 Solution

Interpret $\mathbf{P}=0$ as the zero-pseudomomentum sector, rather than assuming the mechanical centre of mass is at rest. Define $\mathbf{u}=\mathbf{p}+e^{\ast }\mathbf{B}\times \mathbf{r}$. Hamilton's equations areFor example, differentiating the relative kinetic term with respect to $\mathbf{r}$ gives $-e^{\ast }\mathbf{B}\times \mathbf{u}/\mu$ because the cross-product matrix is antisymmetric. Hamilton's minus sign then gives the positive cross-product term in $\dot{\mathbf{p}}$. The Coulomb force is attractive. A change to a moving frame need not preserve a pure magnetic external field, so setting $\mathbf{P}=0$ is a restriction on the chosen solution family.