Generalised Lorentz force expression from Classical Mechanics by Goldstein

Question

I am reading chapter 7 in the 3rd edition of Goldstein's Classical mechanics textbook and the expression for the Lorentz force is confusing me. I cannot scan it so I am just going to write it out verbatim and formulate my question afterwards. Here is the extract of page 298 from the text:

In terms of $\phi$ and $\mathbf{A}$, the Lorentz force is $$\mathbf{F} = q\{-\nabla\phi+\frac{1}{c}\frac{\partial \mathbf{A}}{\partial t} + 1[v \times(\nabla\times \mathbf{A})]\}\tag{7.67c}.$$ This suggests that we should generalize the force law to $$\frac{dp_{\mu}}{d\tau} = q\left(\frac{\partial (u^\nu A_\nu)}{\partial x^\mu}-\frac{dA_\mu}{d\tau}\right).\tag{7.68}$$

The first equation is the three three dimensional Lorentz force express using the vector and scalar potentials (As a note I think the second term should be $-\frac{\partial A}{\partial t}$ but the above is as written.)

I am unsure howhow you reach the second equation from the first expression, I would appreciate any help in understanding this problem.

Answer 1

TL;DR: The total derivative term $$\frac{dA_\mu}{d\tau}~=~\gamma\frac{dA_\mu}{dt} ~=~\gamma\left(\vec{v}\cdot\vec{\nabla} A_\mu+ \frac{\partial A_\mu}{\partial t}\right)$$ in eq. (7.68) is correct. It should not be a partial derivative.

Before trying to read the relativistic formulation in section 7.6, I would strongly recommend you to fully understand the non-relativistic derivation in section 1.5, which essentially features the same issue.