The action of contact transformations pseudogroup on the second order ODEs which are cubic in second derivative

TL;DR

This paper investigates the local equivalence of second order ODEs cubic in the second derivative under contact transformations, reducing the problem to the equivalence of 2-webs in three-dimensional space.

Contribution

It introduces a method to analyze the equivalence of such ODEs by linking it to the classification of 2-webs under finite-dimensional group actions.

Findings

Reduction of the equivalence problem to 2-webs classification

Identification of the group action preserving the 2-webs

Framework for analyzing contact transformation invariants

Abstract

In the present paper we consider the problem of local equivalence of second order ODEs which are cubic in second derivative under the action of the pseudogroup of contact transformations. We show how it may be reduced to the equivalence problem of 2-webs in $\mathbb{R}^3$ under the action of finite-dimensional group, admitted by the associated second order equation.