Variational principles for conformal geodesics

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.

14 pages, one figure. A gap removed from the proof of Corollary 4.2. Final version, to appear in Letters in Mathematical Physics