Symmetry condition in terms of Lie brackets

TL;DR

This paper investigates the symmetry conditions of systems of PDEs using Lie brackets, showing that the vanishing of certain commutators characterizes generalized symmetries within the jet manifold framework.

Contribution

It establishes a new equivalence between the vanishing of Lie bracket commutators and the existence of generalized symmetries for passive orthonomic PDE systems.

Findings

Lie brackets characterize generalized symmetries in PDE systems

Commutator vanishing is equivalent to symmetry conditions

Framework applies to submanifolds in jet spaces

Abstract

A passive orthonomic system of PDEs defines a submanifold in the corresponding jet manifold, coordinated by so called parametric derivatives. We restrict the total differential operators and the prolongation of an evolutionary vector field v to this submanifold. We show that the vanishing of their commutators is equivalent to v being a generalized symmetry of the system.