On Reduction of Lagrange Systems
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”.
KeywordsMechanical System Invariant Manifold Lagrange Equation Lagrange Function Qualitative Property
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