Point classification of 2nd order ODEs: Tresse classification revisited and beyond

TL;DR

This paper revisits Tresse's 1896 classification of second-order ODE invariants using modern geometric methods, discussing absolute invariants and the equivalence problem to enhance understanding of point transformations.

Contribution

It provides a modern geometric reinterpretation of Tresse's classification and extends the discussion to absolute invariants and the equivalence problem for second-order ODEs.

Findings

Reinterpreted Tresse's invariants using modern geometry

Clarified the role of absolute invariants in classification

Discussed the equivalence problem for second-order ODEs

Abstract

In 1896 Tresse gave a complete description of relative differential invariants for the pseudogroup action of point transformations on the 2nd order ODEs. The purpose of this paper is to review, in light of modern geometric approach to PDEs, this classification and also discuss the role of absolute invariants and the equivalence problem.