自治广义Birkhoff系统的三重组合梯度系统表示
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摘要
研究自治广义Birkhoff系统的三重组合梯度系统表示.首先,给出了4类三重组合梯度系统的定义和微分方程.其次,给出了自治广义Birkhoff系统成为三重组合梯度系统的条件.最后,利用三重组合梯度系统的性质来研究自治广义Birkhoff系统解的稳定性.文末举例说明结果的应用.
This paper has explored triple combined gradient system representations for autonomous generalized Birkhoffian systems. The definitions and differential equations of triple composite gradient systems have been given. The conditions which autonomous generalized Birkhoffian system become triple composite gradient systems have been obtained.The characteristic of triple composite gradient systems can be used to study the stability of autonomous generalized Birkhoffian system. Four examples have been given to illustrate the application of the results.
This paper has explored triple combined gradient system representations for autonomous generalized Birkhoffian systems. The definitions and differential equations of triple composite gradient systems have been given. The conditions which autonomous generalized Birkhoffian system become triple composite gradient systems have been obtained.The characteristic of triple composite gradient systems can be used to study the stability of autonomous generalized Birkhoffian system. Four examples have been given to illustrate the application of the results.
引文
[1]Santilli R M.Foundation of theoretical mechanicsⅠ[M].New York:Springer,1978.
[2]Santilli R M.Foundation of theoretical mechanicsⅡ[M].New York:Springer-Verlag,1983.
[3]Hirch M W,Smale S.Differential equations,dynamical systems and linear algebra[M].New York:Academic Press,1974.
[4]Mclachlan R I,Quispel G R W,Robidoux N.Geometric integration using discrete gradients[J].Phil Tran R Soc Lond A,1999,357:1 021-1 045.DOI:10.1098/rsta.1999.0363.
[5]梅凤翔,吴惠彬.约束力学系统的梯度表示[M].北京:科学出版社,2016.Mei F X,Wu H B.Gradient representations of constrained mechanical systems[M].Beijing:Science Press,2016.
[6]梅凤翔.关于梯度系统[J].力学与实践,2012,34(1):89-90.Mei F X.About gradient system[J].Mechanics in Engineering,2012,34(1):89-90.
[7]梅凤翔,吴惠彬.广义Birkhoff系统的梯度表示[J].动力学与控制学报,2012,10(4):289-292.Mei F X,Wu H B.A gardient representation for generalized Birkhoff system[J].Journal of Dynamics and Control,2012,10(4):289-292.
[8]梅凤翔,吴惠彬.Birkhoff系统的广义斜梯度表示[J].动力学与控制学报,2015,13(5):329-331.Mei F X,Wu H B.Generalized skew-gradient representation for Birkhoff system[J].Journal of Dynamics and Control,2015,13(5):329-331.
[9]Mei F X,Wu H B.Bifurcation for the generalized Birkhoffian system[J].Chin Phys B,2015,24(5):419-420.
[10]Mei F X,Wu H B.Skew-gradient representation of generalized Birkhoffian system[J].Chin Phys B,2015,24(10):312-314.
[11]Mei F X,Cui J C.Skew-gradient representation of constrained mechamical system[J].Applied Mathematics and Mechanics,2015,36(7):873-882.DOI:10.1007/s10483-015-1954-9.
[12]李彦敏,陈向炜,吴惠彬,等.广义Birkhoff系统的两类广义梯度表示[J].物理学报,2016,65(8):080 201.Li Y M,Chen X W,Wu H B,et al.Two kinds of generalized gradient representations for generalized Birkhoff systems[J].Acta Phys Sin,2016,65(8):080 201.
[13]梅凤翔,吴惠彬,李彦敏.Nielsen方程的两类广义梯度表示[J].北京大学学报:自然科学版,2016,52(4):588-591.Mei F X,Wu H B,Li Y M.Two kinds of gradient representations for Nielsen equations[J].Acta Scientiarum Naturalium Universitatis Pekinensis,2016,52(4):588-591.
[14]崔金超,廖翠萃,梅凤翔.自治Birkhoff系统的四类梯度表示[J].华东师范大学学报:自然科学版,2017(3):94-98.Cui J C,Liao C C,Mei F X.Four kinds of gradient representations of autonomous Birkhoffian systems[J].Journal of East China Normal University:Natural Sciences Edition,2017(3):94-98.
[15]陈向炜,张晔,梅凤翔.用具有负定非对称矩阵的梯度系统构造稳定的广义Birkhoff系统[J].力学学报,2017,49(1):149-153.Chen X W,Zhang Y,Mei F X.Stable generalized Birkhoff systems constructed by using a gradient system with non-symmetrical negative-definite matrix[J].Chinese Journal of Theoretical and Applied Mechanics,2017,49(1):149-153.
