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.
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References
L. V. Ovsjannikov:Group Properties of Differential Equations (Novosibirsk, 1962).
L. V. Ovsjannikov:Group Analysis of Differential Equations (Academic Press, New York, N.Y., 1982).
P.J. Olver:Applications of Lie Groups to Differential Equations (Springer, Berlin, 1986).
D. H. Sattinger andO. Weaver:Lie Groups and Algebras, (Springer, New York, N.Y., 1986).
G. W. Bluman andS. Kumei:Symmetries and Differential Equations (Springer, Berlin, 1989).
H. Stephani:Differential Equations. Their Solution Using Symmetries (Cambridge Univ. Press, 1989).
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.
G. Gaeta:Geometrical Symmetries of Nonlinear Equations and Physics, in preparation.
M. Aguirre andJ. Krause:Int. J. Theor. Phys.,30, 495, 1461 (1991).
G. Cicogna andG. Gaeta:Lie point symmetries in bifurcation problems, to be published inAnn. Inst. H. Poincaré.
G. Cicogna:J. Phys. A,23, L1339 (1990).
V. I. Arnold:Mathematical Methods of Classical Mechanics (Springer, Berlin, 1978 and 1989).
R. Courant andD. Hilbert:Methods of Mathematical Physics (Interscience Publ., New York, N. Y., 1962), chapt. II, Appendix 1, § 2.
M. Golubitsky, I. Stewart andD. Schaeffer:Singularity and Groups in Bifurcation Theory, Vol. II (Springer, New York, N.Y., 1988).
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C.N.R. fellow under grant 203.01.48.
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Cicogna, G., Gaeta, G. On Lie point symmetries in mechanics. Nuov Cim B 107, 1085–1096 (1992). https://doi.org/10.1007/BF02727046
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DOI: https://doi.org/10.1007/BF02727046