Weyl's Theory in the Generalized Lie Algebroids Framework

TL;DR

This paper extends Weyl's geometric theory within the framework of generalized Lie algebroids, developing new formulas, identities, and results for mechanical systems using advanced differential geometric tools.

Contribution

It introduces a generalized Lie algebroid framework for Weyl's theory, including new formulas, identities, and Weyl-type results for mechanical systems.

Findings

Formulas of Ricci type and Cartan-Bianchi identities derived

Definition of geodesics and curvature in the generalized Lie algebroid setting

Two new Weyl-type results for mechanical systems obtained

Abstract

The geometry of the Lie algebroid generalized tangent bundle of a generalized Lie algebroid is developed. Formulas of Ricci type and identities of Cartan and Bianchi type are presented. Introducing the notion of geodesic of a mechanical $\left( \rho ,\eta \right) $-system with respect to a $(\rho, \eta)$-spray, the Berwald $(\rho, \eta)$-derivative operator and its mixed curvature, we obtain main results to conceptualize the Weyl's method in this general framework. Finally, we obtain two new results of Weyl type for the geometry of mechanical $\left( \rho ,\eta \right) $-systems.