Bessel方程的Noether对称性和守恒量
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摘要
Birkhoff力学比Hamilton力学更普遍,但只有一些动力系统能够实现Birkhoff化.文章基于Santilli的第一方法,给出经典贝塞尔方程的一种新型Birkhoff化.通过引入Lie群无穷小变换下的不变性,建立Bessel方程的Noether对称性变换与准对称性变换,给出相应的对称性判据.得到Bessel方程Noether定理导致的守恒量,以及Noether逆定理.最后,给出n阶经典Bessel方程的Noether定理导致的一个守恒量,说明本方法的有效性.
The Birkhoffan mechanics is more general than the Hamilton systems,but only for some dynamical systems can be applied by a Birkhoffan formulations. This paper explores a novel Birkhoffan formulations of the classical Bessel equation.Based on the first method of Santilli,Birkhoffan formulation of Bessel equation has been established.The Noether symmetric and quasisymmetric transformations of the Bessel equation have been established by introducing the invariance of the action under infinitesimal transformations,the criteria of corresponding symmetries are proposed.The Noether theorems of Bessel equation have been presented,and the conserved quantities have been obtained and n-th order classical Bessel equation has been studied to show the effectiveness of the proposed method.
The Birkhoffan mechanics is more general than the Hamilton systems,but only for some dynamical systems can be applied by a Birkhoffan formulations. This paper explores a novel Birkhoffan formulations of the classical Bessel equation.Based on the first method of Santilli,Birkhoffan formulation of Bessel equation has been established.The Noether symmetric and quasisymmetric transformations of the Bessel equation have been established by introducing the invariance of the action under infinitesimal transformations,the criteria of corresponding symmetries are proposed.The Noether theorems of Bessel equation have been presented,and the conserved quantities have been obtained and n-th order classical Bessel equation has been studied to show the effectiveness of the proposed method.
引文
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[2]孔楠,朱建青.时间尺度上Lagrange系统Mei对称性及其直接导致的Mei守恒量[J].云南大学学报:自然科学版,2017,39(2):219-224.KONG N,ZHU J Q.On the Mei conserved quantity caused directly by Mei symmetry of the Lagrange system on time scales[J].Journal of Yunnan University:Natural Sciences Edition,2017,39(2):219-224.
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[4]孙现亭,张美玲,张耀宇,等.变质量相对运动力学系统Nielsen方程的Noether守恒量[J].云南大学学报:自然科学版,2014,36(3):360-365.SUN X T,ZHANG M L,ZHANG Y Y,et al.Noether conserved quantity for Nielsen equations in a variable mass holonomic dynamical system of the relative motion[J].Journal of Yunnan University:Natural Sciences Edition,2014,36(3):360-365.
[5]王建波,陈梅,解加芳,等.约束系统广义Tzénoff方程的Mei对称性及其对应的守恒量[J].云南大学学报:自然科学版,2012,34(5):533-539.WANG J B,CHEN M,XIE J F,et al.On the Mei symmetry of the restraint system generalized Tzénoff equations and the corresponding conserved quantities[J].Journal of Yunnan University:Natural Sciences Edition,2012,34(5):533-539.
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[7]郑世旺,陈梅.非完整系统Tzénoff方程的Mei对称性所对应的一种新守恒量[J].云南大学学报:自然科学版,2011,33(4):412-416.ZHENG S W,CHEN M.A new conserved quantity by Mei symmetry of Tzénoff equation for the non-holonomic systems[J].Journal of Yunnan University:Natural Sciences Edition,2011,33(4):412-416.
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[12]梅凤翔,吴惠彬,李彦敏,等.Birkhoff力学的研究进展[J].力学学报,2016,48(2):263-268.MEI F X,WU H B,LI Y M,et al.Advances in research on Birkhoffian mechanics[J].Chinese Journal of Theoretical and Applied Mechanics,2016,48(2):263-268.
[13]DAS J,EVERITT W N,HINTON D B,et al.The fourthorder Besseltype differential equation[J].Appl Anal,2004,83:325-362.
[14]EVERITTA W N,KALFB H.The Bessel differential equation and the Hankel transform[J].J Comput Appl Math,2007,208:3-19.
[15]PARAND K,NIKARYA M.Application of Bessel functions for solving differential and integro-differential equations of the fractional order[J].Appl Math Model,2014,38:4 137-4 147.
[16]YZBA S.Improved Bessel collocation method for linear Volterra integrodi erential equations with piecewise intervals and application of a Volterra population model[J].Appl Math Model,2016,40:5 349-5 363.
