On Reduction of Lagrange Systems

  • Valentin Irtegov
  • Tatyana Titorenko
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6244)


We consider nonlinear conservative Lagrange systems with cyclic coordinates, which by means of the Legendre transformation are reduced to linear Routh systems. The latter allows one to reduce the problem of qualitative analysis for the nonlinear systems of above type to linear systems. Such an approach to investigation of the Lagrange systems is demonstrated by an example of a mechanical system with two cyclic coordinates and three positional coordinates. Some results of analysis of the initial system and the reduced one are given. We propose also a procedure of finding and investigation of qualitative properties of invariant manifolds (IMs) for the Lagrange systems with a nonlinear Routh function. The procedure is based on the analysis of stationary conditions of the “extended” Routh function. The efficiency of the proposed approach is demonstrated by an example of analysis of a concrete mechanical system.

Most part of the computations represented in this paper have been conducted with the aid of the computer algebra system “Mathematica”.


Mechanical System Invariant Manifold Lagrange Equation Lagrange Function Qualitative Property 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Borisov, A.V., Mamayev, I.S.: Poisson Structures and Lie Algebras in Hamiltonian Mechanics. Udmurdsk University, Izhevsk (1999)Google Scholar
  2. 2.
    Elkin, V.I.: Reduction of Nonlinear Control Systems. FAZIS-Computing Center of RAS, Moscow (2003)zbMATHGoogle Scholar
  3. 3.
    Griffits, F.: External Differential Forms and Variational Calculus. NFMI, Novosibirsk (1999)Google Scholar
  4. 4.
    Lurier, A.I.: Analytical Mechanics. GIFML, Moscow (1961)Google Scholar
  5. 5.
    Irtegov, V.D., Titorenko, T.N.: Using the system “Mathematica” in problems of mechanics. Mathematics and Computers in Simulation 3-5(57), 227–237 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Olver, P.J.: Applications of Lie Groups to Differential Equations, 2nd edn. Springer, New York (1993)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Valentin Irtegov
    • 1
  • Tatyana Titorenko
    • 1
  1. 1.Institute for System Dynamics and Control Theory SB RASIrkutskRussia

Personalised recommendations