<PERSON> 1980 AJP “Derivation of Maxwell’s equations from the gauge invariance of classical mechanics” confusion on step to derive equations of motion
In the article <PERSON> 1980 AJP “Derivation of Maxwell’s equations from the gauge invariance of classical mechanics” the author defines the Lagrangian $$\mathcal{L}{0i}=\mathcal{L}_0+\int (\textbf{J} \cdot \textbf{A}-\rho A_0) d^3x$$ where $\textbf{J}$ is the current density of a single particle, $\textbf{A}$ is the magnetic potential, $\rho$ the charge density again for a single particle $A_0$ the scalar potential or electric potential and $\mathcal{L}{0}$ is the Lagrangian for a single particle under the potential $U(\textbf{r})$, to derive the equations of motion he uses <PERSON> principle with the action $$S_{0i}=\int_{t_1}^{t_2} \mathcal{L}{oi}dt$$ and arrives to $$m\ddot{\textbf{r}}=-\nabla U+q(-\nabla A_0-\frac{\partial \textbf{A}}{\partial (ct)})+\frac{q}{c}\dot{\textbf{r}} \times (\nabla \times \textbf{A})$$ im not sure how he arrived to that equation, if he used <PERSON> principle then the Lagrangian should satisfy the equation (im abussing notation here to clariffy that the author varied the action with respect to $\textbf{r}$ ) $$\nabla{\textbf{r}}\mathcal{L}{0i}-\frac{d}{dt}(\nabla{\dot{\textbf{r}}}\mathcal{L}{0i})=0$$ but im stuck here, not for the lagrangian $\mathcal{L}_0$ but for the other part of the expression, i think he integrated and the lagrangian has the form $$\mathcal{L} {0i}=\mathcal{L_0}+q \dot{\textbf{r}} \cdot \textbf{A}-qA_0$$ and used the identity $$\nabla_{\dot{\textbf{r}}}(\dot{\textbf{r}} \cdot \textbf{A})=\textbf{A} \times (\nabla_{\dot{\textbf{r}}} \times \dot{\textbf{r}})+\dot{\textbf{r}} \times (\nabla_{\dot{\textbf{r}}} \times \textbf{A})+(\textbf{A} \cdot \nabla_{\dot{\textbf{r}}})\dot{\textbf{r}}+(\dot{\textbf{r}} \cdot \nabla_{\dot{\textbf{r}}})\textbf{A}$$ and that should equal $\textbf{A}$ cause the vector potential only depends on $\textbf{r},t$ and took the time derivative, also $A_0(\textbf{r},t)$ but i dont see how he arrived to the rest of the equation. Sorry for the possible bad english hope I have made myself understand. 649