Regular and Chaotic Dynamics

, Volume 13, Issue 2, pp 71–80 | Cite as

Lagrange’s identity and its generalizations

Research Articles

Abstract

The famous Lagrange identity expresses the second derivative of the moment of inertia of a system of material points through the kinetic energy and homogeneous potential energy. The paper presents various extensions of this brilliant result to the case 1) of constrained mechanical systems, 2) when the potential energy is quasi-homogeneous in coordinates and 3) of continuum of interacting particles governed by the well-known Vlasov kinetic equation.

Key words

Lagrange’s identity quasi-homogeneous function dilations Vlasov’s equation 

MSC2000 numbers

37A60 82B30 82CXX 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kozlov, V.V., Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat, Rus. J. Nonlin. Dyn., 2006, vol. 3, no. 2, pp. 123–140.Google Scholar
  2. 2.
    Kozlov, V.V. and Kolesnikov, N.N., On Theorems of Dynamics, J. Appl. Math. Mech., 1978, vol. 42, no. 1, pp. 28–33.CrossRefMathSciNetGoogle Scholar
  3. 3.
    Horn, R. and Johnson, C., Matrix Analysis, Cambridge: Cambridge Univ. Press, 1985.CrossRefMATHGoogle Scholar
  4. 4.
    Vedenyapin, V.V., Boltzmann and Vlasov Kinetic Equation, Moscow: Fizmatlit, 2001.Google Scholar
  5. 5.
    Maslov, V.P., Equations of Self-Consistent Field, in Current Problems in Mathematics, Vol. 11, Akad. Nauk SSSR Vsesojuz. Inst. Nauchn. i Tehn. Informacii (VINITI), Moscow, 1978, pp. 153–234.Google Scholar
  6. 6.
    Dobrushin, R.L., Vlasov Equations, Funktsional. Anal. i Prilozhen., 1979, vol. 13, no. 2, pp. 48–58.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© MAIK Nauka 2008

Authors and Affiliations

  1. 1.V.A. Steklov Institute of MathematicsRussian Academy of SciencesMoscowRussia

Personalised recommendations