Simplicial Euclidean Relativistic Lagrangian

TL;DR

This paper introduces a novel simplicial approach to relativistic Lagrangians on Euclidean manifolds, defining paths as 2D simplicial complexes and deriving the relativistic form in the continuum limit.

Contribution

It presents a new geometric framework using simplicial complexes to model relativistic Lagrangians without relying on Lorentz invariance.

Findings

Paths are modeled as 2D simplicial strips.

Relativistic Lagrangian form is obtained in the continuum limit.

The approach connects discrete simplicial structures with continuum relativistic physics.

Abstract

The paths on the {\bf R$^3$} real Euclidean manifold are defined as 2-dimensional simplicial strips; points are replaced by 2-simplexes and the orbits of the action of a one discrete-parameter group on the base manifold becomes a convex polyhedron attached to a 2-dimensional simplicial complex. The Lagrangian of a moving mass is proportional to the width of the path. The special relativistic form of the Lagrangian is recovered in the continuum limit, without relativistic Lorenz invariance considerations.