时间尺度上奇异Chetaev型非完整力学系统的Lie对称性
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摘要
研究了时间尺度上奇异Chetaev型非完整力学系统的Lie对称性与守恒量问题.首先,建立了系统的运动微分方程.其次,基于时间尺度上微分方程在无限小变换下的不变性,给出了时间尺度上奇异Chetaev型非完整系统Lie对称性的确定方程和限制方程.最后,建立时间尺度上奇异Chetaev型非完整系统Lie对称性的结构方程,给出了Lie对称性的守恒量,并举例说明结果的应用.
The Lie symmetry and conserved quantities of singular nonholonomic systems of Chetaev's type on time scales were studied. Firstly, the motion differential equations were established. Secondly, based on the invariance of differential equations under infinitesimal transformation on time scales, we gave the determining equation and the limiting equation of Lie symmetry for singular nonholonomic systems of Chetaev's type on time scales. Finally, a structural equation was established, and the conservation of Lie symmetry was given. At the end of the paper, an example was given to illustrate the application of the theorem.
The Lie symmetry and conserved quantities of singular nonholonomic systems of Chetaev's type on time scales were studied. Firstly, the motion differential equations were established. Secondly, based on the invariance of differential equations under infinitesimal transformation on time scales, we gave the determining equation and the limiting equation of Lie symmetry for singular nonholonomic systems of Chetaev's type on time scales. Finally, a structural equation was established, and the conservation of Lie symmetry was given. At the end of the paper, an example was given to illustrate the application of the theorem.
引文
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[12] 祖启航,朱建青,宋传静.时间尺度上相空间中非Chetaev型非完整系统的Noether理论[J].华中师范大学学报(自然科学版),2017,51(1):23-27.ZU Q H,ZHU J Q,SONG C J.Noether theorem for nonholonomic systems of non-Chetaev’s type in phase space on time scales[J].Journal of Central China Normal University (Natural Sciences),2017,51(1):23-27.(Ch).
[13] ZU Q H,ZHU J Q.Noether theorem for nonholonomic nonconservative mechanical systems in phase space on time scales[J].Journal of Mathematical Physics,2016,57(8):18-56.
[14] 林魏,朱建青.时间尺度上非保守系统的Lie对称性及其守恒量[J].华中师范大学学报(自然科学版),2017,51(6):772-776.LIN W,ZHU J Q.Lie symmetry and conserved quantity for non-conservative systems on time scales[J].Journal of Central China Normal University (Natural Sciences),2017,51(6):772-776.(Ch).
[2] MEI F X,ZHU H P.Lie symmetries and conserved quantities for the singular Lagrange system[J].Journal of Beijing Institute of Technology(English Edition),2000,9(1):11-14.
[3] LI Y C,ZhANG Y,LIANG J H.Lie symmetries and conserved quantities of a type of nonholonomic singular systems[J].Acta Physica Sinica,2002,51(10):2186-2190.
[4] LUO S K.Mei symmetry ,Noether symmetry and Lie symmetry of Hamiltonian canonical equations in a singular system[J].Acta Physica Sinica,2004,53(1):5-10.
[5] 徐超.奇异系统的对称性与守恒量研究[D].北京:中国石油大学,2014.XU C.Symmetries and Conserved Quantities of the Singular Systems[D].Beijing:China University of Petroleum,2014.(Ch).
[6] BOHNER M.Calculus of variations on time scales[J].Dynam Systems Appl,2004,13(12):339-349.
[7] BARTOSIEWICZ Z,TORRES D F M.Noether’s theorem on time scales[J].Journal of Mathematical Analysis and Applications,2008,342(2):1220-1226.
[8] CAI P P,FU J L,GUO Y X.Noether symmetries of the nonconservative and nonholonomic systems on time scales[J].Science China(Physics,Mechanics & Astronomy),2013,56(5):1017-1028.
[9] CAI P P,FU J L,GUO Y X.Lie symmetries and conserved quantities of the constraint mechanical systems on time scales[J].Reports on Mathematical Physics,2017,79(3):279-298.
[10] 张毅.时间尺度上Hamilton系统的Noether理论[J].力学季刊,2016,37(2):214-224.ZHANG Y.Noether theory for Hamiltonian system on time scales[J].Chinese Quarterly of Mechanics,2016,37(2):214-224.(Ch).
[11] SONG C J,ZHANG Y.Noether theorem for Birkhoffian systems on time scales[J].Journal of Mathematical Physics,2015,56(10):1-26.
[12] 祖启航,朱建青,宋传静.时间尺度上相空间中非Chetaev型非完整系统的Noether理论[J].华中师范大学学报(自然科学版),2017,51(1):23-27.ZU Q H,ZHU J Q,SONG C J.Noether theorem for nonholonomic systems of non-Chetaev’s type in phase space on time scales[J].Journal of Central China Normal University (Natural Sciences),2017,51(1):23-27.(Ch).
[13] ZU Q H,ZHU J Q.Noether theorem for nonholonomic nonconservative mechanical systems in phase space on time scales[J].Journal of Mathematical Physics,2016,57(8):18-56.
[14] 林魏,朱建青.时间尺度上非保守系统的Lie对称性及其守恒量[J].华中师范大学学报(自然科学版),2017,51(6):772-776.LIN W,ZHU J Q.Lie symmetry and conserved quantity for non-conservative systems on time scales[J].Journal of Central China Normal University (Natural Sciences),2017,51(6):772-776.(Ch).