is translation invariant also in the $x$-direction, $x$ itself is not a cyclic coordinate and hence $p_x$ is not conserved. Still, using Noether’s theorem one can identify the corresponding conserved quantity as $P_x = p_x + e B y$. Interestingly, the Lagrange function is not invariant under a shift in the $x$-direction but changes by a total derivative (which leaves the classical equations of motion unchanged).

The classical Hamilton function takes the form $$\label{Hamiltonian}
H = \vec p \cdot \vec v - L = 
\frac{1}{2 M}\left[\vec p + e \vec A(\vec x)\right]^2 =
\frac{1}{2 M}\left[p_x^2 + \left(p_y + e B x\right)^2\right].$$ It is straightforward to convince oneself that $H$ has vanishing Poisson brackets, $\{H,P_x\} = \{H,P_y\} = \{H,L\} = 0$, with the three symmetry generators $$P_x = p_x + e B y, \ P_y = p_y, \
L = x \left(p_y + \frac{e B}{2} x\right) - 
y \left(p_x + \frac{e B}{2} y\right).$$ One can identify $P_x$, $P_y$, and $L$ as the gauge-covariant generators of translations and rotations. In particular, one obtains $$\{L,P_x\} = P_y, \ \{L,P_y\} = - P_x,$$ as one would expect for the rotation properties of the vector $(P_x,P_y)$. As is well-known, however, in a magnetic field the 