[16]曹秋鹏,陈向炜.一类非自治广义Birkhoff系统的稳定性和分岔[J].北京大学学报:自然科学版,2016,52(4):653-657.Cao Q P,Chen X W.Stability and bifurcation for a type of non-autonomous generalized Birkhoffian system[J].Acta Scientiarum Naturalium Universitatis Pekinesis,2016,52(4):653-657.
[17]曹秋鹏,张毅,陈向炜.约束自治广义Birkhoff系统平衡稳定性的梯度系统方法[J].云南大学学报:自然科学版,2015,37(2):228-232.Cao Q P,Zhang Y,Chen X W.Stability of equilibrium for autonomous generalized Birkhoffian system with constraints by gradient system[J].Journal of Yunnan University:Natural Sciences Edition,2015,37(2):228-232.
[18]李彦敏,梅凤翔.非自治Birkhoff系统的广义斜梯度表示[J].云南大学学报:自然科学版,2015,37(6):832-836.Li Y M,Mei F X.A generalized skew-gradient representation for non autonomous Birkhoffian system[J].Journal of Yunnan University:Natural Sciences Edition,2015,37(6):832-836.
[19]王嘉航,张毅.非自治广义Birkhoff系统的半负定矩阵梯度系统表示[J].中山大学学报:自然科学版,2018,57(3):21-24.Wang J H,Zhang Y.A semi-negative definite matrix gradient system representation for non-autonomous generalized Birkhoff system[J].Acta scientiarum naturalium universitatis Sunyatseni,2018,57(3):21-24.
[20]Li C,Liu S X,Mei F X.Stability analysis of a simple rheonomic nonholonomic constrained system[J].Chin Phys B,2016,25(12):314-317.
[21]张毅.一类非自治Birkhoff系统的梯度表示[J].苏州科技学院学报:自然科学版,2015,32(4):1-3.DOI:10.3969/j.issn.1672-0687.2015.04.001.Zhang Y.A gradient representation for a type of non-autonomous Birkhoffian system[J].Journal of Suzhou University of Science and Technology:Natural Sciences Edition,2015,32(4):1-3.
[22]梅凤翔,吴惠彬.广义Birkhoff系统与一类组合斜梯度[J].物理学报,2015,64(18):184 501.Mei F X,Wu H B.Generalized Birkhoff system and a kind of combined gradient system[J].Acta Phys.Sin.,2015,64(18):184 501.
[23]陈向炜,梅凤翔.定常Chetaev型非完整系统的组合梯度表示[J].力学季刊,2016,37(1):45-55.Chen X W,Mei F X.Combined gradient representations of stationary nonholonomic system of Chetaev’s type[J].Chinese Quarterly of Mechanics,2016,37(1):45-55.
[24]梅凤翔.分析力学[M].北京:北京理工大学出版社,2013.Mei F X.Analytical mechanics[M].Beijing:Beijing Institute of Technology Press,2013.
[25]梅凤翔.广义Birkhoff系统动力学[M].北京:北京理工大学出版社,2013.Mei F X.Dynamics of generalized Birkhoff systems[M].Beijing:Beijing Institute of Technology Press,2013.
[2]Santilli R M.Foundation of theoretical mechanicsⅡ[M].New York:Springer-Verlag,1983.
[3]Hirch M W,Smale S.Differential equations,dynamical systems and linear algebra[M].New York:Academic Press,1974.
[4]Mclachlan R I,Quispel G R W,Robidoux N.Geometric integration using discrete gradients[J].Phil Tran R Soc Lond A,1999,357:1 021-1 045.DOI:10.1098/rsta.1999.0363.
[5]梅凤翔,吴惠彬.约束力学系统的梯度表示[M].北京:科学出版社,2016.Mei F X,Wu H B.Gradient representations of constrained mechanical systems[M].Beijing:Science Press,2016.
[6]梅凤翔.关于梯度系统[J].力学与实践,2012,34(1):89-90.Mei F X.About gradient system[J].Mechanics in Engineering,2012,34(1):89-90.
[7]梅凤翔,吴惠彬.广义Birkhoff系统的梯度表示[J].动力学与控制学报,2012,10(4):289-292.Mei F X,Wu H B.A gardient representation for generalized Birkhoff system[J].Journal of Dynamics and Control,2012,10(4):289-292.
[8]梅凤翔,吴惠彬.Birkhoff系统的广义斜梯度表示[J].动力学与控制学报,2015,13(5):329-331.Mei F X,Wu H B.Generalized skew-gradient representation for Birkhoff system[J].Journal of Dynamics and Control,2015,13(5):329-331.
[9]Mei F X,Wu H B.Bifurcation for the generalized Birkhoffian system[J].Chin Phys B,2015,24(5):419-420.
[10]Mei F X,Wu H B.Skew-gradient representation of generalized Birkhoffian system[J].Chin Phys B,2015,24(10):312-314.
