约束Hamilton系统的积分因子及其守恒量
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摘要
本文提出了约束Hamilton系统守恒量构成的一般途径.首先,给出了约束Hamilton系统的固有约束,并且建立了约束Hamilton系统正则方程;其次,给出了约束Hamilton系统的积分因子和守恒量定理;然后构建了约束Hamilton系统的广义Killing方程;最后举例说明其应用.显然,这种方法与之前的方法相比较,具有步骤清晰明了、限制条件少、运算简单的优点.
In this paper a general approach to constructing the conservation laws for constrained Hamilton system is presented. Firstly the internal constraint of the constrained Hamilton system is given and the canonical equation of the system is established. Secondly, the definition of integrating factors and the conservation theorem of constrained Hamilton system are given. Then the general Killing equation of the constrained Hamilton system is established. Finally, an example is given to illustrate the application of the integrating factor method. Obviously, compared with previous methods, this method has the advantages of clear calculation steps, less restrictive conditions and simple calculation.
In this paper a general approach to constructing the conservation laws for constrained Hamilton system is presented. Firstly the internal constraint of the constrained Hamilton system is given and the canonical equation of the system is established. Secondly, the definition of integrating factors and the conservation theorem of constrained Hamilton system are given. Then the general Killing equation of the constrained Hamilton system is established. Finally, an example is given to illustrate the application of the integrating factor method. Obviously, compared with previous methods, this method has the advantages of clear calculation steps, less restrictive conditions and simple calculation.
引文
[1]乔永芬,岳庆文,董永安.广义力学中完整非保守系统的Noether守恒律[J].应用数学和力学,1994,15(9):831-837.
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[3]刘端.非完整非保守动力学系统的Nother理论及其逆定理[J].中国科学,1990, 11:1189-1197.
[4]CHEHJ,OLVERPJ,POHJANPELTOJ.Algorithmsfordifferentialinvariantsofsymmetrygroupsof differential equations[M]. New York:Springer-Verlag, 2008, 501-532.
[5]梅凤翔,吴润衡,张永发.非Chetaev型非完整系统的Lie对称性和和守恒量[J].力学学报,1998,30(4):468-473.
[6]刘荣万,傅景礼.非完整非保守力学系统在相对空间的Lie对称性和守恒量[J].应用数学和学,1999,20(6):579-601.
[7]葛伟宽,张毅.变质量完整力学系统的Hojman守恒量[J].力学季刊,2004, 25(4):573-576.
[8]BLUMAN G W, KUMMEI S. Symmetries and differential equations[M]. Berlin:Springer, 1989.
[9]OLVER P J. Applications of Lie group to differential equations[M]. Berlin:Springer, 2000.
[10]OLVER P J. Equivalence, invariants and symmetry[M]. Cambridge:Cambridge University Press, 1995.
[11]MACCALLUMM.Differentialequations:theirsolutionusingsymmetries[M].Cambridge:Cambridge University Press, 1989.
[12]OVSIANNIKOV L V. Group analysis of differential equations[M]. New York:Academic Press, 1982.
[13]GORRINGEVM,LEACHPGL.Liepointsymmetriesforsystemsofsecondorderlinearordinary differential equations[J]. Questions Mathematicae, 1988, 11(1):95-117.
[14]OLVERPJ,POHJANPELTOJ.MovingframesforLiepseudo-groups[J].CanadianJournalof Mathematics, 2013, 60(6):1336-1386.
[15]MEI J Q, WANG H Y. The generating set of the differential invariant algebra and Maurer-Cartan equations ofA(2+1)-dimensionalburgersequation[J].JournalofSystemsScience&Complexity,2013,26(2):281-290.
[16]BEFFAG.Poissongeometryofdifferentialinvariantsofcurvesinsomenonsemisimplehomogeneous spaces[J]. Proceedings of the American Mathematical Society, 2006, 134(3):779-791.
[17]OLCWEPJ,POHJANPELTOJ.DifferentialinvariantalgebrasofLiepseudo-groups[J].Advancesin Mathematics, 2013, 222(5):1746-1792.
[18]LISLEIG,REISGJ.Symmetryclassificationusingnoncommutativeinvariantdifferentialoperators[J].Foundations of Computational Mathematics, 2006, 6(3):33-386.
[19]ZHAOYY.Liesymmetriesandconservedquantitiesofnon-conservativemechanicalsystem(in Chinese)[J]. Chinese Journal of Theoretical and Applied Mechanics, 1994, 26(3):380-384.
[20]MEIFX.ApplicationsofLiegroupsandLiealgebrastoconstrainedmechanicalsystems[M].Beijing:Science Press, 1999.
[21]ZHANGHB.Liesymmetriesandconservedquantitiesofnon-holonomicmechanicalsystemswith unilateral Vacco constraints[J]. Chinese Physics, 2002, 11(1):1-4.
[22]张毅,薛纭.完整力学系统的共性不变性与守恒量[J].力学季刊,2009, 30(2):216-221.
