Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T10:20:57.141Z Has data issue: false hasContentIssue false
  • English
  • Français

Lie Symmetry and Approximate Hojman Conserved Quantity of Lagrange Equations for a Weakly Nonholonomic System

Published online by Cambridge University Press:  08 August 2013

Y.-L. Han ,
X.-X. Wang ,
M.-L. Zhang  and
L.-Q. Jia
Y.-L. Han
Affiliation:
School of Science, Jiangnan University, Wuxi, 214122, P. R., China
X.-X. Wang
Affiliation:
School of Science, Jiangnan University, Wuxi, 214122, P. R., China
M.-L. Zhang
Affiliation:
School of Science, Jiangnan University, Wuxi, 214122, P. R., China
L.-Q. Jia*
Affiliation:
School of Science, Jiangnan University, Wuxi, 214122, P. R., China
*
*jlq0000@163.com
Get access

Abstract

The Lie symmetry and Hojman conserved quantity of Lagrange equations for a weakly nonholonomic system and its first-degree approximate holonomic system are studied. The differential equations of motion for the system are established. Under the special infinitesimal transformations of group in which the time is invariable, the definition of the Lie symmetry for the weakly nonholonomic system and its first-degree approximate holonomic system are given, and the exact and approximate Hojman conserved quantities deduced directly from the Lie symmetry are obtained. Finally, an example is given to study the exact and approximate Hojman conserved quantity for the system.

Keywords

Weakly nonholonomic systemLagrange equationsLie symmetryApproximate Hojman conserved quantity
Type
Research Article
Information
Journal of Mechanics , Volume 30 , Issue 1 , February 2014 , pp. 21 - 27
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2014 

