  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.
