Geometric Properties of Paths in Relativistic Lagrangian Mechanics

TL;DR

This paper explores the geometric properties of paths in relativistic Lagrangian mechanics, introducing covariant Lagrangians and a measure of path deviation to analyze phenomena like the twin paradox.

Contribution

It develops a covariant framework for Lagrangian functions and introduces a deviation measure, providing a rigorous analysis of the twin paradox in relativistic mechanics.

Findings

Covariant Lagrangians are essential for path analysis in relativity.

A covariant measure of path deviation is introduced.

The twin paradox is rigorously resolved using geometric methods.

Abstract

Considering an extension of the principle of covarience to the action along a path in relativistic Lagrangian mechanics, we motivate the use of geometric -- i.e. covariant and parameter invariant -- Lagrangian functions. We then study some properties of geometric Lagrangians, and introduce the notion of deviation of a path, which is a covariant measure of how much a path departs from a geodesic. Finally, we apply this notion of the twin paradox, and provide a rigorous resolution of it.