  In this paper we study 1/k-geodesics, those closed geodesics that minimize on
any subinterval of length $l(\gamma)/k$. We employ energy methods to provide a
relationship between the 1/k-geodesics and what we define as the balanced
points of the uniform energy. We show that classes of balanced points of the
uniform energy persist under the Gromov-Hausdorff convergence of Riemannian
manifolds. Additionally, we relate half-geodesics (1/2-geodesics) to the
Grove-Shiohama critical points of the distance function. This relationship
affords us the ability to study the behavior of half-geodesics via the well
developed field of critical point theory. Along the way we provide a complete
characterization of the differentiability of the Riemannian distance function.
