Charged-particle electromagnetic Lagrangian

For electromagnetic scalar and vector potentials $\varphi$ and $A$,Withthe Euler--Lagrange equations give the Lorentz force $m\ddot{r}=q(E+\dot{r}\times B)$, while the canonical momentum is $p=m\dot{r}+qA$.

The Legendre transform of the charged-particle electromagnetic Lagrangian is

Under $\varphi \mapsto \varphi -{\partial }_{t}f$ and $A\mapsto A+\nabla f$, the charged-particle Lagrangian changes byIt therefore has unchanged equations of motion by total-time-derivative invariance.

For a constant uniform magnetic field $B$, choose the symmetric gauge $A=\frac{1}{2}B\times r$. The coordinate parallel to $B$ is cyclic, so its canonical momentum is conserved. Since $A\cdot B=0$, this component is also the mechanical momentum $m\dot{r}\cdot \hat{B}$.