Application of canonical coordinates for solving single-freedom constraint mechanical systems

doi:10.1007/s10483-014-1849-7

Abstract

Abstract: This paper introduces the canonical coordinates method to obtain the first integral of a single-degree freedom constraint mechanical system that contains conservative and non-conservative constraint homonomic systems. The definition and properties of canonical coordinates are introduced. The relation between Lie point symmetries and the canonical coordinates of the constraint mechanical system are expressed. By this relation, the canonical coordinates can be obtained. Properties of the canonical coordinates and the Lie symmetry theory are used to seek the first integrals of constraint mechanical system. Three examples are used to show applications of the results.

Key words: canonical coordinates, first integral, Lie symmetry, constraint mechanical system

2010 MSC Number:

Fang GAO;Xiao-bo ZHANG;Jing-li FU. Application of canonical coordinates for solving single-freedom constraint mechanical systems . Applied Mathematics and Mechanics (English Edition), 2014, 35(8): 1029-1038.

TrendMD

References

[1] Bluman, G. W. and Kumei, S. Symmetries and Differential Equations, Springer, Berlin (1989)
[2] Olver, P. J. Applications of Lie Groups to Differential Equations, Springer, Berlin (2000)
[3] Olver, P. J. Equivalence, Invariants and Symmetry, Cambridge University Press, Cambridge (1995)
[4] MacCallum, M. Differential Equations: Their Solution Using Symmetries, Cambridge University Press, Cambridge (1989)
[5] Ovsiannikov, L. V. Group Analysis of Differential Equations, Academic Press, New York (1982)
[6] Gorringe, V. M. and Leach, P. G. L. Lie point symmetries for systems of second order linear ordinary differential equations. Quaestiones Mathematicae, 11(1), 95-117 (1988)
[7] Noether, E. Invariante variations probleme. Transport Theory and Statistical Physics, 1(3), 186-207 (1971)
[8] Candotti, E., Palmieri, C., and Vitale, B. On the inversion of Noether's theorem in classical dynamical systems. American Journal of Physics, 40, 424-429 (1972)
[9] Djukic, D. D. S. and Vujanovic, B. D. Noether's theory in classical nonconservative mechanics. Acta Mechanica, 23(1-2), 17-27 (1975)
[10] Lutzky, M. Symmetry groups and conserved quantities for the harmonic oscillator. Journal of Physics A: Mathematical and General, 11(2), 249-258 (1978)
[11] Zhao, Y. Y. Lie symmetries and conserved quantities of non-conservative mechanical system (in Chinese). Chinese Journal of Theoretical and Applied Mechanics, 26(3), 380-384 (1994)
[12] Mei, F. X. Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems, Science Press, Beijing (1999)
[13] Mei, F. X. Lie symmetries and conserved quantities of constrained mechanical systems. Acta Mechanica, 141(3-4), 135-148 (2000)
[14] Zhang, H. B. Lie symmetries and conserved quantities of non-holonomic mechanical systems with unilateral Vacco constraints. Chinese Physics, 11(1), 1-4 (2002)
[15] Mei, F. X. Symmetry and Conserved Quantity of Constrained Mechanical System, Beijing Institute of Technology Press, Beijing (2004)
[16] He, G. and Mei, F. X. Three kinds of symmetry about nonholonomic-nonconservative systems. Journal of Beijing Institute of Technology, 27, 565-567 (2007)
[17] Fu, J. L., Chen, B. Y., and Chen, L. Q. Noether symmetries of discrete nonholonomic dynamical systems. Physics Letters A, 373(4), 409-412 (2009)
[18] Chen, X. W., Chang, L., and Mei, F. X. Conformal invariance and Hojman conserved quantities of first order Lagrange systems. Chinese Physics B, 17(9), 3180-3184 (2008)
