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It's not quite clear how you jump from step 3 to step 4... $\vec{r}\cdot \dot{\vec{r}} = r\dot{r}$ This identity does not always hold. @Loonuh
After further examining the original question, and the source for the question, which was in the book "Electromagnetic Theory" by Ferraro on p.543, I was able to understand the conserved quantity $\vec{J}$ as thus. Considering that with $\vec{B}(\vec{r}) = \frac{g\vec{r}}{r^3}$ the Lorentz force yields: \$\begin{align} m\ddot{\vec{r}} &= q\dot{\vec{r}} ...