Lie对称性和共形不变性及守恒量若干问题的研究
|
摘要
本文基于对称性理论研究了某些力学系统守恒量的若干问题.目前研究的对称性主要有Noether对称性、Lie对称性、Mei对称性以及共形不变性,它们导致的守恒量有Noether守恒量、Hojman守恒量、Mei守恒量.本文将应用这些对称性理论研究某些力学系统的守恒量,研究内容有以下四部分：
在第一部分即本文的第二章中,我们研究了Lagrange系统、Hamilton系统、广义Hamilton系统的Lie对称性两种提法的等价性,对于广义Hamilton系统还研究了在一般无限小变换下的Lie对称性导致的新型守恒量,给出新型守恒量的表达式和证明过程.
在第二部分即本文的第三章中,我们先研究了广义Hamilton系统的共形不变性与Mei对称性的关系,给出共形不变性同时具有Mei对称性的充要条件,并且得到了共形不变性通过Mei对称性导致的Mei守恒量；其次讨论了Lorenz方程的Robbins模型的共形不变性通过Mei对称性导致的Mei守恒量；最后研究了平面Kepler方程的共形不变性与Lie对称性,给出共形不变性通过Lie对称性导致的Hojman守恒量,同时研究了Kepler方程的Mei对称性,得到了与系统的总能量、角动量相互独立的守恒量.
在第三部分即本文的第四章中,我们研究了Nielsen方程的Lie对称性导致的新型守恒量和Appell方程的Mei对称性导致的新型守恒量,给出新型守恒量的具体表达式和证明过程,举例说明结果的应用.
在第四部分即本文的第五章中,我们应用单参数Lie群理论,研究了广义齐次系统的拟齐次多项式首次积分.通过讨论Lotka-Volterra系统的约化问题,给出系统拟齐次多项式首次积分的存在条件和具体表达式.
This dissertation investigates mainly some problems on conserved quantity of me-chanical systems based on the symmetry theory. Firstly, we discuss the equivalent problem of the two different kinds of Lie symmetries for Lagrange system, Hamilton system, and the generalized Hamilton system respectively. Then, we obtain the con-served quantities deduced by the conformal invariance for the generalized Hamilton system. Secondly, we study the conformal invariance, Lie symmetry, Mei symmetry and conserved quantities of the planar Kepler equation. Thirdly, we are concerned with a new conserved quantity of Nielsen equation induced by Lie symmetry and a new conserved quantity of Appell equation induced by Mei symmetry. Finally, we consider reduction problem of the generalized homogeneous system, the condition that Lotka-Volterra system has quasi-homogeneous polynomial first integral is given, and the expression of the first integral is shown.
We are concerned with the following non-autonomous system
We denote the differential operator corresponding to system (1) by
We will take the infinitesimal generator vector of some one-parameter Lie group as follows
Literature [30] pointed out that the necessary and sufficient condition that system (1) admits one-parameter Lie group of the generator (3) is where [X*, V*]=X*V*-V*X*is the operation of Lie bracket.
Introducing the first expansion of infinitesimal transformations for Lagrange sys-tem The necessary and sufficient condition that Lagrange system accepts one-parameter Lie group of the generator (5) is Literature [1] pointed out that the determining equations of Lie symmetry under the transformation (5) for Lagrange system is
Theorem1That Lagrange system admits one-parameter Lie group of the gener-ator (5) is equivalent to the determining equations of Lie symmetry could be expressed as (7), namely,(6) holds if and only if (7) holds.
We have a similar conclusion for Hamilton system. Suppose ζ0,ζs,ηs are in-finitesimal generators. On one hand, Literature [1] derived the determining equations of Lie symmetry for Hamilton system as follows On the other hand, if we denoted the infinitesimal generator vector of a one-parameter Lie group by we have the necessary and sufficient condition that Hamilton system accepts one-parameter Lie group of the generator (9) is
Theorem2That Hamilton system admits one-parameter Lie group of the gener-ator (9) is equivalent to the determining equations of Lie symmetry could be expressed as (8), namely,(10) holds if and only if (8) holds.
We can also prove a similar conclusion for the generalized Hamilton system. Sup-pose ζ0,ζi are infinitesimal generators. On one hand, Literature [1] told us that the
determining equations of Lie symmetry of the generalized Hamilton system had the following expression
On the other hand, if we write the infinitesimal generator vector of some one-parameter Lie group as
we can get the necessary and sufficient condition that the generalized Hamilton system admits one-parameter Lie group of the generator (12) is
Theorem3That the generalized Hamilton system admits one-parameter Lie group of a generator (12) is equivalent to the determining equations of Lie symmetry could be expressed as (11), namely,(13) holds if and only if (11) holds.
Theorem4If the infinitesimal generators ζ0,ζi satisfy (11), and there is a function λ=λ(t,x) satisfying
then we obtain the conserved quantity resulted from the generalized Hamilton system where
Theorem5Suppose the generators under the usual infinitesimal transformation are ζ0,ζi for the generalized Hamilton system, if there exists a matrix Γik satisfying
then the necessary and sufficient condition for the conformal invariance to be Mei symmetry for the generalized Hamilton system is Γki=lik,where lik is the conformal factor of the conformal invariance.
Theorem6For the generalized Hamilton system, if the generators under the usual infinitesimal transformation ζ0,ζi satisfy then the matrix Γik in (17) will be
Theorem6shows that the matrix Γik could be given by the generators, which satisfy the conformal invariance and Mei symmetry simultaneously.
For the planar Kepler equation, we study Hojman conserved quantity resulted from conformal invariance by Lie symmetry. We get the following theorem.
Theorem7Under the time-invariant infinitesimal transformations, If the gen-erators ζs satisfy and there exists a function λ=λ(t,q, q) so that then Hojman conserved quantity induced by the conformal invariance is
We also discuss Mei conserved quantity of Kepler equation deduced by Mei sym-metry. A conserved quantity independent with the system's total energy and angular momentum is obtained.
Theorem8For Nielsen equation, if the infinitesimal generators ζ0,ζs satisfy ζs+2αsζ0-qsζ0=X(1)(αs),(23) and there exists a function λ=λ(t,q, q) satisfying
then a conserved quantity of the Nielsen equation induced by Lie symmetry is
where
Theorem9For complete system Appell equations, If the infinitesimal generators ζ0,ζs satisfy and there is a gauge function GM=GM(t,q, q) satisfying
then a new conserved quantity of the complete system Appell equation deduced by Mei symmetry is
Finally, we consider the following third-order Lotka-Volterra system,
where A, B, C are nonzero real parameters. Theorem10If the systems (29) has quasi-homogeneous polynomial first integral,
then the parameters A, B,C satisfy two of the following four conditions1-A+AC=0,1-C+BC=0,1-B+AB=0, ABC+1=0.(30)
Theorem11When the parameters A,B,C satisfy Theorem10, system (29) has quasi-homogeneous polynomial first integral of degree1Ω1(x,y,z)=Bx-BCy-z.(31)
Theorem12When the parameters A,B,C satisfy Theorem10, system (29) has quasi-homogeneous polynomial first integral of degree2Ω2(x,y,z)=A2B2x2+y2+A2z2+2ABxy-2A2Bxz-2Ayz.(32)
在第一部分即本文的第二章中,我们研究了Lagrange系统、Hamilton系统、广义Hamilton系统的Lie对称性两种提法的等价性,对于广义Hamilton系统还研究了在一般无限小变换下的Lie对称性导致的新型守恒量,给出新型守恒量的表达式和证明过程.
