Symmetries and infinitesimal symmetries of singular differential equations

TL;DR

This paper investigates the geometric and infinitesimal symmetries of linearly singular differential equations, establishing their equivalence to dynamic symmetries and analyzing invariance under vector field flows.

Contribution

It introduces the concept of geometric symmetry for singular differential equations and proves their equivalence to dynamic symmetries, extending symmetry analysis to singular cases.

Findings

Geometric symmetries are equivalent to dynamic symmetries.

Infinitesimal symmetries are characterized similarly to finite symmetries.

Invariance under vector field flows is studied with a focus on infinitesimal variations.

Abstract

The aim of this paper is to study symmetries of linearly singular differential equations, namely, equations that can not be written in normal form because the derivatives are multiplied by a singular linear operator. The concept of geometric symmetry of a linearly singular differential equation is introduced as a transformation that preserves the geometric data that define the problem. It is proved that such symmetries are essentially equivalent to dynamic symmetries, that is, transformations mapping solutions into solutions. Similar results are given for infinitesimal symmetries. To study the invariance of several objects under the flows of vector fields, a careful study of infinitesimal variations is performed, with a special emphasis on infinitesimal vector bundle automorphisms.