Normal Forms of second order Ordinary Differential Equations $y_{xx}=J(x,y,y_{x})$ under Fibre-Preserving Maps

TL;DR

This paper develops a method to classify second order ODEs under fibre-preserving transformations using normal forms, Lie algebra computations, and the Cauchy-Kovalevskaya theorem, providing explicit criteria for equivalence to the simplest form.

Contribution

It introduces a systematic approach to derive normal forms for second order ODEs under fibre-preserving maps, utilizing Lie algebra and Moser's method, with explicit descriptions and existence proofs.

Findings

Normal forms form an ideal in the space of formal power series.

Explicit description of the symmetry Lie algebra for y''=0.

Vanishing of primary invariants implies equivalence to y''=0.

Abstract

We study the equivalence problem of classifying second order ordinary differential equations $y_{xx}=J(x,y,y_{x})$ modulo fibre-preserving point transformations $x\longmapsto \varphi(x)$, $y\longmapsto \psi(x,y)$ by using Moser's method of normal forms. We first compute a basis of the Lie algebra ${\frak{g}}_{{\{y_{xx}=0\}}}$ of fibre-preserving symmetries of $y_{xx}=0$. In the formal theory of Moser's method, this Lie algebra is used to give an explicit description of the set of normal forms $\mathcal{N}$, and we show that the set is an ideal in the space of formal power series. We then show the existence of the normal forms by studying flows of suitable vector fields with appropriate corrections by the Cauchy-Kovalevskaya theorem. As an application, we show how normal forms can be used to prove that the identical vanishing of Hsu-Kamran primary invariants directly imply that the second order differential equation is fibre-preserving point equivalent to $y_{xx}=0$.