Abstract
The conformal invariance and conserved quantity of Mei symmetry for a higher-order nonholonomic mechanical system are presented. Introducing an infinitesimal transformation group and infinitesimal generator vector, the definition of conformal invariance of Mei symmetry and the determining equation for the holonomic system which corresponds to a higher-order nonholonomic system are provided, and the relationship between Mei symmetry and conformal invariance of the system is discussed. The basis of restriction equations and additional restriction equations, the conformal invariance of weak and strong Mei symmetry for the higher-order nonholonomic mechanical system is constructed. With the aid of a structure equation that the gauge function satisfies, the system’s corresponding conserved quantity is derived. Finally, an example is given to illustrate the application of the method and its result.
Similar content being viewed by others
References
Hertz H.R.: Die Prinzipien der Mechanik. Gesammelte Werke, Leipzig (1894)
Li Z.P.: The transformation properties of constrained system. Acta Phys. Sin. 30, 1659–1671 (1981)
Li Z.P.: Classical and Quantal Dynamics of Constrained Systems and Their Symmetrical Properties. Beijing Polytechnic University Press, Beijing (1993)
Pang T., Fang J.H., Zhang M.J., Lin P., Lu K.: A new type of conserved quantity deduced from Mei symmetry of nonholonomic systems in terms of quasi-coordinates. Chin. Phys. B 18, 3150–3154 (2009)
Li Z.P., Jiang J.H.: Symmetries in Constrained Canonical Systems. Science Press, Beijing (2002)
Luo S.K., Guo Y.X., Mei F.X.: Form invariance and Hojman conserved quantity for nonholonomic mechanical systems. Acta Phys. Sin. 53, 2413–2418 (2004)
Li A.M., Zhang Y., Li Z.P.: Poincaré-Cartan integral invariant of a nonholonomic constrained generalized mechanical system. Acta Phys. Sin. 53, 2816–2820 (2004)
Luo S.K.: Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type for disturbed nonholonomic systems. Chin. Phys. Lett. 24, 3017–3020 (2007)
Ge W.K.: Approximate conserved quantity of a weakly nonholonomic system. Acta Phys. Sin. 58, 6729–6731 (2009)
Li, Z.J., Jiang, W.A., Luo, S.K.: Lie symmetries, symmetrical perturbation and a new adiabatic invariant for disturbed nonholonomic systems. Nonlinear Dyn. (2011). doi:10.1007/s11071-011-9993-6
Noether, A.E.: Invariante Variationsprobleme. Nachr. Akad. Wiss. Göttingen, Math-Phys. Kl. II, 235–257 (1918)
Lutzky M.: Dynamical symmetries and conserved quantities. J. Phys. A Math. Gen. 12, 973–981 (1979)
Mei F.X.: Form invariance of Lagrange system. J. Beijing Inst. Technol. 9, 120–124 (2000)
Fan J.H.: Mei symmetry and Lie symmetry of the rotational relativistic variable mass system. Commun. Theor. Phys. 40, 269–272 (2003)
Zhang Y., Ge W.K.: A new conservation law from Mei symmetry for the relativistic mechanical system. Acta Phys. Sin. 54, 1464–1467 (2005)
Mei F.X.: Symmetries and Conserved Quantities of Constrained Mechanical Systems. Beijing Institute of Technology Press, Beijing (2004)
Zhang Y.: Symmetries and Mei conserved quantities for systems of generalized classical mechanics. Acta Phys. Sin. 54, 2980–2984 (2005)
Luo S.K., Zhang Y.F. et al.: Advances in the Study of Dynamics of Constrained Systems. Science Press, Beijing (2008)
Jia L.Q., Xie J.F., Luo S.K.: Mei symmetry and Mei conserved quantity of nonholonomic systems with unilateral Chetaev type in Nielsen style. Chin. Phys. B 17, 1560–1564 (2008)
Jia L.Q., Zhang Y.Y., Luo S.K., Cui J.C.: Mei symmetry and Mei conserved quantity of Nielsen equations for nonholonomic systems of unilateral non-Chetaev’s type in the event space. Acta Phys. Sin. 58, 2141–2146 (2009)
Jia L.Q., Xie Y.L., Luo S.K.: Mei conserved quantity deduced from Mei symmetry of Appell equation in a dynamical system of relative motion. Acta Phys. Sin. 60, 040201 (2011)
Jiang W.A., Li Z.J., Luo S.K.: Mei symmetries and Mei conserved quantities for higher-order nonholonomic constraint systems. Chin. Phys. B 20, 030202 (2011)
Jiang W.A., Luo S.K.: Mei symmetry leading to Mei conserved quantity of generalized Hamiltonian system. Acta Phys. Sin. 60, 060201 (2011)
