Group Invariant Solutions without Transversality and the Principle of Symmetric Criticality

TL;DR

This paper extends Lie's method to find group invariant solutions for non-transverse group actions, identifies obstructions to symmetric criticality, and demonstrates applications with Maxwell's equations.

Contribution

It introduces a generalized approach for non-transverse symmetry reductions and highlights obstructions to the principle of symmetric criticality.

Findings

Extended Lie's method for non-transverse actions

Identified local obstructions to symmetric criticality

Applied to Maxwell's equations examples

Abstract

We extend Lie's classical method for finding group invariant solutions to the case of non-transverse group actions. For this extension of Lie's method we identify a local obstruction to the principle of symmetric criticality. Two examples of non-transverse symmetry reductions for the potential form of Maxwell's equations are then examined.