具Marta势能Hamilton系统的Liouville不可积性
|
摘要
基于Morales-Ramis理论,用理论分析的方法考虑具有Marta势能的Hamilton系统的不可积性问题,证明了该Hamilton系统在Liouville意义下是亚纯不可积的.利用该结果可从不可积性的角度了解该系统的动力学行为及拓扑结构.
Based on the Morales-Ramis theory,we considered the non-integrability of the Hamiltonian system with Marta potential energy by using the theoretical analysis method,and proved that this Hamiltonian system was not meromorphic integrable in sense of Liouville.The results can be used to understand the dynamic behavior and topological structure of the system from the perspective of non-integrability.
Based on the Morales-Ramis theory,we considered the non-integrability of the Hamiltonian system with Marta potential energy by using the theoretical analysis method,and proved that this Hamiltonian system was not meromorphic integrable in sense of Liouville.The results can be used to understand the dynamic behavior and topological structure of the system from the perspective of non-integrability.
引文
[1]焦佳,周庆健,周冉.用中心流形理论研究微分系统的不可积性[J].吉林大学学报(理学版),2014,52(2):269-272.(JIAO Jia,ZHOU Qingjian,ZHOU Ran.Nonintegrability of Differential Systems Based on Central Manifold Theory[J].Journal of Jilin University(Science Edition),2014,52(2):269-272.)
[2]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),2009,47(2):277-280.(ZHAO Hongfa,LI Wenlei,LUO Yucheng.Rational First Integrals of Polynomial Systems[J].Journal of Jilin University(Science Edition),2009,47(2):277-280.)
[3]Ziglin S L.Branching of Solutions and Nonexistence of First Integrals in Hamiltonian Mechanics.Ⅰ[J].Function Analy Appl,1982,16(3):181-189.
[4]Ziglin S L.Branching of Solutions and the Nonexistence of First Integrals in Hamiltonian Mechanics.Ⅱ[J].Function Anal Appl,1983,17(1):6-17.
[5]Yoshida H.Non-integrability of the Truncated Toda Lattice Hamiltonian at Any Order[J].Comm Math Phys,1988,116(4):529-538.
[6]Irigoyen M,SimóC.Non Integrability of the J2 Problem[J].Celestial Mech Dynal Astronomy,1993,55(3):281-287.
[7]Yoshida H.Non-integrability of a Class of Perturbed Kepler Problems in Two Dimensions[J].Phys Lett A,1987,120(8):388-390.
[8]Morales-Ruiz J J,Ramis J P.Galoisian Obstructions to Integrability of Hamiltonian Systems.Ⅰ,Ⅱ[J].Meth Appl Anal,2001,8(1):33-95;97-111.
[9]Singer M F,Ulmer F.Galois Groups of Second and Third Order Linear Differential Equations[J].J Symb Comput,1993,16(1):9-36.
[10]Tsygvintsev A V.Non-existence of New Meromorphic First Integrals in the Planar Three-Body Problem[J].Celestial Mech Dynam Astronom,2003,86(3):237-247.
[11]Morales-Ruiz J J,Sim O C,Simon A S.Algebraic Proof of the Non-integrability of Hills Problem[J].Ergodic Theory Dynam Systems,2005,5(4):1237-1256.
[12]Maciejewski A J,Przybylska M.Non-integrability of ABCFlow[J].Phys Lett A,2002,303(4):265-272.
[13]HUANG Kaiyin,SHI Shaoyun,LI Wenlei.Meromorphic Non-integrability of Several 3D Dynamical Systems[J/OL].Entropy,2017-05-10.doi:10.3390/e19050211.
[14]Aguiar M A M,de,Malta C P,Baranger M,et al.Bifurcations of Periodic Trajectories in Non-integrable Hamiltonian Systems with Two Degrees of Freedom:Numerical and Analytical Results[J].Annal Phys,1987,180(2):167-205.
[15]Kovacic J J.An Algorithm for Solving Second Order Linear Homogeneous Differential Equations[J].J Symbolic Comput,1986,2(1):3-43.
[2]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),2009,47(2):277-280.(ZHAO Hongfa,LI Wenlei,LUO Yucheng.Rational First Integrals of Polynomial Systems[J].Journal of Jilin University(Science Edition),2009,47(2):277-280.)
[3]Ziglin S L.Branching of Solutions and Nonexistence of First Integrals in Hamiltonian Mechanics.Ⅰ[J].Function Analy Appl,1982,16(3):181-189.
[4]Ziglin S L.Branching of Solutions and the Nonexistence of First Integrals in Hamiltonian Mechanics.Ⅱ[J].Function Anal Appl,1983,17(1):6-17.
[5]Yoshida H.Non-integrability of the Truncated Toda Lattice Hamiltonian at Any Order[J].Comm Math Phys,1988,116(4):529-538.
[6]Irigoyen M,SimóC.Non Integrability of the J2 Problem[J].Celestial Mech Dynal Astronomy,1993,55(3):281-287.
[7]Yoshida H.Non-integrability of a Class of Perturbed Kepler Problems in Two Dimensions[J].Phys Lett A,1987,120(8):388-390.
[8]Morales-Ruiz J J,Ramis J P.Galoisian Obstructions to Integrability of Hamiltonian Systems.Ⅰ,Ⅱ[J].Meth Appl Anal,2001,8(1):33-95;97-111.
[9]Singer M F,Ulmer F.Galois Groups of Second and Third Order Linear Differential Equations[J].J Symb Comput,1993,16(1):9-36.
[10]Tsygvintsev A V.Non-existence of New Meromorphic First Integrals in the Planar Three-Body Problem[J].Celestial Mech Dynam Astronom,2003,86(3):237-247.
[11]Morales-Ruiz J J,Sim O C,Simon A S.Algebraic Proof of the Non-integrability of Hills Problem[J].Ergodic Theory Dynam Systems,2005,5(4):1237-1256.
[12]Maciejewski A J,Przybylska M.Non-integrability of ABCFlow[J].Phys Lett A,2002,303(4):265-272.
[13]HUANG Kaiyin,SHI Shaoyun,LI Wenlei.Meromorphic Non-integrability of Several 3D Dynamical Systems[J/OL].Entropy,2017-05-10.doi:10.3390/e19050211.
[14]Aguiar M A M,de,Malta C P,Baranger M,et al.Bifurcations of Periodic Trajectories in Non-integrable Hamiltonian Systems with Two Degrees of Freedom:Numerical and Analytical Results[J].Annal Phys,1987,180(2):167-205.
[15]Kovacic J J.An Algorithm for Solving Second Order Linear Homogeneous Differential Equations[J].J Symbolic Comput,1986,2(1):3-43.