Classification of Second-order Ordinary Differential Equations Admitting Lie Groups of Fibre-preserving Point Symmetries

We use Elie Cartan's method of equivalence to give a complete classification, in terms of differential invariants, of second-order ordinary differential equations admitting Lie groups of fibre-preserving point symmetries. We then apply our results to the determination of all second-order equations which are equivalent, under fibre-preserving transformations, to the free particle equation. In addition we present those equations of Painleve' type which admit a transitive symmetry group. Finally we determine the symmetry group of some equations of physical interest, such as the Duffing and Holmes-Rand equations, which arise as models of non-linear oscillators.