ODEs whose symmetry groups are not fiber-preserving

TL;DR

This paper classifies scalar and system ODEs with non-fiber-preserving symmetries, revealing exceptions to the common fiber-preserving symmetry structure using Lie algebra techniques.

Contribution

It provides a complete description of ODEs with non-fiber-preserving symmetries, including invariants and the limitations of the moving frames method.

Findings

Most symmetric higher order ODEs have fiber-preserving symmetries.

Explicit characterization of exceptions using differential invariants.

Identification of limitations in the method of moving frames.

Abstract

We observe that, up to conjugation, a majority of symmetric higher order ODEs (ordinary differential equations) and ODE systems have only fiber-preserving point symmetries. By exploiting Lie's classification of Lie algebras of vector fields, we describe all the exceptions to this in the case of scalar ODEs and systems of ODEs on a pair of functions. The scalar ODEs whose symmetry algebra is not fiber preserving can be expressed via absolute and relative scalar differential invariants, while a similar description for ODE systems requires us to also invoke conditional differential invariants and vector-valued relative invariants to deal with singular orbits of the action. Investigating prolongations of the actions, we observe some interesting relations between different realizations of Lie algebras. We also note that it may happen that the prolongation of a finite-dimensional Lie algebra acting on a differential equation never becomes free. An example of an underdetermined ODE system for which this phenomenon occurs shows limitations of the method of moving frames.