在第二部分即本文的第三章中,我们先研究了广义Hamilton系统的共形不变性与Mei对称性的关系,给出共形不变性同时具有Mei对称性的充要条件,并且得到了共形不变性通过Mei对称性导致的Mei守恒量；其次讨论了Lorenz方程的Robbins模型的共形不变性通过Mei对称性导致的Mei守恒量；最后研究了平面Kepler方程的共形不变性与Lie对称性,给出共形不变性通过Lie对称性导致的Hojman守恒量,同时研究了Kepler方程的Mei对称性,得到了与系统的总能量、角动量相互独立的守恒量.
在第三部分即本文的第四章中,我们研究了Nielsen方程的Lie对称性导致的新型守恒量和Appell方程的Mei对称性导致的新型守恒量,给出新型守恒量的具体表达式和证明过程,举例说明结果的应用.
在第四部分即本文的第五章中,我们应用单参数Lie群理论,研究了广义齐次系统的拟齐次多项式首次积分.通过讨论Lotka-Volterra系统的约化问题,给出系统拟齐次多项式首次积分的存在条件和具体表达式.
This dissertation investigates mainly some problems on conserved quantity of me-chanical systems based on the symmetry theory. Firstly, we discuss the equivalent problem of the two different kinds of Lie symmetries for Lagrange system, Hamilton system, and the generalized Hamilton system respectively. Then, we obtain the con-served quantities deduced by the conformal invariance for the generalized Hamilton system. Secondly, we study the conformal invariance, Lie symmetry, Mei symmetry and conserved quantities of the planar Kepler equation. Thirdly, we are concerned with a new conserved quantity of Nielsen equation induced by Lie symmetry and a new conserved quantity of Appell equation induced by Mei symmetry. Finally, we consider reduction problem of the generalized homogeneous system, the condition that Lotka-Volterra system has quasi-homogeneous polynomial first integral is given, and the expression of the first integral is shown.
We are concerned with the following non-autonomous system
We denote the differential operator corresponding to system (1) by
We will take the infinitesimal generator vector of some one-parameter Lie group as follows
Literature [30] pointed out that the necessary and sufficient condition that system (1) admits one-parameter Lie group of the generator (3) is where [X*, V*]=X*V*-V*X*is the operation of Lie bracket.
Introducing the first expansion of infinitesimal transformations for Lagrange sys-tem The necessary and sufficient condition that Lagrange system accepts one-parameter Lie group of the generator (5) is Literature [1] pointed out that the determining equations of Lie symmetry under the transformation (5) for Lagrange system is
Theorem1That Lagrange system admits one-parameter Lie group of the gener-ator (5) is equivalent to the determining equations of Lie symmetry could be expressed as (7), namely,(6) holds if and only if (7) holds.
We have a similar conclusion for Hamilton system. Suppose ζ0,ζs,ηs are in-finitesimal generators. On one hand, Literature [1] derived the determining equations of Lie symmetry for Hamilton system as follows On the other hand, if we denoted the infinitesimal generator vector of a one-parameter Lie group by we have the necessary and sufficient condition that Hamilton system accepts one-parameter Lie group of the generator (9) is
Theorem2That Hamilton system admits one-parameter Lie group of the gener-ator (9) is equivalent to the determining equations of Lie symmetry could be expressed as (8), namely,(10) holds if and only if (8) holds.
We can also prove a similar conclusion for the generalized Hamilton system. Sup-pose ζ0,ζi are infinitesimal generators. On one hand, Literature [1] told us that the
determining equations of Lie symmetry of the generalized Hamilton system had the following expression
On the other hand, if we write the infinitesimal generator vector of some one-parameter Lie group as
we can get the necessary and sufficient condition that the generalized Hamilton system admits one-parameter Lie group of the generator (12) is
Theorem3That the generalized Hamilton system admits one-parameter Lie group of a generator (12) is equivalent to the determining equations of Lie symmetry could be expressed as (11), namely,(13) holds if and only if (11) holds.
Theorem4If the infinitesimal generators ζ0,ζi satisfy (11), and there is a function λ=λ(t,x) satisfying
then we obtain the conserved quantity resulted from the generalized Hamilton system where
Theorem5Suppose the generators under the usual infinitesimal transformation are ζ0,ζi for the generalized Hamilton system, if there exists a matrix Γik satisfying
then the necessary and sufficient condition for the conformal invariance to be Mei symmetry for the generalized Hamilton system is Γki=lik,where lik is the conformal factor of the conformal invariance.
Theorem6For the generalized Hamilton system, if the generators under the usual infinitesimal transformation ζ0,ζi satisfy then the matrix Γik in (17) will be
Theorem6shows that the matrix Γik could be given by the generators, which satisfy the conformal invariance and Mei symmetry simultaneously.
For the planar Kepler equation, we study Hojman conserved quantity resulted from conformal invariance by Lie symmetry. We get the following theorem.
Theorem7Under the time-invariant infinitesimal transformations, If the gen-erators ζs satisfy and there exists a function λ=λ(t,q, q) so that then Hojman conserved quantity induced by the conformal invariance is
We also discuss Mei conserved quantity of Kepler equation deduced by Mei sym-metry. A conserved quantity independent with the system's total energy and angular momentum is obtained.
Theorem8For Nielsen equation, if the infinitesimal generators ζ0,ζs satisfy ζs+2αsζ0-qsζ0=X(1)(αs),(23) and there exists a function λ=λ(t,q, q) satisfying
then a conserved quantity of the Nielsen equation induced by Lie symmetry is
where
Theorem9For complete system Appell equations, If the infinitesimal generators ζ0,ζs satisfy and there is a gauge function GM=GM(t,q, q) satisfying
then a new conserved quantity of the complete system Appell equation deduced by Mei symmetry is
Finally, we consider the following third-order Lotka-Volterra system,
where A, B, C are nonzero real parameters. Theorem10If the systems (29) has quasi-homogeneous polynomial first integral,
then the parameters A, B,C satisfy two of the following four conditions1-A+AC=0,1-C+BC=0,1-B+AB=0, ABC+1=0.(30)
Theorem11When the parameters A,B,C satisfy Theorem10, system (29) has quasi-homogeneous polynomial first integral of degree1Ω1(x,y,z)=Bx-BCy-z.(31)
Theorem12When the parameters A,B,C satisfy Theorem10, system (29) has quasi-homogeneous polynomial first integral of degree2Ω2(x,y,z)=A2B2x2+y2+A2z2+2ABxy-2A2Bxz-2Ayz.(32)
引文
[1]梅凤翔.约束力学系统的对称性与守恒量[M].北京理工大学出版社,2004.
