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Abstracts

XVII conference

Euations Lagrange: symmetries, first integrals

Yakovenko G.N.

Yakovenko G.N., Moscow Institute of Physics and Technology, Russia, 141700, Moscow Region., Dolgoprudny, Institutsky str., 9, Department of Theoretical Mechanics Tel.: (495)576-57-33, fax: (495)408-68-69. E-mail: Yakovenko_G@mtu-net.ru

1 pp. (accepted)

The calculation of the first integral of the equations Lagrange on theorem Noether [1, 3] (or Besseli-Hagen [2, 3]) is required that equations allowed one-parameter group variational [3] (or divergent [3]) symmetry. The first integral is generated function Lagrange and infinitesimal: factor under the first degree in decomposition of the equations of the group on parameter. The first generalization: equations Lagrange allow the one-parameter family (not without fall group) variational (or divergent) symmetry [4]. It Is proved; that in this case there is one-parameter family first integral. The family integral is generated function Lagrange and right parts of the system of the common differential equations, which decision is a family symmetry. The function independent integrals can be kept In family first integral. Happens to the example (isolated conservative system), when one-parameter family contains ten function independent first integrals. The second generalization: equations Lagrange allow one-parameter group wanderring symmetry, which are not variational (or divergent) [3]. For calculation of one first integral is required several such symmetry. As example is considered flat moving the charged particle in magnetic field at presence of viscous friction.

References.

1. Noether E. Invariante Variationsprobleme. Nachr. Konig. Gesell. Wissen. Gottingen, Math-Phys. Kl. 1918. S. 235-257.

2. Bessel-Hagen E. Uber die Erhaltungsatze der Electrodynamic // Math. Ann., 1921. Bd. 84. S. 258-276.

3. Yakovenko G.N. The symmetries of the equations Hamilton and Lagrange - M.: MZ press, 2006. 186 s.

4. Yakovenko G.N. The theorem Noether-Besseli-Hagen - concentrated variant // Works IX International Hetaev conferences "Analytical mechanics, stability of the motion and management motion" (Irkutsk, June 12-16 2007), T. 2 / Irkutsk: IDSTU . S. 321-326.



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