Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-19T02:52:51.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Inverse Problems of Mei Symmetry for Nonholonomic Systems with Variable Mass

Published online by Cambridge University Press:  14 April 2015

W.-L. Huang  and
J.-L. Cai
W.-L. Huang
Affiliation:
Department of Physics and Telecom Engineering Hunan City University Yiyang, China
J.-L. Cai*
Affiliation:
College of Science Hangzhou Normal University Hangzhou, China
*
* Corresponding author ([email protected])
Get access

Abstract

The inverse problem of the Mei symmetry for nonholonomic systems with variable mass is studied. Firstly, the authors discuss the Mei symmetry of the holonomic system opposite to a nonholonomic system. Secondly, weak and strong Mei symmetries of a nonholonomic system are concluded through restriction equations and additional restriction equations. Thirdly, the relevant conserved quantity is deduced by means of the structure equation for the gauge function. Fourthly, the inverse problem of the Mei symmetry is obtained by the Noether symmetry. Finally, the paper offers an example to illustrate the application of the research result.

Keywords

Non-holonomic systemVariable massMei symmetryInverse problem
Type
Research Article
Information
Journal of Mechanics , Volume 31 , Issue 5 , October 2015 , pp. 515 - 523
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Mei, F. X., Inverse Problems of Dynamics, National Defense Industry Press, Beijing (2009).Google Scholar
2.Hertz, H. R., “Die Prinzipien Der Mechanik,” Gesammelte Werke, Leibzing, p. 37 (1894).Google Scholar
3.Li, Z. P., Classical and Quantal Dynamics of Constrained Systems and Their Symmetrical Properties, Beijing Polytechnic University Press, Beijing (1993).Google Scholar
4.Luo, S. K., “New Types of the Lie Symmetries and Conserved Quantities for a Relativistic Hamilton System,” Chinese Physics Letters, 20, pp. 597–599 (2003).Google Scholar
5.Luo, S. K., “A Set of Lie Symmetrical Conservation Law for Rotational Relativistic Hamiltonian Systems,” Communications in Theoretical Physics, 40, pp. 265–268 (2003).Google Scholar
6.Cai, J. L., “Conformal Invariance and Conserved Quantity for the Nonholonomic System of Chetaev’s type,” International Journal of Theoretical Physics, 49, pp. 201–211 (2010).CrossRefGoogle Scholar
7.Jiang, W. A., Li, Z. J. and Luo, S. K., “Mei Symmetries and Mei Conserved Quantities for Higher-Order Nonholonomic Constraint Systems,” Chinese Physics B, 20, p. 030202 (2011).CrossRefGoogle Scholar
8.Xia, L. L., Li, Y. C., Wang, J. and Hou, Q. B., “Symmetries and Mei Conserved Quantities of Nonholonomic Controllable Mechanical Systems,” Communications in Theoretical Physics, 46, pp. 415–418 (2006).Google Scholar
9.Noether, A. E., “Invariante Variationsprobleme,” Nachr. d. König. Gesellsch d. Wiss. zu Göttingen, Math-phys. Klasse, pp. 235–257(1918).Google Scholar
10.Lutzky, M., “Dynamical Symmetries and Conserved Quantities,” Journal of Physics A: Mathematical and General, 12, pp. 973–981 (1979).CrossRefGoogle Scholar
11.Hojman, S. A., “A New Conservation Law Constructed Without Using Either Lagrangians or Hamiltonians,” Journal of Physics A: Mathematical and General, 25, pp. L291–L295 (1992).CrossRefGoogle Scholar
12.Mei, F. X., “Form Invariance of Lagrange System,” Journal of Beijing Institute of Technology, 9, pp. 120–124 (2000).Google Scholar
13.Fan, J. H., “Mei Symmetry and Lie Symmetry of the Rotational Relativistic Variable Mass System,” Communications in Theoretical Physics, 40, pp. 269–272 (2003).Google Scholar
14.Jia, L. Q., Xie, J. F. and Luo, S. K., “Mei Symmetry and Mei Conserved Quantity of Nonholonomic Systems with Unilateral Chetaev Type in Nielsen Style,” Chinese Physics B, 17, pp. 1560–1564 (2008).Google Scholar
15.Xia, L. L. and Chen, L. Q., “Mei Symmetries and Conserved Quantities for Non-Conservative Hamiltonian Difference Systems with Irregular Lattices,” Nonlinear Dynamics, 70, pp. 1223–1230 (2012).Google Scholar
16.Jiang, W. A. and Luo, S. K., “Mei Symmetry Leading to Mei Conserved Quantity of Generalized Hamiltonian System,” Acta Physica Sinica, 60, p. 060201 (2011).CrossRefGoogle Scholar
