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动力学逆问题是星际航行学、火箭动力学、规划运动学理论的基本问题. Mei对称性是力学系统的动力学函数在群的无限小变换下仍然满足系统原来的运动微分方程的一种新的不变性. 本文研究广义坐标下一般完整系统的Mei对称性以及与Mei对称性相关的动力学逆问题. 首先, 给出系统动力学正问题的提法和解法. 引入时间和广义坐标的无限小单参数变换群, 得到无限小生成元向量及其一次扩展. 讨论由n个广义坐标确定的一般完整力学系统的运动微分方程, 将其Lagrange函数和非势广义力作无限小变换, 给出系统运动微分方程的Mei对称性定义, 在忽略无限小变换的高阶小量的情况下得到Mei对称性的确定方程, 借助规范函数满足的结构方程导出系统Mei对称性导致的Noether守恒量. 其次, 研究系统Mei对称性的逆问题. Mei对称性的逆问题的提法是: 由已知守恒量来求相应的Mei对称性. 采取的方法是将已知积分当作由Mei对称性导致的Noether守恒量, 由Noether逆定理得到无限小变换的生成元, 再由确定方程来判断所得生成元是否为Mei对称性的. 然后, 讨论生成元变化对各种对称性的影响. 结果表明, 生成元变化对Noether和Lie对称性没有影响, 对Mei 对称性有影响, 但在调整规范函数时, 若满足一定条件, 生成元变化对Mei对称性也可以没有影响. 最后, 举例说明结果的应用.Inverse problems in dynamics are the basic problems in astronautics, rocket dynamics, and motion planning theory, etc. Mei symmetry is a kind of new symmetry where the dynamical function in differential equations of motion still satisfies the equation's primary form under infinitesimal transformations of the group. Mei symmetry and its inverse problem of dynamics for a general holonomic system in generalized coordinates are studied. Firstly, the direct problem of dynamics of the system is proposed and solved. Introducing a one-parameter infinitesimal transformation group with respect to time and coordinates, the infinitesimal generator vector and its first prolonged vector are obtained. Based on the discussion of the differential equations of motion for a general holonomic system determined by n generalized coordinates, their Lagrangian and non-potential generalized forces are made to have an infinitesimal transformation, the definition of Mei symmetry about differential equation of motion for the system is then provided. Ignoring the high-order terms in the infinitesimal transformation, the determining equation of Mei symmetry is given. With the aid of a structure equation which the gauge function satisfies, the system's corresponding conserved quantities are derived. Secondly, the inverse problem for the Mei symmetry of the system is studied. The formulation of the inverse problem of Mei symmetry is that we use the known conserved quantity to seek the corresponding Mei symmetry. The method is: considering a given integral as a Noether conserved quantity obtained by Mei symmetry, the generators of the infinitesimal transformations can be obtained by the inverse Noether theorem. Then the question whether the obtained generators are Mei symmetrical or not is verified by the determining equation, and the effect of generators' changes on the symmetries is discussed. It has been shown from the studies that the changes of the generators have no effect on the Noether and Lie symmetries, but have effects on the Mei symmetry. However, under certain conditions, while adjusting the gauge function, changes of generators can also have no effect on the Mei symmetry. In the end of the paper, an example for the system is provided to illustrate the application of the result.
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Keywords:
- general holonomic system /
- Mei symmetry /
- conserved quantity /
- inverse problem
[1] Galiullin A S 1986 Methods of solution of inverse problems of dynamics (Moscow: Nauka) (in Russian)
[2] Mei F X 2009 Inverse Problems of Dynamics(Beijing: National Defense Industry Press) (in Chinese) [梅凤翔 2009 动力学逆问题 (北京: 国防工业出版社)]
[3] Mei F X 2000 J. Beijing Inst. Technol. 9 120
[4] Jia L Q, Xie J F, Luo S K 2008 Chin. Phys. B 17 1560
[5] Cai J L 2009 Acta Phys. Pol. A 115 854
[6] Wang P, Fang J H, Wang X M 2009 Chin. Phys. B 18 1312
[7] Cai J L 2010 Chin. J. Phys. 48 728
[8] Jiang W A, Luo S K 2011 Acta Phys. Sin. 60 060201 (in Chinese) [姜文安, 罗绍凯 2011 物理学报 60 060201]
[9] Mei F X 2004 Symmetries and Conserved Quantities of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔 2004 约束力学系统的对称性与守恒量 (北京: 北京理工大学出版社)]
