Hamiltonian for a free electron in electromagnetic field

[athosanian]

hello, how to derive the hamiltonian for a free electron in electromagnetic field mathematically ?
for a first step what is the lagrangian for a free electron in the EM field in classical mechanics ?
the physics textbook always like to give the results directly.

 

[blue_leaf77]

Hope it helps:
http://www.tcm.phy.cam.ac.uk/~bds10/aqp/lec5_compressed.pdf

 

[athosanian]

thanks, I understand how to derive the Hamiltonian from Lagrangian.,
however, the file gives the Lagrangian directly without any reason.

 

[blue_leaf77]

Up to now, all literatures I have read always deduce the Lagrangian from the equation of motion (Lorentz force law) by, apparently, inspection - kind of trying to seek for the right form of L and check it by plugging it into the equation of motion of L. I have never seen a reference doing the reverse way, that is deriving the Lagrangian from the energies like we usually do in systems involving gravitational potential only. Perhaps there are such references but I just never come across them. Anyway, just for a check you can plug in the final form of the Lagrangian you found there into the Lagrange-Euler equation (the equation of motion), I think you should end up with the expression of Lorenty force.

 

[athosanian]

yes, I put the L into the E-L equation, could derive the Newton's equaiton of motion. But I think it is very smart to work out the lagrangian.