On generalized forces in higher derivative Lagrangian theory

TL;DR

This paper explores higher derivative Lagrangians extending electromagnetic and gravitational interactions, proposing a generalized induction principle linking higher fields and currents, and reflecting on new concepts of force and inertia.

Contribution

It introduces a framework for higher derivative Lagrangians that generalize electromagnetic and gravitational interactions, proposing a new induction principle for higher fields and currents.

Findings

Odd order Lagrangians resemble electromagnetism.

Even order Lagrangians resemble gravity.

Proposes a generalized induction principle.

Abstract

In this article, we introduce higher derivative Lagrangians of this form $\alpha_1 A_{\mu}(x)\dot{x}^\mu$, $\alpha_2 G_{\mu}(x)\ddot{x}^\mu$, $\alpha_3 B_{\mu}(x)\dddot{x}^\mu$, $\alpha_4 K_{\mu}(x)\ddddot{x}^\mu$, $\cdots$, that generalize the electromagnetic interaction to higher order derivatives. We show that odd order Lagrangians describe interactions analog to electromagnetism while even order Lagrangians are similar to gravitational interaction. From this analogy, we formulate the concept of the generalized induction principle assuming the coupling between the higher fields $U_{(n),\mu}(x),\ n\geq1$ and the higher currents $j^{(n)\mu}=\rho(x)d^nx^{\mu}/ds^n$, where $\rho(x)$ is the spatial density of mass ($n$ even) or of electric charge ($n$ odd). In short, this article is an invitation to reflect on a generalization of the concept of force and of inertia. We discuss the implications of these paradigms more in depth in the last section of the paper.