Second order time dependent tangent bundles and their applications

TL;DR

This paper develops a geometric framework for time-dependent Lagrangian mechanics using second order tangent bundles, introducing new concepts like time dependent connections and semisprays, and analyzing their applications to mechanical systems.

Contribution

It introduces the concepts of time dependent connections and semisprays on manifolds, and explores their role in the geometry of time-dependent Lagrangian systems.

Findings

Established a geometric structure for time-dependent mechanics.

Analyzed interactions of regular Lagrangians with second order tangent bundles.

Provided examples illustrating the theoretical framework.

Abstract

The aim of this paper is to geometrize time dependent Lagrangian mechanics in a way that the framework of second order tangent bundles plays an essential role. To this end, we first introduce the concepts of time dependent connections and time dependent semisprays on a manifold $M$ and their induced vector bundle structures on the second order time dependent tangent bundle $\R\times T^2M$. Then we turn our attention to regular time Lagrangians and their interaction with $\R\times T^2M$ in different situations such as mechanical systems with potential fields, external forces and holonomic constraints. Finally we propose an examples to support our theory.