[2]罗绍凯,张永发等.约束系统动力学研究进展[M].科学出版社,2008.
[3]C. G. Jacobi. Vorlesungen Uber Dynamik[M]. Berlin:Reimer.1866.
[4]J. R. Schiitz. Prinzip der absoluten Erhaltung der Energie[J]. Nachr. Konig. Gesell. Wissen. Gottingen, Math. Phys. KI1897,110-123.
[5]A. Einstein. Zur Elektrodynamik bewegter Korper[J]. Ann. Phys.1905,17:891-921.
[6]F. Engel. Uber die zehn allgemeinen Integrale der Klassischen Mechanik[J]. Nachr. Konig. Gesell. Wissen. Gottingen, Math. Phys. KI1916,270-275.
[7]E. Noether. Invariante Variationsprobleme[J]. Nachr. Konig. Gesell. Wissen. Gottingen, Math. Phys. KI1918,235-257.
[8]李子平.约束系统的变换性质[J].物理学报,1981,20(12)：1659-1671.
[9]梅凤翔.关于Noether对称性、Lie对称性和形式不变性[J].北京理工大学学报,2001,21(4)：535-536.
[10]葛伟宽,梅凤翔.作战微分方程模型的Noether对称性[J].兵工学报,2001,22(2)：241-243.
[11]梅凤翔.关于对称性的注释[J].力学与实践,2004,26(2)：70-71.
[12]Qiao Y F, Li R J, and Ma Y S. Non-Nother Conserved Quantity for Relativistic Non-holonomic System with Variable[J]. Commun. Theor. Phys.,2005,43(2):197-200.
[13]吴惠彬,梅凤翔.关于Noether对称性的两种理解[J].物理学报,2006,55(8)：3825-3828.
[14]Mei F X, Xie J F, Gang T Q. Weakly Noether Symmetry for Nonholonomic Systems of Chetave's Type[J]. Commun. Theor. Phys.,2008,49(6):1413-1416.
[15]I. C. Moreira, O. M. Ritter, and F. C. Santos. Lie Symmetries for charge-monopole problem[J]. J. Phys. A:math. Gen.,1985,18:L427-L430.
[16]Sergio A Hojman. A new conservation law constructed without using either Lagrangians or Hamiltionians[J]. J. Phys. A:Math. Gen.,1992,25:L291-L295.
[17]赵跃宇,梅凤翔.关于力学系统的对称性与不变量[J].力学进展,1993,23(3)：360-372.
[18]Mei F X, Zhang Y F, Shang M. Lie Symmetries and Conserved Quantities of Birkhoffian System[J]. Mechanics research Communications,1999,26(1):7-12.
[19]罗绍凯,梅凤翔.非完整力学系统的非Noether守恒量—Hojman守恒量[J].物理学报,2004,53(3)：666-670.
[20]张宏彬,陈立群,刘荣万,顾书龙.广义Hojman定理[J].物理学报,2005,54(6)：2489-2493.
[21]Luo S K, Jia L Q, and Cai J L. Noether Symmetry Can Lead to Non-Noether Conserved Quantity of Holonomic Non conservative Systems in General Lie Transformations [J]. Commun. Theor. Phys.,2005,43(2):193-196.
[22]Zhang P Y, and Fang J H. Lie Symmetrical Non-Noether Conserved Quantities of Poincare-Chetaev Equations[J]. Commun. Theor. Phys.,2005,44(2):223-225.
[23]楼智美.非中心力场中经典粒子的轨道参数方程与对称性[J].物理学报,2005,54(4)：1460-1463.
[24]吴惠彬,张永发,梅凤翔.求解微分方程的Hojman方法[J].物理学报,2006,55(10)：4987-4990.
[25]Xie J F, Mei F X, and Gang T Q. Weak Noether Symmetry and non-Noether Conserved Quantities for Genaral Holonomic Systems [J]. Commun. Theor. Phys.,2008,50(4):844-846.
[26]Pang T, Fang J H, Lin P, Zhang M J, and Lu K. New Type of Conserved Quantities of Lie Symmetry Nonholonomic Mechanical Systems in Phase Space[J]. Commun. Theor. Phys.,2009,52(6):977-980.
[27]张宏彬,吕洪升,顾书龙.完整约束力学系统保Lie对称性差分格式[J].物理学报,2010,59(8)：5213-5218.
[28]Liu H Z, Li J B. Lie Symmetries, Conservation Laws and Exact Solutions for Two Rod Equations[J]. Acta. Appl. Math.,2010,110:573-587.
[29]Fu J L, Fu H, Liu R W. Hojman conserved quantities of discrete mechanico-electrical systems constructed by continuous symmetries [J]. Physics Letters A,2010,374:1812-1818.
[30]Guan K Y, Liu S, and Lei J Z. Lie Algebra Admitted by an Ordinary Differential Equation System[J]. Ann.of Diff.Eqs.,1998,14(2):131-142.
[31]Hu Y X, Guan K Y. Techniques for Searching First Integrals by Lie Group and Appli-cation to Gyroscope System[J]. Science in China(Series A),2005,48(8):1135-1143.
[32]Mei F X. Form Invariance of Lagrange System[J]. Journal of Beijing Institute of Tech-nology,2000,9(2):120-124.
[33]Mei F X. Form Invariance of Appell Equation[J]. Chin. Phys.2001,10(3):177-180.
[34]Mei F X, Xu X J, Zhang Y F. A unified symmetry of lagrangian systems[J]. Acta. Mech. Sin.2004,20(6):668-671.
[35]葛伟宽,张毅.完整力学系统的Lie-形式不变性[J].物理学报,2005,54(11)：4985-4988.
[36]Fang J H, Wang P, and Ding N. Noether-Mei Symmetry of Mechanical System in Phase Space[J]. Chin. Phys. B,2006,45(5):882-884.
[37]Fang J H, Ding N, and Wang P. Unified Symmetry and Conserved Quantities of Me-chanical System in Phase Space[J]. Commun. Theor. Phys.,2006,46(1):97-100.
[38]李元成,夏丽莉,赵伟,后其宝,王静,荆宏星.机电系统的统一对称性[J].物理学报,2007,56(9)：5037-5040.
[39]Zhang X N, Fang J H, Wang P, and Ding N. Noether-Lie Symmetry of Generalized Classical Mechanical Systems [J]. Commun. Theor. Phys.,2008,49(2):305-307.
A.S.Galiullin, G.G.Gafarov, R.P.Malaishka, and A.M.Khwan. Analytical Dynamics of Helmholtz, Birkhoff and Nambu Systems[M]. UFN, Moscow1997.
[41]蔡建乐,梅凤翔.Lagrange系统Lie点变换下的共形不变性与守恒量[J].物理学报,2008,57(9)：5369-5373.
