Advertisement

Acta Mechanica

, Volume 141, Issue 3–4, pp 135–148 | Cite as

Lie symmetries and conserved quantities of constrained mechanical systems

  • F. X. Mei
Original Papers

Summary

The Lie symmetries and conserved quantities of constrained mechanical systems are studied. Using the invariance of the ordinary differential equations under the infinitesimal transformations, the determining equations and the restriction equations of the Lie symmetries of the systems are established. The structure equation and the form of conserved quantities are obtained. We find the corresponding conserved quantity from a known Lie symmetry, that is a direct problem of the Lie symmetries. And then, the inverse problem of the Lie symmetries-finding the corresponding Lie symmetry from a known conserved quantity-is studied. Finally, the relation between the Lie symmetry and the Noether symmetry is given.

Keywords

Mechanical System Direct Problem Nonholonomic System Restriction Equation Determine Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Djukić, Dj. S., Vujanović, B. D.: Noether's theory in classical nonconservative mechanics. Acta Mech.23, 17–27 (1975).MathSciNetCrossRefGoogle Scholar
  2. [2]
    Li, Z. P.: The transformation properties of constrained system. Acta Phys. Sinica (in Chinese)30, 1659–1671 (1981).MathSciNetGoogle Scholar
  3. [3]
    Bahar, L. Y., Kwatny, H. G.: Extension of Noether's theorem to constrained nonconservative dynamical systems. Int. J. Non-Linear Mech.22, 125–138 (1987).zbMATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    Liu, D.: Noether's theorem and its inverse of non-holonomic nonconservative dynamical systems. Science in China34, 419–429 (1991).zbMATHGoogle Scholar
  5. [5]
    Lutzky, M.: Dynamical symmetries and conserved quantities. J. Phys. A Math. Gen.12, 973–981 (1979).zbMATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    Zhao, Y. Y.: Conservative quantities and Lie symmetries of nonconservative dynamical systems. Acta Mech. Sinica (in Chinese)26, 380–384 (1994).Google Scholar
  7. [7]
    Bluman, G. W., Kumei, S.: Symmetries and differential equations. New York: Springer 1989.Google Scholar
  8. [8]
    Hamel, G.: Theoretische Mechanik. Berlin: Springer 1949.Google Scholar
  9. [9]
    Mei, F. X.: Foundations of mechanics of non-holonomic systems. Beijing: Beijing Institute of Technology Press 1985 (in Chinese).Google Scholar
  10. [10]
    Noveselov, V. S.: Variational methods in mechanics. Leningrad: LGU 1966 (in Russian).Google Scholar

Copyright information

© Springer-Verlag 2000

Authors and Affiliations

  • F. X. Mei
    • 1
  1. 1.Department of Applied MechanicsBeijing Institute of TechnologyBeijingChina

Personalised recommendations