electromagnetism - Is the canonical momentum conserved when a particle moves in magnetic field?

Here is a question about the canonical momentum that I had asked some days ago, but I still have one point that I am not understand.

Considering a particle moves in a magnetic field with charge $q$ and mass $m$, its hamiltonian is $$H=\frac{\vec{P}^2}{2m}=\frac{(\vec{p}+q\vec{A})^2}{2m}$$ where $\vec{p}$ is the momentum of the particle, $\vec{A}$ is the vector potential of the magnetic field and $\vec{P}$ is the canonical momentum of the particle.

I think, because of the expression of the hamiltonian, the canonical momentum $\vec{P}$ is a conserved quantity.

But by the answer in the previous link, it seems that the canonical momentum is not conserved even in a simple example that a particle moves in a homogeneous magnetic field.

I am confused about this question. Is the canonical momentum conserved when a particle moves in magnetic field?

Answer

As Jan noted, the Hamiltonian should have a minus sign:

$H=\frac{(p-qA)^2}{2m}$

where $p$ is the canonical momentum, and the expression $p-qA$ is the kinetic momentum $P$.

A homogenous magnetic field is an interesting case, because the vector potential in a given gauge does not exhibit translation invariance, but the physical system clearly does. The solution to this dilemma is that you can preserve translational invariance by changing the gauge as you move the coordinates.

There is a conserved quantity associated with this symmetry, but it turns out not to be the canonical momentum (or the kinetic momentum, either). I don't know if it has a particular name, and since it is gauge-dependent there is no universal expression you can write for it. But, for example, in the gauge where $A=\frac{B}{2}(-y,x)$, it is just $(p+qA)$. If anyone has an insight into any physical interpretation of this quantity I would be interested to hear it.

There is a very nice example of all this, worked out concretely, in these notes.