[23]韩广才,张耀良.一类广义力学系统的能量方程[J].哈尔滨工业大学学报,2001, 22(4):67-71.
[24]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.
[25]梅凤翔.非完整力学基础[M].北京:北京工业学院出版社,1985.
[26]梅凤翔.常群和常代数对约束力学系统的应用[M].北京:科学出版社,1999.
[27]张毅.Hamilton系统的一类新型守恒律[J].力学季刊,2002, 23(3):392-396.
[28]MEI F X. Lie symmetries and conserved quantities of constrained mechanical systems[J].ActaMechanica,2000, 141(3-4):135-148.
[29]MEIFX.Symmetryandconservedquantityofconstrainedmechanicalsystem[M].Beijing:Instituteof Technology Press, 2004.
[30]CHEN X W, CHANG L, MEI F X. Conformal invariance and Hojman conserved quantities of first order Lagrange systems[J]. Chinese Physics B, 2008, 17(9):3180-3184.
[31]DJUKICD,SUTELAT.Integratingfactorsandconservationlawsfornon-conservativedynamical systems[J]. International Journal of Non-Linear Mechanics, 1984, 19(4):331-339.
[32]乔永芬,张耀良,赵淑红.完整非保守系统Raitzin正则运动方程的积分因子和守定理[J].物理学报,2002, 51(8):1661-1665.
[33]乔永芬,张耀良,赵淑红.具有单面约束的非完整动力学系统的积分因子和守恒定理[J].长沙大学学报,2002, 16(2):11-15.
[34]乔永芬,张耀良,韩广才.变质量非完整Vacco动力学系统的积分因子和守恒定理[J].商丘师范学院学报,2003, 19(20):1-6.
[35]乔永芬,赵淑红,李仁杰.时间空间中非完整非保守系统的守恒量存在定理及其逆定理[J].物理学报,2006, 55(1):5585-5589.
[36]乔永芬,韩广才,张耀良.Chaplygin方程的积分因子和守恒定理[J].信阳师范学院学报,2003,16(2):153-156.
[37]张毅,薛纭.Birkhoff系统的积分因子和守恒量[J].力学季刊,2003, 24(2):280-285.
[38]张毅,范存新.事件空间中约束Brikhoff系统的守恒定理[C]//中国数学力学物理学高新技术交叉研究学会第十二届学术年会,2008.
[39]张毅,薛纭.Birkhoff系统的积分因子与守恒定理[J].力学季刊,2003, 24(2):280-285.
[40]束方平,张毅.用积分因子方法研究广义Birkhoff系统的守恒律[J].华中师范大学学报,2014,48(1):42-45.
[41]束方平,张毅,朱建青.基于分数阶模型的Lagrange系统的积分因子和守恒量[J].苏州科技学院学报,2015, 2:1-5.
[42]张耀良,朱正兵.变质量非完整系统Hamilton正则方程积分因子和守恒定理[J].哈尔滨工程大学学报,2002, 23(4):118-121.
[43]郑世旺,乔永芬.准坐标下广义非保守系统Lagrange方程的积分因子与守恒定理[J].物理学报,2006,55(7):3241-3245.
[44]周景润.约束Hamilton系统的Lie对称性及其在场论中的应用[D].杭州:浙江理工大学,2017.
[2]UTZKYM.Dynamicalsymmetriesandconservedquantities[J].JournalofPhysicsAGeneralPhysics,1979, 12(17):973-981.
[3]刘端.非完整非保守动力学系统的Nother理论及其逆定理[J].中国科学,1990, 11:1189-1197.
[4]CHEHJ,OLVERPJ,POHJANPELTOJ.Algorithmsfordifferentialinvariantsofsymmetrygroupsof differential equations[M]. New York:Springer-Verlag, 2008, 501-532.
[5]梅凤翔,吴润衡,张永发.非Chetaev型非完整系统的Lie对称性和和守恒量[J].力学学报,1998,30(4):468-473.
[6]刘荣万,傅景礼.非完整非保守力学系统在相对空间的Lie对称性和守恒量[J].应用数学和学,1999,20(6):579-601.
[7]葛伟宽,张毅.变质量完整力学系统的Hojman守恒量[J].力学季刊,2004, 25(4):573-576.
[8]BLUMAN G W, KUMMEI S. Symmetries and differential equations[M]. Berlin:Springer, 1989.
[9]OLVER P J. Applications of Lie group to differential equations[M]. Berlin:Springer, 2000.
[10]OLVER P J. Equivalence, invariants and symmetry[M]. Cambridge:Cambridge University Press, 1995.
[11]MACCALLUMM.Differentialequations:theirsolutionusingsymmetries[M].Cambridge:Cambridge University Press, 1989.
[12]OVSIANNIKOV L V. Group analysis of differential equations[M]. New York:Academic Press, 1982.
[13]GORRINGEVM,LEACHPGL.Liepointsymmetriesforsystemsofsecondorderlinearordinary differential equations[J]. Questions Mathematicae, 1988, 11(1):95-117.