[19] Yang, S. P., Chen, L. Q., and Li, S. H. Dynamics of Vehicle-Road Coupled System (in Chinese), Science Press, Beijing (2012)
[20] Lu, Y. J., Yang, S. P., Li, S. H., and Chen, L. Q. Numerical and experimental investigation on stochastic dynamic load of a heavy duty vehicle. Applied Mathematical Modelling, 34, 2698-2710 (2010)
[21] Khalique, M. C. and Anjan, B. A Lie symmetry approach to nonlinear Schrödinger's equation with non-Kerr law nonlinearity. Communications in Nonlinear Science and Numerical Simulation, 14(12), 4033-4040 (2009)
[22] Fu, J. L. and Chen, B. Y. Hojman conserved quantities and Lie symmetries of discrete nonconservative systems. Modern Physics Letters B, 23(10), 1315-1322 (2009)
[23] Fu, J. L., Fu, H., and Liu, R. W. Hojman conserved quantities of discrete mechanico-electrical systems constructed by continuous symmetries. Physics Letters A, 374(17), 1812-1818 (2010)
[24] Fang, J, H. A new type of conserved quantity of Lie symmetry for the Lagrange system. Chinese Physics B, 19(4), 21-24 (2010)
[25] Fu, J. L., Li, X. W., Li, C. R., Zhao, W. J., and Chen, B. Y. Symmetries and exact solutions of discrete nonconservative systems. Science China Physics, Mechanics & Astronomy, 53(9), 1668-1706 (2010)
[26] Fu, J. L., Chen, L. Q., and Chen, B. Y. Noether-type theorem for discrete nonconservative dynamical systems with nonregular lattices. Science China Physics, Mechanics & Astronomy, 53(3), 544-554 (2010)
[27] Yang, S. P., Chen, L. Q., and Li, S. H. Dynamics of Vehicle-Road Coupled System, Science Press, Beijing (2012)
[28] Zhou, S., Fu, H., and Fu, J. L. Symmetry theories of Hamiltonian systems with fractional derivatives. Science China Physics, Mechanics & Astronomy, 54(10), 1846-1852 (2011)
[29] Fu, J. L., Chen, B. Y., Fu, H., Zhao, G. L., Liu, R. W., and Zhu, Z. Y. Velocity-dependent symmetries and non-Noether conserved quantities of electromechanical systems. Science China Physics, Mechanics & Astronomy, 54(2), 288-295 (2011)
[30] Fu, J. L., Zhao,W. J., and Chen, B. Y. Energy-work connection integration schemes for mechanicoelectrical systems. Nonlinear Dynamics, 70(1), 755-765 (2012)
[31] Mustafa, M. T. and Masood, K. Symmetry solutions of a nonlinear elastic wave equation with third-order an harmonic corrections. Applied Mathematics and Mechanics (English Edition), 30(8), 1017-1026 (2009) DOI 10.1007/s10483-009-0808-z
[32] Rosmila, A. B., Kandasamy, R., and Muhaimin, I. Lie symmetry group transformation for MHD natural convection flow of nanofluid over linearly porous stretching sheet in presence of thermal stratification. Applied Mathematics and Mechanics (English Edition), 33(5), 593-604 (2012) DOI 10.1007/s10483-012-1573-9
[33] Fujii, T., Kudo, Y., Ohmura, Y., Suzuki, K., Kogo, J., Mizuno, Y., Kita, Z., and Sawada, M. Polarization properties of state-of-art lithography optics represented by first canonical coordinate of Lie group. Proceedings of the SPIE, 6520, 1-6 (2007)
[34] Tucker, J. D., Azimi-Sadjadi, M. R., and Dobeck, G. J. Canonical coordinates for detection and classification of underwater objects from sonar imagery. OCEANS 2007-Europe, Aberdeen, 1-6 (2007)
[35] Arnal, D., Currey, B., and Dali, B. Construction of canonical coordinates for exponential Lie groups. Transactions of the American Mathematical Society, 361, 6283-6348 (2009)
[36] Pakdemirli, M. and Aksoy, Y. Group classification for path equation describing minimum drag work and symmetry reductions. Applied Mathematics and Mechanics (English Edition), 31(7), 911-916 (2010) DOI 10.1007/s10483-010-1325-x
[37] Dillen, F., Nistor, A. I., and Munteanu, M. I. Canonical coordinates and principal directions for surfaces in H2 ? R. Taiwanese Journal of Mathematics, 15(5), 2265-2289 (2011)