Comment on higher derivative Lagrangians in relativistic theory

We discuss the consequences of higher derivative Lagrangians of the form $α_1 A_μ(x)\dot{x}^μ$, $α_2 G_μ(x)\ddot{x}^μ$, $α_3 B_μ(x)\dddot{x}^μ$, $α_4 K_μ(x)\ddddot{x}^μ$, $\cdots$, $U_{(n)μ}(x)x^{(n)μ}$ 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 $U_{(n),μ}(x),\ n\geq1$ with the higher currents $j^{(n)μ}=ρ(x)x^{(n)μ}$, where $ρ(x)$ is the spatial density of mass or of electric charge. In addition, we discuss the analogy of the field $G_μ(x)$ 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.

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