Variational principles for conformal geodesics

TL;DR

This paper develops a variational formulation for conformal geodesics using an enlarged variation class, addressing the challenge posed by their third-order equations and exploring associated fourth-order systems with spiral solutions.

Contribution

It introduces a conformally invariant third-order Lagrangian for conformal geodesics by enlarging the variation class, and analyzes the resulting fourth-order ODE system.

Findings

Established a variational principle for conformal geodesics.

Derived a conformally invariant fourth-order ODE system.

Identified spiral solutions within the fourth-order system.

Abstract

Conformal geodesics are solutions to a system of third order of equations, which makes a Lagrangian formulation problematic. We show how enlarging the class of allowed variations leads to a variational formulation for this system with a third--order conformally invariant Lagrangian. We also discuss the conformally invariant system of fourth order ODEs arising from this Lagrangian, and show that some of its integral curves are spirals.