Energy-momentum conservation laws in Finsler/Kawaguchi Lagrangian formulation

TL;DR

This paper introduces a geometric reformulation of Lagrangian mechanics using Finsler and Kawaguchi geometry, providing a unified, reparameterisation invariant framework that expresses symmetries and conservation laws, including energy-momentum, in a geometric manner.

Contribution

It develops a novel geometric formalism for Lagrangian theories using Finsler and Kawaguchi metrics, unifying symmetries and conservation laws in a reparameterisation invariant way.

Findings

Applied to scalar, Dirac, electromagnetic fields, and gravity with fluid coupling.

Reformulates energy-momentum conservation as part of Euler-Lagrange equations.

Proposes an alternative geometric definition of gravitational energy-momentum.

Abstract

We reformulate the standard Lagrangian formalism to a reparameterisation invariant Lagrangian formalism by means of Finsler and Kawaguchi geometry. In our formalism, various types of symmetries that appears in theories of physics are expressed geometrically by symmetries of Finsler (Kawaguchi) metric, and the conservation law of energy-momentum is a part of Euler-Lagrange equations. The application to scalar field, Dirac field, electromagnetic field and general relativity coupled to perfect fluid (added: ver.3) are discussed. By this formalism, we try to propose an alternative definition of energy-momentum current of gravity.