Choas.2003,13(4):995-1002
[10]Hu Yanxia. On the Integrability of Quasihomogeneous and Quasidegenerate Infinity System [J]. Advances in Difference Equations, Volume 2007, Article ID 98427,10 pages.
[11]刘许成.复合型积分因子的存在定理及其应用[J].数学的实践与认识,2004,34(2)：168-171.
[12]程惠东,孟新柱.积分因子存在定理的一般充要条件[J].数学的实践与认识,2006,36(8)：309-312.
[13]G. W. Bluman, S. Kumei. Symmetrics and Differential Equations [M],2ed. New York:Springer-Verlag,1989.
[14]GUnal. Algebraic integrability and generalized symmetries of dynamical systems [J]. Physics Letters A 260,1999.352-359.
[15]E. S. Cheb-Terrab, A. D. Roche. Integrating Factors for Second-order ODEs [J]. JGurnal of Symbolic Computation,Volume 27, Issue 5, May 1999, Pages 501-519.
[16]Sibusiso Moyo, P.GL.Leach. Symmetry Properties of Autonomous Integrating Factors [J]. Symmetry, Integrability and Geometry:Methods and Applications, Vol.1 (2005), Paper 024,12 pages.
[17]Sibusiso Moyo, P.GL.Leach. A Note on the Integrability of a Class of Nonlinear Ordinary Differential Equations [J]. Journal of Nonlinear Mathematical Physics Volume 15, Supplement 1 (2008),159-164.
[18]Olver P. J. Applications of Lie Group to Differential Equations [M],2ed. New York:Springer-Verlag,1989.
[19]Guan Keying, Liu Sheng, Lei Jinzhi. Lie Algebra Admitted by an Ordinary Equation System [J]. Ann. of Diff. Eqs.,1998,14(2):131-142.
[20]胡彦霞,管克英.常微分方程组接受Lie群的判定条件[J].北京交通大学学报,2004,28(6)：19-21.
[21]刘胜,管克英.二阶非自治系统首次积分的一种构造方法[J].内蒙古大学学报(自然科学版),1999,30(2)：135-139.
[22]刘洪伟,管克英.用两个单参数Lie群求3阶自治系统的首次积分[J].应用数学学报,2006,29(3),567-573.
[23]刘洪伟.用n-1个单参数李群求n阶自治系统的首次积分[J].应用数学学报,2006,29(3)：567-573.
[24]Yanxia Hu, Xiaozhong Yang. A Method for Obtaining First Integrals and Integrating Factors of Autonmous Systems and Application to Euler-Poisson Equations [J]. Reports On Mathematical Physics.2006,58(1),41-50.
[25]史少云.非线性系统有理首次积分的不存在性和部分存在性[J].数学物理学报,2008,28A(4)：603—611.
[26]赵红发,黎文磊,骆昱成.多项式系统的有理首次积分[J].吉林大学学报(理学版).2009年3月.第47卷第2期：77-280.
[27]焦佳,许志国.周期系统的形式首次积分[J].吉林大学学报(理学版),2009年3月,第47卷第2期：289—291.
[28]杨恩浩.微分方程可积性的群分析理论[M].广州：暨南大学出版社,1990.
[29]韩锋.二阶恰当自治系统所接受的Lie群研究[J].大庆师范学院学报,2006(02)：16-18.
[30]V. I. Arnold. Mathematical Methods of Classical mechanics [M]. New York: Springer-Verlag,1978.
[31]Cooke. R. The Mathematics of Sonya Kovalevskaya [M], New York: Springer-Verlag,1984.