[14]OLVERPJ,POHJANPELTOJ.MovingframesforLiepseudo-groups[J].CanadianJournalof Mathematics, 2013, 60(6):1336-1386.
[15]MEI J Q, WANG H Y. The generating set of the differential invariant algebra and Maurer-Cartan equations ofA(2+1)-dimensionalburgersequation[J].JournalofSystemsScience&Complexity,2013,26(2):281-290.
[16]BEFFAG.Poissongeometryofdifferentialinvariantsofcurvesinsomenonsemisimplehomogeneous spaces[J]. Proceedings of the American Mathematical Society, 2006, 134(3):779-791.
[17]OLCWEPJ,POHJANPELTOJ.DifferentialinvariantalgebrasofLiepseudo-groups[J].Advancesin Mathematics, 2013, 222(5):1746-1792.
[18]LISLEIG,REISGJ.Symmetryclassificationusingnoncommutativeinvariantdifferentialoperators[J].Foundations of Computational Mathematics, 2006, 6(3):33-386.
[19]ZHAOYY.Liesymmetriesandconservedquantitiesofnon-conservativemechanicalsystem(in Chinese)[J]. Chinese Journal of Theoretical and Applied Mechanics, 1994, 26(3):380-384.
[20]MEIFX.ApplicationsofLiegroupsandLiealgebrastoconstrainedmechanicalsystems[M].Beijing:Science Press, 1999.
[21]ZHANGHB.Liesymmetriesandconservedquantitiesofnon-holonomicmechanicalsystemswith unilateral Vacco constraints[J]. Chinese Physics, 2002, 11(1):1-4.
[22]张毅,薛纭.完整力学系统的共性不变性与守恒量[J].力学季刊,2009, 30(2):216-221.
[23]韩广才,张耀良.一类广义力学系统的能量方程[J].哈尔滨工业大学学报,2001, 22(4):67-71.
[24]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.
[25]梅凤翔.非完整力学基础[M].北京:北京工业学院出版社,1985.
[26]梅凤翔.常群和常代数对约束力学系统的应用[M].北京:科学出版社,1999.
[27]张毅.Hamilton系统的一类新型守恒律[J].力学季刊,2002, 23(3):392-396.
[28]MEI F X. Lie symmetries and conserved quantities of constrained mechanical systems[J].ActaMechanica,2000, 141(3-4):135-148.
[29]MEIFX.Symmetryandconservedquantityofconstrainedmechanicalsystem[M].Beijing:Instituteof Technology Press, 2004.
[30]CHEN X W, CHANG L, MEI F X. Conformal invariance and Hojman conserved quantities of first order Lagrange systems[J]. Chinese Physics B, 2008, 17(9):3180-3184.
[31]DJUKICD,SUTELAT.Integratingfactorsandconservationlawsfornon-conservativedynamical systems[J]. International Journal of Non-Linear Mechanics, 1984, 19(4):331-339.
[32]乔永芬,张耀良,赵淑红.完整非保守系统Raitzin正则运动方程的积分因子和守定理[J].物理学报,2002, 51(8):1661-1665.
[33]乔永芬,张耀良,赵淑红.具有单面约束的非完整动力学系统的积分因子和守恒定理[J].长沙大学学报,2002, 16(2):11-15.
[34]乔永芬,张耀良,韩广才.变质量非完整Vacco动力学系统的积分因子和守恒定理[J].商丘师范学院学报,2003, 19(20):1-6.
[35]乔永芬,赵淑红,李仁杰.时间空间中非完整非保守系统的守恒量存在定理及其逆定理[J].物理学报,2006, 55(1):5585-5589.
[36]乔永芬,韩广才,张耀良.Chaplygin方程的积分因子和守恒定理[J].信阳师范学院学报,2003,16(2):153-156.
[37]张毅,薛纭.Birkhoff系统的积分因子和守恒量[J].力学季刊,2003, 24(2):280-285.
[38]张毅,范存新.事件空间中约束Brikhoff系统的守恒定理[C]//中国数学力学物理学高新技术交叉研究学会第十二届学术年会,2008.
[39]张毅,薛纭.Birkhoff系统的积分因子与守恒定理[J].力学季刊,2003, 24(2):280-285.
[40]束方平,张毅.用积分因子方法研究广义Birkhoff系统的守恒律[J].华中师范大学学报,2014,48(1):42-45.
[41]束方平,张毅,朱建青.基于分数阶模型的Lagrange系统的积分因子和守恒量[J].苏州科技学院学报,2015, 2:1-5.
[42]张耀良,朱正兵.变质量非完整系统Hamilton正则方程积分因子和守恒定理[J].哈尔滨工程大学学报,2002, 23(4):118-121.
[43]郑世旺,乔永芬.准坐标下广义非保守系统Lagrange方程的积分因子与守恒定理[J].物理学报,2006,55(7):3241-3245.
[44]周景润.约束Hamilton系统的Lie对称性及其在场论中的应用[D].杭州:浙江理工大学,2017.