The emergence of the relativistic Lagrangian from the non-relativistic multiplicative Lagrangian

TL;DR

This paper demonstrates how the relativistic Lagrangian naturally emerges from a statistical analysis of the multiplicative Lagrangian, revealing deep connections between classical and relativistic physics.

Contribution

It introduces a novel multiplicative Lagrangian framework that unifies classical and relativistic mechanics through a statistical perspective.

Findings

Relativistic Lagrangian arises from statistical averaging of multiplicative Lagrangian.

The formalism reveals a deeper connection between spacetime and action.

The approach offers a new foundation linking classical and relativistic physics.

Abstract

The multiplicative Lagrangian and Hamiltonian introduce an additional parameter that, despite its variation, results in identical equations of motion as those derived from the standard Lagrangian. This intriguing property becomes even more striking in the case of a free particle. By manipulating the parameter and integrating out, the statistical average of the multiplicative Lagrangian and Hamiltonian naturally arises. Astonishingly, from this statistical viewpoint, the relativistic Lagrangian and Hamiltonian emerge with remarkable elegance. On the action level, this formalism unveils a deeper connection: the spacetime of Einstein's theory reveals itself from a statistical perspective through the action associated with the multiplicative Lagrangian. This suggests that the multiplicative Lagrangian/Hamiltonian framework offers a profound and beautiful foundation, one that reveals the underlying unity between classical and relativistic descriptions in a way that transcends traditional formulations. In essence, the multiplicative approach introduces a richer and more intricate structure to our understanding of physics, bridging the gap between different theoretical realms through a statistical perspective.