Neutral Coulomb pair in a uniform magnetic field ($\mathbf{K}=M\dot{\mathbf{R}}-e\mathbf{B}\times \mathbf{r}$)

For charges $+e,-e$ and relative coordinate $\mathbf{r}={\mathbf{r}}_2-{\mathbf{r}}_1$, total magnetic pseudomomentum is conserved in a uniform magnetic field. In an appropriate symmetric-gauge Lagrangian differing by a total derivative, it is the canonical centre-of-mass momentum. Mechanical centre-of-mass momentum instead equals $\mathbf{K}+e\mathbf{B}\times \mathbf{r}$ and need not be constant. The sector $\mathbf{K}=0$ is not generally a stationary-centre-of-mass frame.

In the zero-pseudomomentum sector, a planar relative circular orbit has two signed angular-frequency branches. Here $\mu =m_1{m}_2/(m_1+m_2)$, $e^{\ast }=e(m_1-m_2)/[2(m_1+m_2)]$, $R$ is separation, and the Coulomb constant is absorbed into $e^2$. The centre of mass also circles: ${\mathbf{R}}_{\mathrm{cm}}={\mathbf{R}}_0+eB\mathbf{r}/[(m_1+m_2)\omega ]$. This distinction is essential when reconstructing individual particle orbits.