[17]CORTS J C,JDAR L,VILLAFUERTE L.Mean square solution of Bessel differential equation with uncertainties[J].J Comput Appl Math,2017,309:383-395.
[2]孔楠,朱建青.时间尺度上Lagrange系统Mei对称性及其直接导致的Mei守恒量[J].云南大学学报:自然科学版,2017,39(2):219-224.KONG N,ZHU J Q.On the Mei conserved quantity caused directly by Mei symmetry of the Lagrange system on time scales[J].Journal of Yunnan University:Natural Sciences Edition,2017,39(2):219-224.
[3]张晔,张毅,陈向炜.事件空间中Birkhoff系统的Mei对称性与守恒量[J].云南大学学报:自然科学版,2016,38(3):406-411.ZHANG Y,ZHANG Y,CHEN X W.Mei symmetry and conserved quantity of a Birkhoffian system in event space[J].Journal of Yunnan University:Natural Sciences Edition,2016,38(3):406-411.
[4]孙现亭,张美玲,张耀宇,等.变质量相对运动力学系统Nielsen方程的Noether守恒量[J].云南大学学报:自然科学版,2014,36(3):360-365.SUN X T,ZHANG M L,ZHANG Y Y,et al.Noether conserved quantity for Nielsen equations in a variable mass holonomic dynamical system of the relative motion[J].Journal of Yunnan University:Natural Sciences Edition,2014,36(3):360-365.
[5]王建波,陈梅,解加芳,等.约束系统广义Tzénoff方程的Mei对称性及其对应的守恒量[J].云南大学学报:自然科学版,2012,34(5):533-539.WANG J B,CHEN M,XIE J F,et al.On the Mei symmetry of the restraint system generalized Tzénoff equations and the corresponding conserved quantities[J].Journal of Yunnan University:Natural Sciences Edition,2012,34(5):533-539.
[6]张美玲,王肖肖,韩月林,等.变质量完整系统相对运动Nielsen方程的Mei对称性和Mei守恒量[J].云南大学学报:自然科学版,2012,34(6):664-668.ZHANG M L,WANG X X,HAN Y L,et al.On Mei symmetry and Mei conserved quantity of Nielsen equations for a variable mass holonomic system of relative motion.[J].Journal of Yunnan University:Natural Sciences Edition,2012,34(6):664-668.
[7]郑世旺,陈梅.非完整系统Tzénoff方程的Mei对称性所对应的一种新守恒量[J].云南大学学报:自然科学版,2011,33(4):412-416.ZHENG S W,CHEN M.A new conserved quantity by Mei symmetry of Tzénoff equation for the non-holonomic systems[J].Journal of Yunnan University:Natural Sciences Edition,2011,33(4):412-416.
[8]SANTILLI R M.Foundations of theoretical mechanics II[M].New York:Springer,1983.
[9]BROUCKE R.Equations of motion in phase space[J].Hadronic J,1979,2:1 122-1 158.
[10]LUMSDEN C J,TRAINOR L E H.On the statistical mechanics of constrained biophysical systems[J].J Statistical Phys,1979,20:657-669.
[11]MEI F X.On the Birkhoffian mechanics[J].Int J Nonlinear Mech,2001,36:817-834.
[12]梅凤翔,吴惠彬,李彦敏,等.Birkhoff力学的研究进展[J].力学学报,2016,48(2):263-268.MEI F X,WU H B,LI Y M,et al.Advances in research on Birkhoffian mechanics[J].Chinese Journal of Theoretical and Applied Mechanics,2016,48(2):263-268.
[13]DAS J,EVERITT W N,HINTON D B,et al.The fourthorder Besseltype differential equation[J].Appl Anal,2004,83:325-362.
[14]EVERITTA W N,KALFB H.The Bessel differential equation and the Hankel transform[J].J Comput Appl Math,2007,208:3-19.
[15]PARAND K,NIKARYA M.Application of Bessel functions for solving differential and integro-differential equations of the fractional order[J].Appl Math Model,2014,38:4 137-4 147.
[16]YZBA S.Improved Bessel collocation method for linear Volterra integrodi erential equations with piecewise intervals and application of a Volterra population model[J].Appl Math Model,2016,40:5 349-5 363.
[17]CORTS J C,JDAR L,VILLAFUERTE L.Mean square solution of Bessel differential equation with uncertainties[J].J Comput Appl Math,2017,309:383-395.