Magnetism and lagrange formulation of mechanics

[Nick R]

Hello, I am aware that magnetic forces can do no work.

I am also aware that, in a conservative system, equations of motion that minimize the "action" (which are the true equations of motion) can be found with the euler-lagrange equation. The only information the euler-lagrange equation needs about the system is the potential energy as a function of position and kinetic energy as a function of the first time derivative of position.

So take a system in which there is a steady current (in some direction) flowing to infinity along some axis, and a charged particle moving orthonally to the steady current.

The charged particle will experience a central force (magnetic) that will cause it to orbit the current at some radius.

So my question is, how is it possible for the lagrangian formuation of mechanics to account for this simple sitauation? The magnetic force can do no work, so it can't affect the potential energy field (right?), and the kinetic energy is always just (1/2)mv^2. It would seem that the lagrangian can contain no information relevant to the magnetic force (the lagrangian being defined as T - U ; which is the only item having any information relevant to the physical situation in the euler-lagrange equation).

 

[DeShark]

The potential is a velocity dependant one. It is talked about more fully in another post on this forum. https://www.physicsforums.com/showthread.php?t=190047

EDIT:
It is also mentioned https://www.physicsforums.com/showthread.php?t=116098
This explains (ish) why it has this form. And also shows that the absolute value of the momentum (and hence k.e.) is conserved (i.e. no work is done)