Comment on higher derivative Lagrangians in relativistic theory

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.