Finsler connection for general Lagrangian systems

TL;DR

This paper introduces a simplified Finsler non-linear connection applicable to both regular and singular Lagrangian systems, offering easier calculations and potential applications in physics, especially for constrained systems.

Contribution

A new simplified definition of Finsler non-linear connection that applies to both regular and singular cases, expressed in point-Finsler space, facilitating calculations and physical applications.

Findings

Provides a unified connection framework for regular and singular Finsler systems.

Offers formulas that simplify calculations compared to traditional methods.

Demonstrates applicability through examples of constrained systems.

Abstract

We give a Finsler non-linear connection by a new simplified definition for not only regular case but also singular case. In regular case, it corresponds to non-linear connection part of Berwald's connection, but our connection is expressed not in line element space but in point-Finsler space. In this view we recognize Finsler metric L(x,dx) as a non-linear form, which is a natural generalisation of Riemannian metric having original expression, \sqrt{g_{ab}(x)dx^a dx^b}. Furthermore our formulae provide easier calculation rather than conventional treatments, so we think that they suits to application to physics. Our definition can be used in the singular case of Finsler metric, which correspond to gauged constraint systems in mechanics. Here we give some non trivial examples of constraint systems for exposition of validity of our connection.