Fu J.L., Chen L.Q., Chen B.Y.: Noether-type theorem for discrete nonconservative dynamical systems with nonregular lattices. Sci. China Phys. Mech. Astron. 53, 545–554 (2010)
Fu J.L., Chen L.Q., Chen B.Y.: Noether-type theory for discrete mechanico-electrical dynamical systems with nonregular lattices. Sci. China Phys. Mech. Astron. 53, 1687–1698 (2010)
Jiang, W.A., Luo, S.K.: A new type of non-Noether exact invariants and adiabatic invariants of generalized Hamiltonian systems. Nonlinear Dyn. (2011). doi:10.1007/s11071-011-9996-3
Jiang, W.A., Li, L., Li, Z.J., Luo, S.K.: Lie symmetrical perturbation and a new type of non-Noether adiabatic invariants for disturbed generalized Birkhoffian systems. Nonlinear Dyn. (2011). doi:10.1007/s11071-011-0051-1
Galiullin A.S., Gafarov G.G., Malaishka R.P., Khwan A.M.: Analytical Dynamics of Helmholtz, Birkhoff and Nambu Systems. UFN, Moscow (1997)
Cai J.L., Mei F.X.: Conformal invariance and conserved quantity of Lagrange systems under Lie point transformation. Acta Phys. Sin. 57, 5369–5373 (2008)
Cai J.L., Luo S.K., Mei F.X.: Conformal invariance and conserved quantity of Hamilton systems. Chin. Phys. B 17, 3170–3174 (2008)
Cai J.L.: Conformal invariance and conserved quantities of general holonomic systems. Chin. Phys. Lett. 25, 1523–1526 (2008)
Cai J.L.: Conformal invariance and conserved quantity for the nonholonomic system of Chetaev’s type. Int. J. Theor. Phys. 49, 201–211 (2010)
He G., Mei F.X.: Conformal invariance and integration of first-order differential equations. Chin. Phys. B 17, 2764–2765 (2008)
Mei F.X., Xie J.F., Gang T.Q.: A conformal invariance for generalized Birkhoff equations. Acta Mech. Sin. 24, 583–585 (2008)
Xia L.L., Cai J.L., Li Y.C.: Conformal invariance and conserved quantities of general holonomic systems in phase space. Chin. Phys. B 18, 3158–3162 (2009)
Luo Y.P.: Generalized conformal symmetries and its application of Hamilton systems. Int. J. Theor. Phys. 48, 2665–2671 (2009)
Li Y.C., Xia L.L., Wang X.M.: Conformal invariance and generalized Hojman conserved quantities of mechanico-electrical systems. Chin. Phys. B 18, 4643–4649 (2009)
Chen X.W., Li Y.M., Zhao Y.H.: Conformal invariance and conserved quantities of dynamical system of relative motion. Chin. Phys. B 18, 3139–3144 (2009)
Zhang M.J., Fang J.H., Lin P., Lu K., Pang T.: Conformal invariance and a new type of conserved quantities of mechanical systems with variable mass in phase space. Commun. Theor. Phys. 52, 561–564 (2009)
Zhang Y.: Conformal invariance and Noether symmetry, Lie symmetry of holonomic mechanical systems in event space. Chin. Phys. B 18, 4636–4642 (2009)
Zhang Y.: Conformal invariance and Noether symmetry, Lie symmetry of Birkhoffian systems in event space. Commun. Theor. Phys. 53, 166–170 (2010)
Luo Y.P., Fu J.L.: Conformal invariance and Hojman conserved quantities for holonomic systems with quasi-coordinates. Chin. Phys. B 19, 090303 (2010)
Cai J.L.: Conformal invariance and conserved quantities of Mei symmetry for Lagrange systems. Acta Phys. Pol. A 115, 854–856 (2009)
Cai J.L.: Conformal invariance and conserved quantities of Mei symmetry for general holonomic systems. Acta Phys. Sin. 58, 22–27 (2009)
Luo Y.P., Fu J.L.: Conformal invariance and conserved quantities of Appell systems under second-class Mei symmetry. Chin. Phys. B 19, 090304 (2010)
Cai J.L.: Conformal invariance and conserved quantity of Hamilton system under second-class Mei symmetry. Acta Phys. Pol. A 117, 445–448 (2010)
Luo Y.P., Fu J.L.: Conformal invariance and conserved quantities of Birkhoff systems under second-class Mei symmetry. Chin. Phys. B 20, 021102 (2011)
Mei F.X.: Foundations of Mechanics of Nonholonomic Systems. Beijing Institute of Technology Press, Beijing (1985)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Huang, WL., Cai, JL. Conformal invariance and conserved quantity of Mei symmetry for higher-order nonholonomic system. Acta Mech 223, 433–440 (2012). https://doi.org/10.1007/s00707-011-0573-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-011-0573-0