Newtonian mechanics in a Riemannian manifold

TL;DR

This paper reviews the generalization of Newtonian mechanics from Euclidean space to Riemannian manifolds, highlighting how classical mechanics can be extended to more abstract geometric settings.

Contribution

It provides a comprehensive review of how Newtonian mechanics is formulated within the framework of Riemannian geometry, expanding classical concepts to curved spaces.

Findings

Mechanics can be formulated on Riemannian manifolds.

Classical equations are adapted to curved geometries.

The review bridges Newtonian mechanics and differential geometry.

Abstract

The work done by Isaac Newton more than three hundred years ago, continues being a path to increase our knowledge of Nature. To better understand all the ideas behind it, one of the finest ways is to generalize them to wider situations. In this report we make a review of one of these enlargements, the one that bears the mechanical systems from the elementary homogeneous three dimensional Euclidean space to the more abstract geometry of a Riemannian manifold.