[42]刘畅,梅凤翔,郭永新.Lagrange系统的共形不变性与Hojman守恒量[J].物理学报,2008,57(11)：6704-6708.
[43]Liu C, Zhu N, Mei F X, and Guo Y X. Conformal Invariance and Noether Conserved Quantities of First-Order Lagrange Systems [J]. Commun. Theor. Phys.,2008,50(5):1065-1068.
[44]Mei F X, Zhu H P. Lie symmetries and Conserved Quantities for the Singular Lagrange System[J]. Journal of Beijing Institute of Technology,2000,9(1):11-14.
[45]梅凤翔.关于Emden方程的对称性[J].力学与实践,2003,25(2)：69-70.
[46]方建会,彭勇,廖永潘.关于Lagrange系统和Hamilton系统的Mei对称性[J].物理学报,2005,54(2)：496-499.
[47]顾书龙,张宏彬.Emden方程的Mei对称性、Lie对称性和Noether对称性[J].物理学报,2006,55(11)：5594-5598.
[48]梅凤翔,解加芳,江铁强.求Emden-Fowler方程积分的分析力学方法[J].物理学报,2007,56(9)：5041-5044.
[49]Ding N, Fang J H, Wang P, and Zhang X N. Peturbation to Lie symmetry and Adiabatic Invariants for General Holonomic Mechanical Systems [J]. Commun. Theor. Phys.,2007,48(1):19-22.
[50]Zhao S H, and Liang L F. On Form Invariance, Lie Symmetry and Three Kinds of Conserved Quantities of Generalized Lagrange's Equations[J]. Commun. Theor. Phys.2007,48(1):37-42.
[51]Ding N, Fang J H, Wang P, and Zhang X N. A New Type of Conserved Quantity of Mei Symmetry for Lagrange Systems [J]. Commun. Theor. Phys.,2007,48(5):799-800.
[52]Mei F X, Wu H B. Symmetry of Lagrangians of nonholomic sysem[J]. Physsics Letters A,2008,372:2141-2147.
[53]方建会.Lagrange系统Mei对称性直接导致的一种守恒量[J].物理学报,2009,58(9)：3617-3619.
[54]顾书龙.Emden方程的Lie对称性与新型守恒量[J].四川兵工学报,2009,30(10)：4-5.
[55]钱华峰,王其申.LRC电路系统的对称性与守恒量[J].巢湖学院学报,2010,12(6)：57-61.
[56]贾利群,张耀宇,杨新芳,崔金超,解银丽.Lagrange系统Mei对称性的Ⅲ型结构方程和Ⅲ型Mei守恒量[J].物理学报,2010,59(5)：2939-2941.
[57]Fang J H, Zhang M J, Zhang W W. A new type conserved quantity induced by sym-metries of Lagrange sysem[J]. Physsics Letters A,2010,374:1806-1811.
[58]乔磊,雷惠方,贾石海,梁景辉.事件空间中Emden方程的Mei对称性与守恒量[J].江西科学,2011,29(5)：552-554.
[59]贾利群,张全顺,孙现亭,王肖肖,张美玲,解银丽.Lagrange系统的特殊统一对称性和特殊守恒量[J].河南师范大学学报(自然科学版),2011,39(6)：35-38.
[60]贾利群,孙现亭,王肖肖,张美玲,解银丽,田燕宁.Lagrange系统的特殊Noether-Lie对称性和特殊守恒量[J].云南大学学报(自然科学版),2011,33(3)：294-296.
[61]罗绍凯.Hamilton系统的Mei对称性、Noether对称性和Lie对称性[J].物理学报,2003,52(12)：2941-2944.
[62]罗绍凯.奇异系统Hamilton正则方程的Mei对称性、Noether对称性和Lie对称性[J].物理学报,2004,53(1)：5-10.
[63]楼智美.哈密顿Ermakov系统的形式不变性[J].物理学报,2005,54(5)：1969-1971.
[64]Xue X J, Qin M C, and Mei F X. Unified Symmetry of Hmailton Systems [J]. Commun. Theor. Phys.,2005,44(5):769-772.
[65]方建会,丁宁,王鹏.Hamilton系统Mei对称性的一种新守恒量[J].物理学报,2007,56(6)：3039-3042.
[66]梅凤翔.广义Hamilton系统的Lie对称性与守恒量[J].物理学报,2003,52(5)：1048-1050.
[67]吴惠彬.广义Hamilton系统的一类不变性与守恒量[J].北京理工大学学报,2004,24(1)：20-22.
[68]贾利群,郑世旺.带有附加项的广义Hamilton系统的Mei对称性[J].物理学报,2006,55(8)：3829-3832.
[69]刘畅,刘世兴,梅凤翔,郭永新.广义Hamilton系统的共形不变性与Hojman守恒量[J].物理学报,2008,57(11)：6709-6713.
[70]姜文安,罗绍凯.广义Hamilton系统的Mei对称性导致的Mei守恒量[J].物理学报,2011,60(6)：060201.
[71]Ayse KARASU, and Hasan YILDIRIM. On the Lie Symmetries of Kepler-Ermakov Systems [J]. Journal of Nonlinear Mathematical Physics,2002,9(4):475-482.
[72]吴惠彬,许学军,王树勇,梅凤翔.完整系统形式不变性导致的新守恒量[J].北京理工大学学报,2004,24(6)：469-471.
[73]P. G. L. Leach. and A. Karasu. The Lie Algebra sl(2,R) and so-called Kepler-Ermakov Systems [J]. Journal of Nonlinear Mathematical Physics,2004,269-275.
[74]梅凤翔.同一类型对称性可生成不同类型守恒量[J].力学与实践,2005,27(3)：84-85.
[75]Wang p, Fang J H, and Ding N. Two Type of New Conserved Quantities and Mei Symmetry of Mechanical System in Phase Space[J]. Commun. Theor. Phys.,2007,48(6):993-995.
[76]P. G. L. Leach. and K. Andriopoulos. The Ermakov Equation A Commentary [J]. Appl. Anal. Discrete Math.,2008,2:146-157.
[77]Zhang X N, and Fang J H. A New Type of Conserved Quantity Deduced from Mei Sym-metry for Relativistic Mechanical System in Phase Space[J]. Commun. Theor. Phys.2008,49(6):1421-1424.
[78]蔡建乐.一般完整系统Mei对称性的共形不变性与守恒量[J].物理学报,2009,58(1)：22-27.
[79]Fang J H, Zhang M J, Lu K. A New Type of Conserved Quantity of Mei Symmetry for Holonomic Mechanical System[J]. Chin. Phys. Lett.,2009,26(11):110202.
[80]Zhang M J, Fang J H, Lin P, Lu K, and Pang T. Conformal Invariance and a New Type of Conserved Quantities of Mechanical Systems with Variable Mass in Phase Space [J]. Commun. Theor. Phys.,2009,53(4):561-564.
[81]顾书龙,张宏彬.Kepler方程的Noether对称性与Ho j man守恒量[J].物理学报,2010,59(2)：716-718.
