Abstract
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”.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Borisov, A.V., Mamayev, I.S.: Poisson Structures and Lie Algebras in Hamiltonian Mechanics. Udmurdsk University, Izhevsk (1999)
Elkin, V.I.: Reduction of Nonlinear Control Systems. FAZIS-Computing Center of RAS, Moscow (2003)
Griffits, F.: External Differential Forms and Variational Calculus. NFMI, Novosibirsk (1999)
Lurier, A.I.: Analytical Mechanics. GIFML, Moscow (1961)
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)
Olver, P.J.: Applications of Lie Groups to Differential Equations, 2nd edn. Springer, New York (1993)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Irtegov, V., Titorenko, T. (2010). On Reduction of Lagrange Systems. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2010. Lecture Notes in Computer Science, vol 6244. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15274-0_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-15274-0_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15273-3
Online ISBN: 978-3-642-15274-0
eBook Packages: Computer ScienceComputer Science (R0)