17.Mei, F. X., Symmetries and Conserved Quantities of Constrained Mechanical Systems, Beijing Institute of Technology Press, Beijing (2004).Google Scholar
18.Luo, S. K. and Zhang, Y. F., Advances in the Study of Dynamics of Constrained Systems, Science Press, Beijing (2008).Google Scholar
19.Wu, H. B. and Mei, F. X., “Symmetry of Lagrangians of Holonomic Variable Mass System,” Chinese Physics B, 21, p. 064501 (2012).Google Scholar
20.Jiang, W. A., Li, L., Li, Z. J. and Luo, S. K., “Lie Symmetrical Perturbation and a New Type of Non-Noether Adiabatic Invariants for Disturbed Generalized Birkhoffian Systems,” Nonlinear Dynamics, 67, pp. 1075–1081 (2012).CrossRefGoogle Scholar
21.Han, Y. L., Wang, X. X., Zhang, M. L. and Jia, L. Q., “Special Lie Symmetry and Hojman Conserved Quantity of Appell Equations for a Chetaev Nonholonomic System,” Nonlinear Dynamics, 73, pp. 357–361 (2013).Google Scholar
22.Jia, L. Q., Wang, X. X., Zhang, M. L. and Han, Y. L., “Special Mei Symmetry and Approximate Conserved Quantity of Appell Equations for a Weakly Nonholonomic System,” Nonlinear Dynamics, 69, pp. 1807–1812 (2012).Google Scholar
23.Luo, Y. P., “Generalized Conformal Symmetries and its Application of Hamilton Systems,” International Journal of Theoretical Physics, 48, pp. 2665–2671 (2009).Google Scholar
24.Jia, L. Q., Zheng, S. W and Zhang, Y. Y., “Mei Symmetry and Mei Conserved Quantity of Nonholonomic Systems of Non-Chetaev’s Type in Event Space,” Acta Physica Sinica, 56, pp. 5575–5579 (2007).Google Scholar
25.Jiang, W. A. and Luo, S. K., “A New Type of Non-Noether Exact Invariants and Adiabatic Invariants of Generalized Hamiltonian Systems,” Nonlinear Dynamics, 67, pp. 475–482 (2012).Google Scholar
26.Jiang, W A., Li, L., Li, Z. J. and Luo, S. K., “Lie Symmetrical Perturbation and a New Type of Non-Noether Adiabatic Invariants for Disturbed Generalized Birkhoffian Systems,” Nonlinear Dynamics, 67, pp. 1075–1081 (2012).CrossRefGoogle Scholar
27.Ibort, L. A. and Solano, J. M., “On the Inverse Problem of the Calculus of Variations for a Class of Coupled Dynamical Systems,” Inverse Problems, 7, pp. 713–725 (1991).Google Scholar
28.Liu, F. L. and Mei, F. X., “Formulation and Solution for Inverse Problem of Nonholonomic Dynamics,” Applied Mathematics and Mechanics (English Edition), 14, pp. 327–332 (1993).Google Scholar
29.Li, G. C. and Mei, F. X., “An Inverse Problem in Analytical Dynamics,” Chinese Physics B, 15, pp. 1669–1671 (2006).Google Scholar
30.Mei, F. X., Me, J. F. and Gang, T. Q., “An Inverse Problem of Dynamics of a Generalized Birkhoff System,” Acta Physica Sinica, 57, pp. 4649–4651 (2008).Google Scholar
31.Ding, G T., “New Kind of Inverse Problems of Noether’s Theory for Hamiltonian Systems,” Acta Physica Sinica, 59, pp. 1423–1427 (2010).Google Scholar
32.Menini, L. and Tornambe, A., “A Lie Symmetry Approach for the Solution of the Inverse Kinematics Problem,” Nonlinear Dynamics, 69, pp. 1965–1977 (2012).CrossRefGoogle Scholar
33.Fang, J. H., “Study of the Lie Symmetries of a Relativistic Variable Mass System,” Chinese Physics, 11, pp. 313–318 (2002).Google Scholar
34.Xia, L. L. and Li, Y. C., “Non-Noether Conserved Quantity for Relativistic Nonholonomic Controllable Mechanical System with Variable Mass,” Acta Physica Sinica, 57, pp. 4652–4656 (2008).Google Scholar
35.Cai, J. L., “Conformal Invariance of Mei Symmetry for the Holonomic System with Variable Mass,” Chinese Journal of Physics, 48, pp. 728–735 (2010).Google Scholar
36.Cui, J. C., Zhang, Y. Y., Yang, X. F. and Jia, L. Q., “Mei Symmetry and Mei Conserved Quantity of Appell Equations for Avariable Mass Holonomic System,” Chinese Physics B, 19, p. 030304 (2010).Google Scholar
37.Cai, J. L., “Conformal Invariance of Mei Symmetry for the Non-Holonomic Systems of Non-Chetaev’s type,” Nonlinear Dynamics, 69, pp. 487–493 (2012).CrossRefGoogle Scholar
38.Wang, S. Y. and Mei, F. X., “Form Invariance and Lie Symmetry of Equations of Non-Holonomic Systems,” Chinese Physics B, 11, pp. 5–8 (2002).Google Scholar
39.Zhang, H. B. and Chen, L. Q., “The Unified Form of Hojman’s Conservation Law and Lutzky’s Conservation Law,” Journal of the Physical Society of Japan, 74, p. 905 (2005).Google Scholar
40.Santilli, R. M., Foundations of Theoretical Mechanics I, Springer, NewYork, p. 72 (1978).Google Scholar