[82]Cai J L. Conformal Invariance and Conserved Quantity for the Nonholonomic System of Chetaev's Type[J]. Int. J. Theor. Phys.,2010,49:201-211.
[83]Wang P, Xue Y, and Liu Y L. Mei symmetry and conserved quantities in Kirchhoff thin elastic rod statics[J]. Chin. Phys. B,2012,21(7):070203.
[84]贾利群,罗绍凯,张耀宇.非完整系统Nielsen方程的Mei对称性与Mei守恒量[J].物理学报,2008,57(4)：2006-2010.
[85]贾利群,张耀宇,罗绍凯,崔金超.事件空间中单面非Chetaev型非完整系统Nielsen方程的Mei对称性与Mei守恒量[J].物理学报,2009,58(4)：2141-2146.
[86]Yang X F, Jia L Q, Cui J C, and Luo S K. Mei symmetry and Mie conserved quantity of Nielsen equaions for a non-holonomic system of Chetaev's type with variable mass[J]. Chin. Phys. B,2010,19(3):030305.
[87]李元成,王晓明,夏丽莉.完整系统Nielsen方程的统一对称性与守恒量[J].物理学报,2010,59(5)：2935-2938.
[88]张美玲,王肖肖,贾利群,田燕宁.Nielsen系统Mei对称性的新型守恒量[J].河南教育学院学报(自然科学版),2011,20(3)：19-21.
[89]王肖肖,张美玲,贾利群,田燕宁.Nielsen系统的特殊统一对称性和特殊守恒量[J].河南教育学院学报(自然科学版),2011,20(3)：22-23.
[90]解银丽,贾利群,杨新芳。相对运动力学系统Nielsen方程Lie对称性与Hojman守恒量[J].物理学报,2011,60(3)：030201.
[91]贾利群,孙现亭,张美玲,王肖肖,解银丽.Nielsen方程Mei对称性导致的一种新型守恒量[J].物理学报,2011,60(8)：084501。
[92]Zhang M L, Sun X T, Wang X X, Xie Y L, and Jia L Q. Lie symmetry and the gener-alized Hojman conserved quantity of Nielsen equations for a variable mass holonomic system of relative motion [J]. Chin. Phys. B,2011,20(11):110202.
[93]Wang X X, Sun X T, Zhang M L, Xie Y L, and Jia L Q. Lie symmetry and Hojman conserved quantity of a Nielsen equation in a dynamical system of relative motion weith Chetaev-tpe nonholonomic constraint [J]. Chin. Phys. B,2011,20(12):124501.
[94]王肖肖,孙现亭,张美玲,解银丽,贾利群。Chetaev型约束的相对运动动力学系统Nielsen方程的Noether对称性与Noether守恒量[J].物理学报,2012,61(2)：064501。
[95]王肖肖,张美玲,韩月林,贾利群,Chetaev型非完整约束相对运动动力学系统Nielsen方程的Mei对称性和Mei守恒量[J].物理学报,2012,61(20)：200203.
[96]Cui J C, Han Y L, and Jia L Q. A type of conserved quantity of Mei symmetry of Nielsen equation for a holonomic system[J]. Chin. Phys. B,2012,21(8):080201.
[97]Jia L Q, Xie J F,and Zheng S W. Structure equation and Mei conserved quantity for Mei symmetry of Appell equation[J]. Chin. Phys. B,2008,17(1):17-22.
[98]贾利群,张耀宇,崔金超.完整系统Appell方程Mei对称性的结构方程和Mei守恒量[J].云南大学学报(自然科学版),2009,31(1)：52-56。
[99]贾利群,崔金超,张耀宇,罗绍凯。Chetaev型约束力学系统Appell方程的Lie对称性与守恒量[J].物理学报。2009,58(1)：16-21.
[100]Jia L Q, Cui J C, Lug S K, Yang X F. Special Lie Symmetry and Hojman Conserved Quantity of Appell Equations for a Holonomic System[J]. Chin. Phys. Lett.,2009,26(3):030303.
[1[101]梅凤翔。关于Appell的理论力学[J].力学与实践,2010,32(3)111-112.
[102]Cui J C, Zhang Y Y, Yang X F, and Jia L Q. Mei symmetry and Mei conserved quantity of Appell equations for a variable mass holonomic system[J]. Chin. Phys. B,2010,19(3):030304.
[103]Jia L Q, Xie Y L, Zhang Y Y, and Yang X F. A type of new conserved quantity deduced from Mei symmetry for Appell equations in a holonomic system with unilateral constraints [J]. Chin. Phys. B,2010,19(11):110301.
[104]Xie Y L, Jia L Q. Special Lie-Mei Symmetry and Conserved Quantities of Appell Equations Expressed by Appell Function [J]. Chin. Phys. Lett.,2010,27(12):120201.
[105]贾利群,解银丽,张耀宇,崔金超,杨新芳.Appell方程Mei对称性导致的一种新型守恒量[J].物理学报,2010,59(11)：7552-7555.
[106]Xie Y L, Jia L Q, and Luo S K. Special Lie symmetry and Hojman conserved quantity of Appell equations in a dynamical of relative motion [J]. Chin. Phys. B,2011,20(1):010203.
[107]贾利群,解银丽,罗绍凯.相对运动力学系统Appell方程Mei对称性导致的Mei守恒量[J].物理学报,2011,60(4)：040201.
[108]杨新芳,孙现亭,王肖肖,张美玲,贾利群.变质量Chetave型非完整系统Appell方程的Mei对称性和Mei守恒量[J].物理学报,2011,60(11)：111101.
[109]Jia L Q, Zhang M L, Wang X X, and Han Y L. Form invariance and approximate consrevd quantity of Appell equations for a weakly nonholonomic system[J]. Chin. Phys. B,2012,21(7):070204.
[110]孙现亭,韩月林,王肖肖,张美玲,贾利群.完整系统Appell方程Mei对称性的一种新的守恒量[J].物理学报,2012,61(20)：200204.
[111]B. Grammaticos, J. Moulin Ollagnier, A. Ramani, J.-M. Strelcyn, S. Wojciechowski, In-tegrals of quadratic ordinary differential equations in R3:the Lotka-Volterra system[J]. Physica-A,1990,163:683-722.
[112]Furta S D. On Non-integrability of General Systems of Differential Equations [J]. Math.Phys.,1996,47:112-131.
[113] J. Moulin Ollagnier. Polynomial first integrals of Lotka-Volterra system[J]. Bul-l.Sci.Math.,1997,121:463-476.
[114] Tsygvintsev A. On the existence of polynomial first integrals of quadratic homogeneoussystemsof ordinary diferential equations[J]. J. Phys. A:Math. Gen.,2001,34(11):2185-2193.
[115] Libre J, Zhang X. Polynomial First Integrals For Quasi-homogeneous Polynomial Dif-ferential Systems[J]. Nonlinearity,2002,15:1269-1280.
[116] Liu M H, Guan K Y. Reduction of Quasi-Homogeneous Autonomous Systems and Re-duced Kowalevskaya Exponent[J]. Acta Mathematicae Applicatae Sinica,2008,(31)4:729-743.
