Physical meaning of magnetic vector potential Can anyone give me an intuition of why the magnetic vector potential $A$ is sometimes interpreted  as the electromagnetic momentum ?
I don't know analytical mechanics, just classical electromagnetism  
 A: In Lagrangian mechanics, one usually introduces the "generalized momentum" as 
$$
p_{\phi_i}:=\frac{\partial L}{\partial \phi_i}
$$where $L$ is the Lagrangian depending on a bunch of generalized coordinates $\phi_1\dots\phi_n$. If you introduce electromagnetic forces, you can derive, that the Lagrangian in the presence of an electromagnetic field has a form like
$$
L=\frac{m}{2}\left(\vec{p}-\frac{q}{c}\vec{A}\right)^2+q\Phi
$$
where $A$ is your vector potential and $\Phi$ your scalar potential, therefore your generalized momentum will be shifted by $A$, i.e. if you write down the respective Newton equation, instead of $\dot{p}$ you will have this shifted by $A$.
The physical meaning behind this formal stuff is, that the electromagnetic field actually carries a momentum, which can be translated into a mechanical momentum. For example, I remember an experiment, where you put two spherical shells into a magnetic field and then turn the field off, the shells will get an angular momentum (I searched for it, didnt find it yet, will update if found).
