On generalized forces in higher derivative Lagrangian theory

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.