Variationality of conformal geodesics in dimension 3

TL;DR

This paper proves that the equation for unparametrized conformal geodesics in three-dimensional conformal manifolds is variational, addressing an open problem in the field.

Contribution

It establishes that conformal geodesics in 3D are governed by a variational principle, expanding understanding of their geometric and physical significance.

Findings

Conformal geodesics in 3D are described by a third-order differential equation.

The paper demonstrates that this equation is derivable from a variational principle.

This result confirms the variational nature of conformal geodesics in the simplest non-trivial dimension.

Abstract

Conformal geodesics form an invariantly defined family of unparametrized curves in a conformal manifold generalizing unparametrized geodesics/paths of projective connections. The equation describing them is of third order, and it was an open problem whether they are given by an Euler--Lagrange equation. In dimension 3 (the simplest, but most important from the viewpoint of physical applications) we demonstrate that the equation for unparametrized conformal geodesics is variational.