Variationality of geodesic circles in two dimensions

TL;DR

This paper explores the variational properties of geodesic circles in two-dimensional Riemannian geometry, revealing how curvature influences the Lagrange derivative and introduces the concept of quasiclassical spin in pseudo-Riemannian settings.

Contribution

It derives the variational differential equation for geodesic circles and analyzes the impact of curvature on the Lagrange derivative, connecting geometry with spin concepts.

Findings

Derived the variational differential equation for geodesic circles

Showed curvature's influence on the Lagrange derivative

Linked curvature effects to the emergence of quasiclassical spin

Abstract

This note treats the notion of Lagrange derivative for the third order mechanics in the context of covariant Riemannian geometry. The variational differential equation for geodesic circles in two dimensions is obtained. The influence of the curvature tensor on the Lagrange derivative leads to the emergence of the notion of quasiclassical spin in the pseudo-Riemannian case.