Quantum Mechanics for Engineers 

© Leon van Dommelen 

D.6 Lorentz force derivation
To derive the given Lorentz force from the given Lagrangian, plug the
canonical momentum and the Lagrangian into the Lagrangian equation of
motion. That gives
This uses the Einstein convention that summation over is to be
understood. Reorder to get
The first parenthetical expression is the electric field as claimed.
The quantity in the second parenthetical expression may be rewritten by
expanding out the sums over to give
where follows in the cyclic sequence
and precedes it. The first two
terms drop out and the others can be recognized as component number
of . (For example,
just write out the first component of
and compare it the expression above for
2 and 3.) Defining as
, the Lorentz force law results.