The south-pointing chariot on a surface

TL;DR

This paper analyzes the motion of the south-pointing chariot on surfaces, deriving conditions for parallel transport of its pointer, including exact solutions on a sphere, advancing understanding of geometric transport phenomena.

Contribution

It provides an exact analysis of the south-pointing chariot's motion on a surface, extending previous approximations to exact solutions on a sphere.

Findings

Pointer is parallel transported in the zero-distance limit.

Exact motion of the chariot on a sphere is derived.

Pointer remains parallel transported exactly on the sphere.

Abstract

We examine closely the motion of the south-pointing chariot on a surface by taking into account the fact that both wheels have to roll without slipping. We firstly develop a procedure of successive approximations. In the limit where the distance between the wheels tends to zero, we find again the result due to Santander that the pointer is parallel transported. This is longer true generally within the first-order approximation. We secondly determine in an exact manner the motion of the south-pointing chariot on the sphere. Then, we define a pointer and we prove that this one is parallel transported exactly.