### 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.