Scalar autonomous second order ordinary differential equations
This paper deals with differential equations of the form \[ \begin{aligned} &\dot u = v,\\ &\dot v = F(u,v), \end{aligned} \] whereby \((u,v)\in{\mathbb R}^2\) and \(F\) an infinitely often differentiable function. The author's main concern, in the case when \(F(u,-v)=F(u,v)\), is to provide conditions so that there exist an infinitely often differentiable change of coordinates in \({\mathbb R}^2\), \((u,v)\mapsto (x,y)\) that transforms the above equation into \[ \begin{aligned} &\dot x = y,\\ &\dot y = g(x), \end{aligned} \] for an appropriate function \(g\).
scientific article; zbMATH DE number 6150881