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

REFERENCES

1.Neimark, J. I. and Fufaev, N. A., “Dynamics of Nonholonomic Systems,” (Providence, RI: AMS) (1972).Google Scholar
2.Mei, F. X., Foundations of Mechanics of Nonholonomic Systems, Beijing Institute of Technology Press, Beijing (1985).Google Scholar
3.Mei, F. X., Liu, D. and Luo, Y., Advanced Analytical Mechanics, Beijing Institute of Technology Press, Beijing (1991).Google Scholar
4.Ostrovskaya, S. and Angels, J., “Nonholonomic Systems Revisited Within the Frame Work of Analytical Mechanics,” Applied Mechanics Review, ASME, 51, pp. 415–433 (1998).Google Scholar
5.Mei, F. X., “Nonholonomic Mechanics,” Applied Mechanics Reviews, ASME, 53, pp. 283–305 (2000).Google Scholar
6.Luo, S. K., Chen, X. W. and Guo, Y. X., “Theory of Symmetry for a Rotational Relativistic Birkhoff System,” Chinese Physics B, 11, pp. 429–436 (2002).Google Scholar
7.Guo, Y. X., Luo, S. K. and Mei, F. X., “Progress of Geometric Dynamics of Non-Holonomic Constrained Mechanical Systems: Lagrange theory and others,” Advances in Mechanics, 34, pp. 477–492 (2004).Google Scholar
8.Zegzhda, S. A., Soltakhanov, Sh. Kh and Yushkov, M. P., “Equations of Motion of Nonholonomic Systems and Variational Principles of Mechanics,” New kind of Control Problems. (Moscow: FIMAT- LIT) (in Russian) (2005).Google Scholar
9.Zhang, Y. and Mei, F. X., “Effects of Constraints on Noether Symmetries and Conserved Quantities in a Birkhoffian System,” Acta Physica Sinica, 53, pp. 2419–2423 (2004).Google Scholar
10.Fu, J. L., Nie, N. M. and Huang, J. F., “Noether Conserved Quantities and Lie Point Symmetries of Difference Lagrange-Maxwell Equations and Lattices,” Chinese Physics B, 18, pp. 2634–2641 (2009).Google Scholar
11.Xie, Y. L., Yang, X. F. and Jia, L. Q., “Noether Symmetry and Noether Conserved Quantity of Nielsen Equation for Dynamical Systems of the Relative Motion,” Communications in Theoretical Physics, 55, pp. 111–114 (2011).CrossRefGoogle Scholar
12.Wang, X. X., Sun, X. T., Zhang, M. L., Xie, Y. L. and Jia, L. Q., “Noether Symmetry and Noether Conserved Quantity of Nielsen Equation in a Dynamical System of the Relative Motion with Non-holonomic Constraint of Chetaev's Type,” Acta Physica Sinica, 61, p. 064501 (2012).Google Scholar
13.Luo, S. K., “A New Type of Lie Symmetrical Non-Noether Conserved Quantity for Nonholonomic Systems,” Chinese Physics, 13, pp. 2182–2186 (2004).Google Scholar
14.Mei, F. X., Symmetries and Conserved Quantities of Constrained Mechanical Systems, Beijing Institute of Technology Press, Beijing (2004).Google Scholar
15.Luo, S. K., Guo, Y. X. and Mei, F. X., “Form Invariance and Hojman Conserved Quantity for Nonholonomic Mechanical Systems,” Acta Physica Sinica, 53, pp. 2413–2418 (2004).Google Scholar
16.Xu, X. J., Mei, F. X. and Zhang, Y. F., “Lie Symmetry and Conserved Quantity of a System of First-Order Differential Equations,” Chinese Physics, 15, pp. 19–21 (2006).Google Scholar
17.Luo, S. K., “A New Type of Non-Noether Adiabatic Invariants for Disturbed Lagrangian Systems: Adiabatic Invariants of Generalized Lutzky Type,” Chinese Physics Letters, 24, pp. 2463–2466 (2007).Google Scholar
18.Cai, J. L. and Mei, F. X., “Conformal Invariance and Conserved Quantity of Lagrange Systems Under Lie Point Transformation,” Acta Physica Sinica, 57, pp. 5369–5373 (2008).Google Scholar
19.Chen, X. W., Liu, C. and Mei, F. X., “Conformal Invariance and Hojman Conservedquantities of First Order Lagrange Systems,” Chinese Physics B, 17, pp. 3180–3184 (2008).Google Scholar
20.Cai, J. L., Luo, S. K. and Mei, F. X., “Conformal Invariance and Conserved Quantity of Hamilton Systems,” Chinese Physics B, 17, pp. 3170–3174 (2008).Google Scholar
21.Fang, J. H., “A New Type of Conserved Quantity of Lie Symmetry for the Lagrange System,” Chinese Physics B, 19, p. 040301 (2010).Google Scholar
22.Ge, W. K., “Mei Symmetry and Conserved Quantity of a Holonomic System,” Acta Physica Sinica, 57, pp. 6714–6717 (2008).Google Scholar
23.Wu, H. B. and Mei, F. X., “Symmetry of Lagrangians of Holonomic Systems in Terms of Quasi-Coordinates,” Chinese Physics B, 18, pp. 3145–3149 (2009).Google Scholar
24.Mei, F. X. and Wu, H. B., “Form Invariance and New Conserved Quantity of Generalised Birkhoffian System,” Chinese Physics B, 19, pp. 050301 (2010).Google Scholar
25.Zheng, S. W., Xie, J. F., Chen, X. W. and Du, X. L., “Another Kind of Conserved Quantity Induced Directly from Mei Symmetry of Tzénoff Equations for Holonomic Systems,” Acta Physica Sinica, 59, pp. 5209–5212 (2010).Google Scholar
26.Cui, J. C., Jia, L. Q. and Zhang, Y. Y., “Mei Symmetry and Mei Conserved Quantity for a Non- Holonomic System of Non-Chetaev's Type with Unilateral Constraints in Nielsen Style,” Communications in Theoretical Physics, 52, pp. 7–11 (2009).Google Scholar
27.Cai, J. L., “Conformal Invariance and Conserved Quantities of Mei Symmetry for Lagrange Systems,” Acta Physica Polonica A, 115, pp. 854–856 (2009).CrossRefGoogle Scholar
28.Yang, X. F., Jia, L. Q., Cui, J. C. and Luo, S. K., “Mei Symmetry and Mei Conserved Quantity of Nielsen Equations for a Nonholonomic System of Chetaev's Type with Variable Mass,” Chinese Physics B, 19, p. 030305 (2010).Google Scholar
29.Cai, J. L., “Conformal Invariance of Mei Symmetry for the Nonholonomic System of Non-Chetaev's Type,” Nonlinear Dynamics, 69, pp. 487–493. (2012).Google Scholar
30.Li, Y. C., Wang, X. M. and Xia, L. L., “Unified Symmetry and Conserved Quantities of Nielsen Equation for a Holonomic Mechanical System,” Acta Physica Sinica, 59, pp. 2935–2938 (2010).Google Scholar
31.Jiang, W. A., Li, Z. J. and Luo, S. K., “Mei Symmetries and Mei Conserved Quantities for HigherOrder Nonholonomic Systems,” Chinese Physics B, 20, p. 03020239 (2011).Google Scholar
32.Cai, J. L., “Conformal Invariance and Conserved Quantities of General Holonomic Systems,” Chinese Physics Letters, 25, pp. 1523–1526 (2008).Google Scholar
33.Mei, F. X., “Equations of Motion for Weak Non-holonomic Systems and Their Approximate Solution,” Transactions of Beijing Institute of Technology, 9, pp. 10–17 (1989).Google Scholar
34.Mei, F. X., “Canonical Transformation of Weak Nonholonomic Systems,” Chinese Science Bulletin, 37, pp. 1180–1183 (1992).Google Scholar
35.Mei, F. X., “On the Stability of One Type of Weakly Nonholonomic Systems,” Transactions of Beijing Institute of Technology, 15, pp. 237–242 (1995).Google Scholar
36.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).CrossRefGoogle Scholar