Second order variational problem and 2-dimensional concircular geometry

TL;DR

This paper establishes a variational framework for geodesic circles in two-dimensional geometry, providing explicit formulations and demonstrating uniqueness in Euclidean space, while also introducing a novel concept of 'spin' force from the variation process.

Contribution

It introduces a variational description for geodesic circles in 2D and explores its uniqueness in Euclidean geometry, also discovering a new 'spin' force concept.

Findings

Variational description of geodesic circles in 2D is established.

Explicit form of the variational functional is provided.

A new 'spin' force concept is introduced from the variation process.

Abstract

It is proved that the set of geodesic circles in two dimensions may be given a variational description and the explicit form of it is presented. In the limit case of the Euclidean geometry a certain claim of uniqueness of such description is proved. A formal notion of 'spin' force is discovered as a by-product of the variation procedure involving the acceleration.