[2]罗绍凯,张永发等.约束系统动力学研究进展[M].科学出版社,2008.
[3]C. G. Jacobi. Vorlesungen Uber Dynamik[M]. Berlin:Reimer.1866.
[4]J. R. Schiitz. Prinzip der absoluten Erhaltung der Energie[J]. Nachr. Konig. Gesell. Wissen. Gottingen, Math. Phys. KI1897,110-123.
[5]A. Einstein. Zur Elektrodynamik bewegter Korper[J]. Ann. Phys.1905,17:891-921.
[6]F. Engel. Uber die zehn allgemeinen Integrale der Klassischen Mechanik[J]. Nachr. Konig. Gesell. Wissen. Gottingen, Math. Phys. KI1916,270-275.
[7]E. Noether. Invariante Variationsprobleme[J]. Nachr. Konig. Gesell. Wissen. Gottingen, Math. Phys. KI1918,235-257.
[8]李子平.约束系统的变换性质[J].物理学报,1981,20(12)：1659-1671.
[9]梅凤翔.关于Noether对称性、Lie对称性和形式不变性[J].北京理工大学学报,2001,21(4)：535-536.
[10]葛伟宽,梅凤翔.作战微分方程模型的Noether对称性[J].兵工学报,2001,22(2)：241-243.
[11]梅凤翔.关于对称性的注释[J].力学与实践,2004,26(2)：70-71.
[12]Qiao Y F, Li R J, and Ma Y S. Non-Nother Conserved Quantity for Relativistic Non-holonomic System with Variable[J]. Commun. Theor. Phys.,2005,43(2):197-200.
[13]吴惠彬,梅凤翔.关于Noether对称性的两种理解[J].物理学报,2006,55(8)：3825-3828.
[14]Mei F X, Xie J F, Gang T Q. Weakly Noether Symmetry for Nonholonomic Systems of Chetave's Type[J]. Commun. Theor. Phys.,2008,49(6):1413-1416.
[15]I. C. Moreira, O. M. Ritter, and F. C. Santos. Lie Symmetries for charge-monopole problem[J]. J. Phys. A:math. Gen.,1985,18:L427-L430.
[16]Sergio A Hojman. A new conservation law constructed without using either Lagrangians or Hamiltionians[J]. J. Phys. A:Math. Gen.,1992,25:L291-L295.
[17]赵跃宇,梅凤翔.关于力学系统的对称性与不变量[J].力学进展,1993,23(3)：360-372.
[18]Mei F X, Zhang Y F, Shang M. Lie Symmetries and Conserved Quantities of Birkhoffian System[J]. Mechanics research Communications,1999,26(1):7-12.
[19]罗绍凯,梅凤翔.非完整力学系统的非Noether守恒量—Hojman守恒量[J].物理学报,2004,53(3)：666-670.
[20]张宏彬,陈立群,刘荣万,顾书龙.广义Hojman定理[J].物理学报,2005,54(6)：2489-2493.
[21]Luo S K, Jia L Q, and Cai J L. Noether Symmetry Can Lead to Non-Noether Conserved Quantity of Holonomic Non conservative Systems in General Lie Transformations [J]. Commun. Theor. Phys.,2005,43(2):193-196.
[22]Zhang P Y, and Fang J H. Lie Symmetrical Non-Noether Conserved Quantities of Poincare-Chetaev Equations[J]. Commun. Theor. Phys.,2005,44(2):223-225.
[23]楼智美.非中心力场中经典粒子的轨道参数方程与对称性[J].物理学报,2005,54(4)：1460-1463.
[24]吴惠彬,张永发,梅凤翔.求解微分方程的Hojman方法[J].物理学报,2006,55(10)：4987-4990.
[25]Xie J F, Mei F X, and Gang T Q. Weak Noether Symmetry and non-Noether Conserved Quantities for Genaral Holonomic Systems [J]. Commun. Theor. Phys.,2008,50(4):844-846.
[26]Pang T, Fang J H, Lin P, Zhang M J, and Lu K. New Type of Conserved Quantities of Lie Symmetry Nonholonomic Mechanical Systems in Phase Space[J]. Commun. Theor. Phys.,2009,52(6):977-980.
[27]张宏彬,吕洪升,顾书龙.完整约束力学系统保Lie对称性差分格式[J].物理学报,2010,59(8)：5213-5218.
[28]Liu H Z, Li J B. Lie Symmetries, Conservation Laws and Exact Solutions for Two Rod Equations[J]. Acta. Appl. Math.,2010,110:573-587.
[29]Fu J L, Fu H, Liu R W. Hojman conserved quantities of discrete mechanico-electrical systems constructed by continuous symmetries [J]. Physics Letters A,2010,374:1812-1818.
[30]Guan K Y, Liu S, and Lei J Z. Lie Algebra Admitted by an Ordinary Differential Equation System[J]. Ann.of Diff.Eqs.,1998,14(2):131-142.
[31]Hu Y X, Guan K Y. Techniques for Searching First Integrals by Lie Group and Appli-cation to Gyroscope System[J]. Science in China(Series A),2005,48(8):1135-1143.
[32]Mei F X. Form Invariance of Lagrange System[J]. Journal of Beijing Institute of Tech-nology,2000,9(2):120-124.
[33]Mei F X. Form Invariance of Appell Equation[J]. Chin. Phys.2001,10(3):177-180.
[34]Mei F X, Xu X J, Zhang Y F. A unified symmetry of lagrangian systems[J]. Acta. Mech. Sin.2004,20(6):668-671.
[35]葛伟宽,张毅.完整力学系统的Lie-形式不变性[J].物理学报,2005,54(11)：4985-4988.
[36]Fang J H, Wang P, and Ding N. Noether-Mei Symmetry of Mechanical System in Phase Space[J]. Chin. Phys. B,2006,45(5):882-884.
[37]Fang J H, Ding N, and Wang P. Unified Symmetry and Conserved Quantities of Me-chanical System in Phase Space[J]. Commun. Theor. Phys.,2006,46(1):97-100.
[38]李元成,夏丽莉,赵伟,后其宝,王静,荆宏星.机电系统的统一对称性[J].物理学报,2007,56(9)：5037-5040.
[39]Zhang X N, Fang J H, Wang P, and Ding N. Noether-Lie Symmetry of Generalized Classical Mechanical Systems [J]. Commun. Theor. Phys.,2008,49(2):305-307.
A.S.Galiullin, G.G.Gafarov, R.P.Malaishka, and A.M.Khwan. Analytical Dynamics of Helmholtz, Birkhoff and Nambu Systems[M]. UFN, Moscow1997.
[41]蔡建乐,梅凤翔.Lagrange系统Lie点变换下的共形不变性与守恒量[J].物理学报,2008,57(9)：5369-5373.
[42]刘畅,梅凤翔,郭永新.Lagrange系统的共形不变性与Hojman守恒量[J].物理学报,2008,57(11)：6704-6708.