[10] Luo S K, Zhang Y F 2008 Advances in the Study of Dynamics of Constrained Systems (Beijing: Science Press) (in Chinese)[罗绍凯, 张永发 2008 约束系统动力学研究进展 (北京: 科学出版社)]
[11] Jia L Q, Zheng S W, Zhang Y Y 2007 Acta Phys. Sin. 56 5575 (in Chinese) [贾利群, 郑世旺, 张耀宇 2007 物理学报 56 5575]
[12] Cai J L, Luo S K, Mei F X 2008 Chin. Phys. B 17 3170
[13] Luo Y P 2009 Int. J. Theor. Phys. 48 2665
[14] Cai J L 2010 Acta Phys. Pol. A 117 445
[15] Cui J C, Zhang Y Y, Yang X F, Jia L Q 2010 Chinese Physics B 19 030304
[16] Cai J L 2010 Int. J. Theor. Phy. 49 201
[17] Jiang W A, Luo S K 2012 Nonlinear Dyn. 67 475
[18] Cai J L 2012 Nonlinear Dyn. 69 487
[19] Xia L L, Chen L Q 2012 Nonlinear Dynamics 70 1223
[20] Jia L Q, Wang X X, Zhang M L, Han Y L 2012 Nonlinear Dyn. 69 1807
[21] Jiang W A, Li L, Li Z J, Luo S K 2012 Nonlinear Dyn. 67 1075
[22] Cai J L, Shi S S, Fang H J, Xu J 2012 Meccanica 47 63
[23] Cai J L, Mei F X 2012 J. Mech. 28 589
[24] Cai J L, Shi S S 2012 Acta Phys. Sin. 61 030201 (in Chinese) [蔡建乐, 史生水 2012 物理学报 61 030201]
[25] Jiang W A, Li L, Li Z J, Luo S K 2012 Nonlinear Dyn. 67 1075
[26] Jiang W A, Li Z J, Luo S K 2011 Chinese Physics B 20 030202
[27] Jia L Q, Wang X X, Zhang M L, Han Y L 2012 Nonlinear Dyn. 69 1807
[28] Liu F L, Mei F X 1993 Appl. Math. Mech. -Engl. Ed. 14 327
[29] Li G C, Mei F X 2006 Chin. Phys. B 15 1669
[30] Ibort L A, Solano J M 1991 Inverse Problems. 7 713
[31] Menini L, Tornambe A 2012 Nonlinear Dyn. 69 1965
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[1] Galiullin A S 1986 Methods of solution of inverse problems of dynamics (Moscow: Nauka) (in Russian)
[2] Mei F X 2009 Inverse Problems of Dynamics(Beijing: National Defense Industry Press) (in Chinese) [梅凤翔 2009 动力学逆问题 (北京: 国防工业出版社)]
[3] Mei F X 2000 J. Beijing Inst. Technol. 9 120
[4] Jia L Q, Xie J F, Luo S K 2008 Chin. Phys. B 17 1560
[5] Cai J L 2009 Acta Phys. Pol. A 115 854
[6] Wang P, Fang J H, Wang X M 2009 Chin. Phys. B 18 1312
[7] Cai J L 2010 Chin. J. Phys. 48 728
[8] Jiang W A, Luo S K 2011 Acta Phys. Sin. 60 060201 (in Chinese) [姜文安, 罗绍凯 2011 物理学报 60 060201]
[9] Mei F X 2004 Symmetries and Conserved Quantities of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔 2004 约束力学系统的对称性与守恒量 (北京: 北京理工大学出版社)]
[10] Luo S K, Zhang Y F 2008 Advances in the Study of Dynamics of Constrained Systems (Beijing: Science Press) (in Chinese)[罗绍凯, 张永发 2008 约束系统动力学研究进展 (北京: 科学出版社)]
[11] Jia L Q, Zheng S W, Zhang Y Y 2007 Acta Phys. Sin. 56 5575 (in Chinese) [贾利群, 郑世旺, 张耀宇 2007 物理学报 56 5575]
[12] Cai J L, Luo S K, Mei F X 2008 Chin. Phys. B 17 3170
[13] Luo Y P 2009 Int. J. Theor. Phys. 48 2665
[14] Cai J L 2010 Acta Phys. Pol. A 117 445
[15] Cui J C, Zhang Y Y, Yang X F, Jia L Q 2010 Chinese Physics B 19 030304
[16] Cai J L 2010 Int. J. Theor. Phy. 49 201
[17] Jiang W A, Luo S K 2012 Nonlinear Dyn. 67 475
[18] Cai J L 2012 Nonlinear Dyn. 69 487
[19] Xia L L, Chen L Q 2012 Nonlinear Dynamics 70 1223
[20] Jia L Q, Wang X X, Zhang M L, Han Y L 2012 Nonlinear Dyn. 69 1807
[21] Jiang W A, Li L, Li Z J, Luo S K 2012 Nonlinear Dyn. 67 1075
[22] Cai J L, Shi S S, Fang H J, Xu J 2012 Meccanica 47 63
[23] Cai J L, Mei F X 2012 J. Mech. 28 589
[24] Cai J L, Shi S S 2012 Acta Phys. Sin. 61 030201 (in Chinese) [蔡建乐, 史生水 2012 物理学报 61 030201]
[25] Jiang W A, Li L, Li Z J, Luo S K 2012 Nonlinear Dyn. 67 1075
[26] Jiang W A, Li Z J, Luo S K 2011 Chinese Physics B 20 030202
[27] Jia L Q, Wang X X, Zhang M L, Han Y L 2012 Nonlinear Dyn. 69 1807
[28] Liu F L, Mei F X 1993 Appl. Math. Mech. -Engl. Ed. 14 327
[29] Li G C, Mei F X 2006 Chin. Phys. B 15 1669
[30] Ibort L A, Solano J M 1991 Inverse Problems. 7 713
[31] Menini L, Tornambe A 2012 Nonlinear Dyn. 69 1965
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