On the use of connections in the calculus of variations in Finsler geometry

TL;DR

This paper introduces a unified approach to connections in Finsler geometry, clarifying why classical Finslerian connections yield identical formulas in the calculus of variations.

Contribution

It provides a simplified, unified framework for understanding connections in sprays and Finsler metrics, explaining their role in the calculus of variations.

Findings

Unified approach to connections in Finsler geometry

Clarification of classical connections' equivalence in formulas

Simplified explanation of calculus of variations in Finsler settings

Abstract

We propose a unified approach to the theory of connections in the geometry of sprays and Finsler metrics which, in particular, gives a simple explanation of the well-known fact that all the classical Finslerian connections provide exactly the same formulas appearing in the calculus of variations.