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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 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 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 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 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 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 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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