[43]Liu C, Zhu N, Mei F X, and Guo Y X. Conformal Invariance and Noether Conserved Quantities of First-Order Lagrange Systems [J]. Commun. Theor. Phys.,2008,50(5):1065-1068.
[44]Mei F X, Zhu H P. Lie symmetries and Conserved Quantities for the Singular Lagrange System[J]. Journal of Beijing Institute of Technology,2000,9(1):11-14.
[45]梅凤翔.关于Emden方程的对称性[J].力学与实践,2003,25(2)：69-70.
[46]方建会,彭勇,廖永潘.关于Lagrange系统和Hamilton系统的Mei对称性[J].物理学报,2005,54(2)：496-499.
[47]顾书龙,张宏彬.Emden方程的Mei对称性、Lie对称性和Noether对称性[J].物理学报,2006,55(11)：5594-5598.
[48]梅凤翔,解加芳,江铁强.求Emden-Fowler方程积分的分析力学方法[J].物理学报,2007,56(9)：5041-5044.
[49]Ding N, Fang J H, Wang P, and Zhang X N. Peturbation to Lie symmetry and Adiabatic Invariants for General Holonomic Mechanical Systems [J]. Commun. Theor. Phys.,2007,48(1):19-22.
[50]Zhao S H, and Liang L F. On Form Invariance, Lie Symmetry and Three Kinds of Conserved Quantities of Generalized Lagrange's Equations[J]. Commun. Theor. Phys.2007,48(1):37-42.
[51]Ding N, Fang J H, Wang P, and Zhang X N. A New Type of Conserved Quantity of Mei Symmetry for Lagrange Systems [J]. Commun. Theor. Phys.,2007,48(5):799-800.
[52]Mei F X, Wu H B. Symmetry of Lagrangians of nonholomic sysem[J]. Physsics Letters A,2008,372:2141-2147.
[53]方建会.Lagrange系统Mei对称性直接导致的一种守恒量[J].物理学报,2009,58(9)：3617-3619.
[54]顾书龙.Emden方程的Lie对称性与新型守恒量[J].四川兵工学报,2009,30(10)：4-5.
[55]钱华峰,王其申.LRC电路系统的对称性与守恒量[J].巢湖学院学报,2010,12(6)：57-61.
[56]贾利群,张耀宇,杨新芳,崔金超,解银丽.Lagrange系统Mei对称性的Ⅲ型结构方程和Ⅲ型Mei守恒量[J].物理学报,2010,59(5)：2939-2941.
[57]Fang J H, Zhang M J, Zhang W W. A new type conserved quantity induced by sym-metries of Lagrange sysem[J]. Physsics Letters A,2010,374:1806-1811.
[58]乔磊,雷惠方,贾石海,梁景辉.事件空间中Emden方程的Mei对称性与守恒量[J].江西科学,2011,29(5)：552-554.
[59]贾利群,张全顺,孙现亭,王肖肖,张美玲,解银丽.Lagrange系统的特殊统一对称性和特殊守恒量[J].河南师范大学学报(自然科学版),2011,39(6)：35-38.
[60]贾利群,孙现亭,王肖肖,张美玲,解银丽,田燕宁.Lagrange系统的特殊Noether-Lie对称性和特殊守恒量[J].云南大学学报(自然科学版),2011,33(3)：294-296.
[61]罗绍凯.Hamilton系统的Mei对称性、Noether对称性和Lie对称性[J].物理学报,2003,52(12)：2941-2944.
[62]罗绍凯.奇异系统Hamilton正则方程的Mei对称性、Noether对称性和Lie对称性[J].物理学报,2004,53(1)：5-10.
[63]楼智美.哈密顿Ermakov系统的形式不变性[J].物理学报,2005,54(5)：1969-1971.
[64]Xue X J, Qin M C, and Mei F X. Unified Symmetry of Hmailton Systems [J]. Commun. Theor. Phys.,2005,44(5):769-772.
[65]方建会,丁宁,王鹏.Hamilton系统Mei对称性的一种新守恒量[J].物理学报,2007,56(6)：3039-3042.
[66]梅凤翔.广义Hamilton系统的Lie对称性与守恒量[J].物理学报,2003,52(5)：1048-1050.
[67]吴惠彬.广义Hamilton系统的一类不变性与守恒量[J].北京理工大学学报,2004,24(1)：20-22.
[68]贾利群,郑世旺.带有附加项的广义Hamilton系统的Mei对称性[J].物理学报,2006,55(8)：3829-3832.
[69]刘畅,刘世兴,梅凤翔,郭永新.广义Hamilton系统的共形不变性与Hojman守恒量[J].物理学报,2008,57(11)：6709-6713.
[70]姜文安,罗绍凯.广义Hamilton系统的Mei对称性导致的Mei守恒量[J].物理学报,2011,60(6)：060201.
[71]Ayse KARASU, and Hasan YILDIRIM. On the Lie Symmetries of Kepler-Ermakov Systems [J]. Journal of Nonlinear Mathematical Physics,2002,9(4):475-482.
[72]吴惠彬,许学军,王树勇,梅凤翔.完整系统形式不变性导致的新守恒量[J].北京理工大学学报,2004,24(6)：469-471.
[73]P. G. L. Leach. and A. Karasu. The Lie Algebra sl(2,R) and so-called Kepler-Ermakov Systems [J]. Journal of Nonlinear Mathematical Physics,2004,269-275.
[74]梅凤翔.同一类型对称性可生成不同类型守恒量[J].力学与实践,2005,27(3)：84-85.
[75]Wang p, Fang J H, and Ding N. Two Type of New Conserved Quantities and Mei Symmetry of Mechanical System in Phase Space[J]. Commun. Theor. Phys.,2007,48(6):993-995.
[76]P. G. L. Leach. and K. Andriopoulos. The Ermakov Equation A Commentary [J]. Appl. Anal. Discrete Math.,2008,2:146-157.
[77]Zhang X N, and Fang J H. A New Type of Conserved Quantity Deduced from Mei Sym-metry for Relativistic Mechanical System in Phase Space[J]. Commun. Theor. Phys.2008,49(6):1421-1424.
[78]蔡建乐.一般完整系统Mei对称性的共形不变性与守恒量[J].物理学报,2009,58(1)：22-27.
[79]Fang J H, Zhang M J, Lu K. A New Type of Conserved Quantity of Mei Symmetry for Holonomic Mechanical System[J]. Chin. Phys. Lett.,2009,26(11):110202.
[80]Zhang M J, Fang J H, Lin P, Lu K, and Pang T. Conformal Invariance and a New Type of Conserved Quantities of Mechanical Systems with Variable Mass in Phase Space [J]. Commun. Theor. Phys.,2009,53(4):561-564.
[81]顾书龙,张宏彬.Kepler方程的Noether对称性与Ho j man守恒量[J].物理学报,2010,59(2)：716-718.
[82]Cai J L. Conformal Invariance and Conserved Quantity for the Nonholonomic System of Chetaev's Type[J]. Int. J. Theor. Phys.,2010,49:201-211.
