A galilean puzzle in electromagnetism

[phoenix85]

A "galilean puzzle" in electromagnetism

Well, I still didn't managed to find the answer, since the puzzle seems a little bit more involved than I first tought...to perform the computations, you only need to know a little bit about lagrangian/hamiltonian dynamics in electromagnetism.

Consider a charged particle moving in a plane perpendicular to a homogeneous (constant) magnetic field (B=[0,0,B]), and with an electric field lying in the plane (E=[Ex,Ey,0]). One can write down the related lagrangian (using the symmetric gauge) and get the equations of motion. If the electric field was not present, we get the cyclotron motion, but in presence of such an electric field, one can show that the particle will be drifting with the speed of the "guiding center", v=ExB/B².

Now, the question : compute the transformed lagrangian under the galilean boost defined by the speed of this guiding center, in order to simplify the electric field components. Will such a galilean boost eliminate the electric field terms from the lagrangian ?

Well, I tried to do it on the paper, but from the galilean boost I get a term depending on time in the new lagrangian (because of the magnetic potential) !?

 

[Bob_for_short]

Well, I still didn't managed to find the answer, since the puzzle seems a little bit more involved than I first tought...to perform the computations, you only need to know a little bit about lagrangian/hamiltonian dynamics in electromagnetism.

Consider a charged particle moving in a plane perpendicular to a homogeneous (constant) magnetic field (B=[0,0,B]), and with an electric field lying in the plane (E=[Ex,Ey,0]). One can write down the related lagrangian (using the symmetric gauge) and get the equations of motion. If the electric field was not present, we get the cyclotron motion, but in presence of such an electric field, one can show that the particle will be drifting with the speed of the "guiding center", v=ExB/B².

Now, the question : compute the transformed lagrangian under the galilean boost defined by the speed of this guiding center, in order to simplify the electric field components. Will such a galilean boost eliminate the electric field terms from the lagrangian ?

Well, I tried to do it on the paper, but from the galilean boost I get a term depending on time in the new lagrangian (because of the magnetic potential) !?

To obtain a complete cancelling you have to make the Lorentz transformation (not Galilean) that involves the time variable too.

Bob.

 

[phoenix85]

To obtain a complete cancelling you have to make the Lorentz transformation (not Galilean) that involves the time variable too.

Bob.

This is also what I thought, but I've been told that the initial lagrangian is galilean invariant, since this is a non-relativistic system. To be more precise about that, when computing the new lagrangian, one gets the old lagrangian plus the total time derivative of a given scalar function (the latter is relevant to enforce the invariance of the Schrödinger equation under the galilean boosts, since the function itself will have to appear as a phase into the transformed wave function).

 

[Bob_for_short]

This is also what I thought, but I've been told that the initial lagrangian is galilean invariant, since this is a non-relativistic system. To be more precise about that, when computing the new lagrangian, one gets the old lagrangian plus the total time derivative of a given scalar function (the latter is relevant to enforce the invariance of the Schrödinger equation under the galilean boosts, since the function itself will have to appear as a phase into the transformed wave function).

Then, maybe, the additional term you get could be neglected in the non-relativistic limit.

Bob.

P.S. If the curve is not a perfect spiral, you cannot eliminate the electric field, I am afraid.