SOPDEs and Nonlinear Connections

TL;DR

This paper explores the relationship between nonlinear connections and SOPDEs within the k-symplectic Lagrangian framework, providing a geometric characterization on the extended tangent bundle.

Contribution

It introduces a geometric characterization of nonlinear connections on the k-tangent bundle and links them to SOPDEs in the context of G"unther's formalism.

Findings

Characterization of nonlinear connections via k-tangent structures

Connection between SOPDEs and nonlinear connections in Lagrangian formalism

Framework for analyzing PDEs using geometric structures

Abstract

The canonical k-tangent structure on $T^1_kQ=TQ\oplus\stackrel{k}...\oplus TQ$ allows us to characterize nonlinear connections on $T^1_kQ$ and to develop G\"unther's (k-symplectic) Lagrangian formalism. We study the relationship between nonlinear connections and second-order partial differential equations (SOPDEs), which appear in G\"unther's Lagrangian formalism.