[83]Wang P, Xue Y, and Liu Y L. Mei symmetry and conserved quantities in Kirchhoff thin elastic rod statics[J]. Chin. Phys. B,2012,21(7):070203.
[84]贾利群,罗绍凯,张耀宇.非完整系统Nielsen方程的Mei对称性与Mei守恒量[J].物理学报,2008,57(4)：2006-2010.
[85]贾利群,张耀宇,罗绍凯,崔金超.事件空间中单面非Chetaev型非完整系统Nielsen方程的Mei对称性与Mei守恒量[J].物理学报,2009,58(4)：2141-2146.
[86]Yang X F, Jia L Q, Cui J C, and Luo S K. Mei symmetry and Mie conserved quantity of Nielsen equaions for a non-holonomic system of Chetaev's type with variable mass[J]. Chin. Phys. B,2010,19(3):030305.
[87]李元成,王晓明,夏丽莉.完整系统Nielsen方程的统一对称性与守恒量[J].物理学报,2010,59(5)：2935-2938.
[88]张美玲,王肖肖,贾利群,田燕宁.Nielsen系统Mei对称性的新型守恒量[J].河南教育学院学报(自然科学版),2011,20(3)：19-21.
[89]王肖肖,张美玲,贾利群,田燕宁.Nielsen系统的特殊统一对称性和特殊守恒量[J].河南教育学院学报(自然科学版),2011,20(3)：22-23.
[90]解银丽,贾利群,杨新芳。相对运动力学系统Nielsen方程Lie对称性与Hojman守恒量[J].物理学报,2011,60(3)：030201.
[91]贾利群,孙现亭,张美玲,王肖肖,解银丽.Nielsen方程Mei对称性导致的一种新型守恒量[J].物理学报,2011,60(8)：084501。
[92]Zhang M L, Sun X T, Wang X X, Xie Y L, and Jia L Q. Lie symmetry and the gener-alized Hojman conserved quantity of Nielsen equations for a variable mass holonomic system of relative motion [J]. Chin. Phys. B,2011,20(11):110202.
[93]Wang X X, Sun X T, Zhang M L, Xie Y L, and Jia L Q. Lie symmetry and Hojman conserved quantity of a Nielsen equation in a dynamical system of relative motion weith Chetaev-tpe nonholonomic constraint [J]. Chin. Phys. B,2011,20(12):124501.
[94]王肖肖,孙现亭,张美玲,解银丽,贾利群。Chetaev型约束的相对运动动力学系统Nielsen方程的Noether对称性与Noether守恒量[J].物理学报,2012,61(2)：064501。
[95]王肖肖,张美玲,韩月林,贾利群,Chetaev型非完整约束相对运动动力学系统Nielsen方程的Mei对称性和Mei守恒量[J].物理学报,2012,61(20)：200203.
[96]Cui J C, Han Y L, and Jia L Q. A type of conserved quantity of Mei symmetry of Nielsen equation for a holonomic system[J]. Chin. Phys. B,2012,21(8):080201.
[97]Jia L Q, Xie J F,and Zheng S W. Structure equation and Mei conserved quantity for Mei symmetry of Appell equation[J]. Chin. Phys. B,2008,17(1):17-22.
[98]贾利群,张耀宇,崔金超.完整系统Appell方程Mei对称性的结构方程和Mei守恒量[J].云南大学学报(自然科学版),2009,31(1)：52-56。
[99]贾利群,崔金超,张耀宇,罗绍凯。Chetaev型约束力学系统Appell方程的Lie对称性与守恒量[J].物理学报。2009,58(1)：16-21.
[100]Jia L Q, Cui J C, Lug S K, Yang X F. Special Lie Symmetry and Hojman Conserved Quantity of Appell Equations for a Holonomic System[J]. Chin. Phys. Lett.,2009,26(3):030303.
[1[101]梅凤翔。关于Appell的理论力学[J].力学与实践,2010,32(3)111-112.
[102]Cui J C, Zhang Y Y, Yang X F, and Jia L Q. Mei symmetry and Mei conserved quantity of Appell equations for a variable mass holonomic system[J]. Chin. Phys. B,2010,19(3):030304.
[103]Jia L Q, Xie Y L, Zhang Y Y, and Yang X F. A type of new conserved quantity deduced from Mei symmetry for Appell equations in a holonomic system with unilateral constraints [J]. Chin. Phys. B,2010,19(11):110301.
[104]Xie Y L, Jia L Q. Special Lie-Mei Symmetry and Conserved Quantities of Appell Equations Expressed by Appell Function [J]. Chin. Phys. Lett.,2010,27(12):120201.
[105]贾利群,解银丽,张耀宇,崔金超,杨新芳.Appell方程Mei对称性导致的一种新型守恒量[J].物理学报,2010,59(11)：7552-7555.
[106]Xie Y L, Jia L Q, and Luo S K. Special Lie symmetry and Hojman conserved quantity of Appell equations in a dynamical of relative motion [J]. Chin. Phys. B,2011,20(1):010203.
[107]贾利群,解银丽,罗绍凯.相对运动力学系统Appell方程Mei对称性导致的Mei守恒量[J].物理学报,2011,60(4)：040201.
[108]杨新芳,孙现亭,王肖肖,张美玲,贾利群.变质量Chetave型非完整系统Appell方程的Mei对称性和Mei守恒量[J].物理学报,2011,60(11)：111101.
[109]Jia L Q, Zhang M L, Wang X X, and Han Y L. Form invariance and approximate consrevd quantity of Appell equations for a weakly nonholonomic system[J]. Chin. Phys. B,2012,21(7):070204.
[110]孙现亭,韩月林,王肖肖,张美玲,贾利群.完整系统Appell方程Mei对称性的一种新的守恒量[J].物理学报,2012,61(20)：200204.
[111]B. Grammaticos, J. Moulin Ollagnier, A. Ramani, J.-M. Strelcyn, S. Wojciechowski, In-tegrals of quadratic ordinary differential equations in R3:the Lotka-Volterra system[J]. Physica-A,1990,163:683-722.
[112]Furta S D. On Non-integrability of General Systems of Differential Equations [J]. Math.Phys.,1996,47:112-131.
[113] J. Moulin Ollagnier. Polynomial first integrals of Lotka-Volterra system[J]. Bul-l.Sci.Math.,1997,121:463-476.
[114] Tsygvintsev A. On the existence of polynomial first integrals of quadratic homogeneoussystemsof ordinary diferential equations[J]. J. Phys. A:Math. Gen.,2001,34(11):2185-2193.
[115] Libre J, Zhang X. Polynomial First Integrals For Quasi-homogeneous Polynomial Dif-ferential Systems[J]. Nonlinearity,2002,15:1269-1280.
[116] Liu M H, Guan K Y. Reduction of Quasi-Homogeneous Autonomous Systems and Re-duced Kowalevskaya Exponent[J]. Acta Mathematicae Applicatae Sinica,2008,(31)4:729-743.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |