Conformal geodesics are not variational in higher dimensions

TL;DR

This paper proves that conformal geodesics do not derive from a variational principle in dimensions higher than three, contrasting with the three-dimensional case where un-parametrized conformal geodesics are variational.

Contribution

It demonstrates the failure of variationality for conformal geodesics in higher dimensions, extending previous results from three dimensions.

Findings

Variationality holds for un-parametrized conformal geodesics in 3D.

Variationality fails for both parametrized and un-parametrized conformal geodesics in higher dimensions.

Variational principle may determine the physical dimension in geometry and physics.

Abstract

Variationality of the equation of conformal geodesics is an important problem in geometry with applications to general relativity. Recently it was proven that, in three dimensions, this system of equations for un-parametrized curves is the Euler-Lagrange equations of a certain conformally invariant functional, while the parametrized system in three dimensions is not variational. We demonstrate that variationality fails in higher dimensions for both parametrized and un-parametrized conformal geodesics, indicating that variational principle may be the selection principle for the physical dimension.