Cartan projective connection spaces and group-theoretic analysis of systems of second-order ordinary differential equations
This note constitutes a nice addition to the geometric approach (in the sense of Élie Cartan) to the symmetry problem of systems of ODEs. Following some recent work on the geometric theory of differential systems [the authors, Sb. Math. 197, No.~7, 951--975 (2006); translation from Mat. Sb. 197, No.~7, 3--28 (2006; Zbl 1143.53310)], the authors focus on second order systems of ODEs of the special type \[ \ddot{x}^{\alpha}+a_{\beta\gamma}\dot{x}^{\alpha}\dot{x}^{\beta}\dot{x}^{\gamma}+b^{\alpha}_{\beta\gamma}\dot{x}^{\beta}\dot{x}^{\gamma}+c^{\alpha}_{\beta}\dot{x}^{\beta}+d^{\alpha}=0, \] where \(\alpha=1,\dots, n-1\) and \(a,b,c,d\) are arbitrary functions. More specifically, those systems that admit the four dimensional symmetry algebra \(VI_{2}\) (following the Kryuchkovich-Petrov notation for solvable Lie algebras [\textit{A. Z. Petrov}, Neue Methoden in der allgemeinen Relativitätstheorie (Russian). Moskva: Nauka (1966; Zbl 0146.23901), chapter 4]) are classified and linearization criteria are proposed, where the linearization procedure refers to any reduction of the system to another expressed by three-parameter linear equations or two linear equations with constant coefficients (and the integral curves of which are lines). It is shown that only four systems with symmetry group \(VI_{2}\) exist, and for those being linearizable, the corresponding changes of variables reducing the system to linear form are presented.
scientific article; zbMATH DE number 5915279
scientific article