Variationality of conformal geodesics in dimension 3

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.

A remark about conformally invariant Lagrangian is supplied at the end. More references added. Ancillary files can be accessed through version 1

References in corpus (6)

• Some examples of the behaviour of conformal geodesics
• The Curvatures of Regular Curves and Euclidean Invariants of their Derivatives
• Chains in CR geometry as geodesics of a Kropina metric
• A comparison of Ashtekar's and Friedrich's formalisms of spatial infinity
• On the geometry of curves and conformal geodesics in the Mobius space
• The inverse problem in the calculus of variations: new developments