Comment on higher derivative Lagrangians in relativistic theory

TL;DR

This paper explores the implications of higher derivative Lagrangians in relativistic particle dynamics, proposing a generalized induction principle, and discusses their potential connection to gravity and inertia concepts.

Contribution

It introduces a framework for higher derivative fields and currents in relativistic theory, extending the concept of inertia and linking higher derivatives to gravitational analogies.

Findings

Derived equations of motion for particles with higher derivative couplings

Proposed a generalized induction principle for higher fields and currents

Discussed the analogy between higher derivative fields and gravity

Abstract

We discuss the consequences of higher derivative Lagrangians of the form $\alpha_1 A_{\mu}(x)\dot{x}^\mu$, $\alpha_2 G_{\mu}(x)\ddot{x}^\mu$, $\alpha_3 B_{\mu}(x)\dddot{x}^\mu$, $\alpha_4 K_{\mu}(x)\ddddot{x}^\mu$, $\cdots$, $U_{(n)\mu}(x)x^{(n)\mu}$ in relativistic theory. After establishing the equations of the motion of particles in these fields, we introduce the concept of the generalized induction principle assuming the coupling between the higher fields $U_{(n),\mu}(x),\ n\geq1$ with the higher currents $j^{(n)\mu}=\rho(x)x^{(n)\mu}$, where $\rho(x)$ is the spatial density of mass or of electric charge. In addition, we discuss the analogy of the field $G_\mu(x)$ with the gravitational field and its inclusion in the general relativity framework in the last section. This letter is an invitation to reflect on a generalisation of the concept of inertia and we also discuss this problem in the last section.