Cartan projective connection spaces and group-theoretic analysis of systems of second-order ordinary differential equations

• Published in: Journal of mathematical sciences (New York, N.Y.) N.Y.), 2010-09, Vol.169 (3), p.282-296
• Main Authors: Aminova, A. V., Aminov, N. A.-M.
• Format: Article
• Language: English
• Subjects: Analysis; Differential equations; Mathematics; Mathematics and Statistics
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-010-0049-0

In the framework of the projective-geometric theory of systems of differential equations developed by the authors (see [9], etc.), we study the group-theoretic properties of systems of two second-order ordinary differential equations (resolved with respect to the second derivatives) whose right-hand sides are third-degree polynomials in derivatives of the unknown functions. We give a classification of those systems which admit four-dimensional solvable symmetry groups of the Lie-Petrov type[VI.sub.2]. For each such system, a (solution) linearization criterion is obtained, i.e., the conditions under which, by a change of variables, the system can be reduced to a differential system whose integral curves are straight lines and are expressed by three-parametric linear equations or two linear equations with constant coefficients. For all linearizable systems, we show linearizing changes of variables. under which, by a change of variables, the system can be reduced to a differential system whose integral curves are straight lines and are expressed by three-parametric linear equations or two linear equations with constant coefficients. For all linearizable systems, we show linearizing changes of variables.