Path integral for a particle coupled to a magnetic field

Hi all,

I am currently having trouble with an exercise: writing the propagator of a particle coupled to a magnetic field.

So the lagrangian is [itex]L_A (\vec{x},\dot{\vec{x}}^2) = \frac{m}{2}\dot{\vec{x}} + e\vec{A}.\dot{\vec{x}}[/itex]
And it says that I should solve it in two different ways:
-by writing [itex]e\vec{A}.\dot{\vec{x}}[/itex] has a derivative relative to t
-by completing the square in the Lagrangian, ie something like [itex]L_A (\vec{x},\dot{\vec{x}}) = \frac{m}{2}(\dot{\vec{x}}+\frac{e}{m}\vec{A})^2 - \frac{e^2}{2m}\vec{A}^2[/itex] I guess.

But still, I develop a few lines more of calculus but I am stuck. I would be really happy if someone could explain how to proceed.

 

Yes, it does look good. But in the second lagrangian, because we don't want to have an electrical field, we have [itex]\dot{\vec{A}}=0[/itex] so the second terms vanishes.