Advertisement

Il Nuovo Cimento B (1971-1996)

, Volume 107, Issue 9, pp 1085–1096 | Cite as

On Lie point symmetries in mechanics

  • G. Cicogna
  • G. Gaeta
Article

Summary

We present some remarks on the existence and the properties of Lie point symmetries of finite dimensional dynamical systems expressed either in Newton-Lagrange or in Hamilton form. We show that the only Lie symmetries admitted by Newton-Lagrange-type problems are essentially linear symmetries, and construct the most general problem admitting such a symmetry. In the case of Hamiltonian problems, we discuss the differences and the relationships between the existence of time-independent Lie point symmetries and the invariance properties of the Hamiltonian under coordinate transformations.

PACS 02.20

Group theory 

PACS 03.20

Classical mechanics of continuous media general mathematical aspects 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    L. V. Ovsjannikov:Group Properties of Differential Equations (Novosibirsk, 1962).Google Scholar
  2. [2]
    L. V. Ovsjannikov:Group Analysis of Differential Equations (Academic Press, New York, N.Y., 1982).Google Scholar
  3. [3]
    P.J. Olver:Applications of Lie Groups to Differential Equations (Springer, Berlin, 1986).CrossRefGoogle Scholar
  4. [4]
    D. H. Sattinger andO. Weaver:Lie Groups and Algebras, (Springer, New York, N.Y., 1986).Google Scholar
  5. [5]
    G. W. Bluman andS. Kumei:Symmetries and Differential Equations (Springer, Berlin, 1989).CrossRefGoogle Scholar
  6. [6]
    H. Stephani:Differential Equations. Their Solution Using Symmetries (Cambridge Univ. Press, 1989).Google Scholar
  7. [7]
    P. Winternitz:Group theory and exact solutions of partially integrable differential systems, inPartially Integrable Evolution Equations in Physics, edited byR. Conte andN. Boccara (Kluwer Academic Publishers, Amsterdam, 1990), p. 515.CrossRefGoogle Scholar
  8. [8]
    G. Gaeta:Geometrical Symmetries of Nonlinear Equations and Physics, in preparation.Google Scholar
  9. [9]
    M. Aguirre andJ. Krause:Int. J. Theor. Phys.,30, 495, 1461 (1991).MathSciNetCrossRefGoogle Scholar
  10. [10]
    G. Cicogna andG. Gaeta:Lie point symmetries in bifurcation problems, to be published inAnn. Inst. H. Poincaré.Google Scholar
  11. [11]
    G. Cicogna:J. Phys. A,23, L1339 (1990).MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    V. I. Arnold:Mathematical Methods of Classical Mechanics (Springer, Berlin, 1978 and 1989).CrossRefGoogle Scholar
  13. [13]
    R. Courant andD. Hilbert:Methods of Mathematical Physics (Interscience Publ., New York, N. Y., 1962), chapt. II, Appendix 1, § 2.Google Scholar
  14. [14]
    M. Golubitsky, I. Stewart andD. Schaeffer:Singularity and Groups in Bifurcation Theory, Vol. II (Springer, New York, N.Y., 1988).CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica 1992

Authors and Affiliations

  • G. Cicogna
    • 1
  • G. Gaeta
    • 2
  1. 1.Dipartimento di Fisica dell’UniversitàPisaItalia
  2. 2.C.Ph.Th., Ecole PolytechniquePalaiseauFrance

Personalised recommendations