A distinguished geometry perspective on multi-time affine quadratic Lagrangians

TL;DR

This paper explores the geometric structure of multi-time affine quadratic Lagrangians using a distinguished geometry approach, constructing key geometric objects like connections, torsion, and curvature.

Contribution

It introduces a geometric framework for multi-time quadratic Lagrangians, detailing the construction of fundamental geometric objects in this context.

Findings

Defined the main Riemann-Lagrange geometric objects for multi-time Lagrangians

Established relations between the non-linear connection and geometric structures

Provided a foundation for further geometric analysis of multi-time Lagrangian systems

Abstract

For a space endowed with a general quadratic multi-time Lagrangian and an associated non-linear connection, the paper constructs the main Riemann-Lagrange distinguished geometric objects (linear connection, torsion and curvature).