Jet Geometrical Objects Depending on a Relativistic Time

TL;DR

This paper develops a geometric framework on the 1-jet space for Lagrangians depending on relativistic time, introducing concepts like d-tensors, semisprays, harmonic curves, and nonlinear connections.

Contribution

It provides a new geometrization approach for relativistic time-dependent Lagrangians using jet geometrical objects and explores their interrelations.

Findings

Defined relativistic jet geometrical objects including d-tensors and semisprays.

Established relations between jet geometrical structures.

Laid groundwork for further geometric analysis of relativistic Lagrangian systems.

Abstract

In this paper we study a collection of jet geometrical concepts, we refer to d-tensors, relativistic time dependent semisprays, harmonic curves and nonlinear connections on the 1-jet space J1(R;M), necessary to the construction of a Miron's-like geometrization for Lagrangians depending on a relativistic time. The geometrical relations between these jet geometrical objects are exposed.