[11]Mei F X,Cui J C.Skew-gradient representation of constrained mechamical system[J].Applied Mathematics and Mechanics,2015,36(7):873-882.DOI:10.1007/s10483-015-1954-9.
[12]李彦敏,陈向炜,吴惠彬,等.广义Birkhoff系统的两类广义梯度表示[J].物理学报,2016,65(8):080 201.Li Y M,Chen X W,Wu H B,et al.Two kinds of generalized gradient representations for generalized Birkhoff systems[J].Acta Phys Sin,2016,65(8):080 201.
[13]梅凤翔,吴惠彬,李彦敏.Nielsen方程的两类广义梯度表示[J].北京大学学报:自然科学版,2016,52(4):588-591.Mei F X,Wu H B,Li Y M.Two kinds of gradient representations for Nielsen equations[J].Acta Scientiarum Naturalium Universitatis Pekinensis,2016,52(4):588-591.
[14]崔金超,廖翠萃,梅凤翔.自治Birkhoff系统的四类梯度表示[J].华东师范大学学报:自然科学版,2017(3):94-98.Cui J C,Liao C C,Mei F X.Four kinds of gradient representations of autonomous Birkhoffian systems[J].Journal of East China Normal University:Natural Sciences Edition,2017(3):94-98.
[15]陈向炜,张晔,梅凤翔.用具有负定非对称矩阵的梯度系统构造稳定的广义Birkhoff系统[J].力学学报,2017,49(1):149-153.Chen X W,Zhang Y,Mei F X.Stable generalized Birkhoff systems constructed by using a gradient system with non-symmetrical negative-definite matrix[J].Chinese Journal of Theoretical and Applied Mechanics,2017,49(1):149-153.
[16]曹秋鹏,陈向炜.一类非自治广义Birkhoff系统的稳定性和分岔[J].北京大学学报:自然科学版,2016,52(4):653-657.Cao Q P,Chen X W.Stability and bifurcation for a type of non-autonomous generalized Birkhoffian system[J].Acta Scientiarum Naturalium Universitatis Pekinesis,2016,52(4):653-657.
[17]曹秋鹏,张毅,陈向炜.约束自治广义Birkhoff系统平衡稳定性的梯度系统方法[J].云南大学学报:自然科学版,2015,37(2):228-232.Cao Q P,Zhang Y,Chen X W.Stability of equilibrium for autonomous generalized Birkhoffian system with constraints by gradient system[J].Journal of Yunnan University:Natural Sciences Edition,2015,37(2):228-232.
[18]李彦敏,梅凤翔.非自治Birkhoff系统的广义斜梯度表示[J].云南大学学报:自然科学版,2015,37(6):832-836.Li Y M,Mei F X.A generalized skew-gradient representation for non autonomous Birkhoffian system[J].Journal of Yunnan University:Natural Sciences Edition,2015,37(6):832-836.
[19]王嘉航,张毅.非自治广义Birkhoff系统的半负定矩阵梯度系统表示[J].中山大学学报:自然科学版,2018,57(3):21-24.Wang J H,Zhang Y.A semi-negative definite matrix gradient system representation for non-autonomous generalized Birkhoff system[J].Acta scientiarum naturalium universitatis Sunyatseni,2018,57(3):21-24.
[20]Li C,Liu S X,Mei F X.Stability analysis of a simple rheonomic nonholonomic constrained system[J].Chin Phys B,2016,25(12):314-317.
[21]张毅.一类非自治Birkhoff系统的梯度表示[J].苏州科技学院学报:自然科学版,2015,32(4):1-3.DOI:10.3969/j.issn.1672-0687.2015.04.001.Zhang Y.A gradient representation for a type of non-autonomous Birkhoffian system[J].Journal of Suzhou University of Science and Technology:Natural Sciences Edition,2015,32(4):1-3.
[22]梅凤翔,吴惠彬.广义Birkhoff系统与一类组合斜梯度[J].物理学报,2015,64(18):184 501.Mei F X,Wu H B.Generalized Birkhoff system and a kind of combined gradient system[J].Acta Phys.Sin.,2015,64(18):184 501.
[23]陈向炜,梅凤翔.定常Chetaev型非完整系统的组合梯度表示[J].力学季刊,2016,37(1):45-55.Chen X W,Mei F X.Combined gradient representations of stationary nonholonomic system of Chetaev’s type[J].Chinese Quarterly of Mechanics,2016,37(1):45-55.
[24]梅凤翔.分析力学[M].北京:北京理工大学出版社,2013.Mei F X.Analytical mechanics[M].Beijing:Beijing Institute of Technology Press,2013.
[25]梅凤翔.广义Birkhoff系统动力学[M].北京:北京理工大学出版社,2013.Mei F X.Dynamics of generalized Birkhoff systems[M].Beijing:Beijing Institute of Technology Press,2013.