Intuitive
The Lorentz force law completes classical electromagnetic and describes the effect of electric and magnetic fields on a point charge.
In addition, the Maxwell equations tells us how charges give rise to electric and magnetic fields.
Concrete
Derivation
The Lagrangian for a charge e with mass m in an electromagnetic potential A is
L(q,˙q)=m|˙q|+eAi˙qi
so we can work out the Euler–Lagrange equations:
pi=∂L∂˙qi=m˙qi|˙q|+eAi=mvi+eAi
where v is the velocity, which we normalize such that |v|=1. An important observation is that here momentum is no longer simply mass times velocity! Continuing the analysis, we find the force
Fi=∂L∂qi=∂∂qi(eAj˙qj)=e∂Aj∂qi˙qj
So the Euler-Lagrange equations give us (using Ai=Aj(q(t)):
˙p=Fddt(mvi+eAi)=e∂Aj∂qi˙qjmdvidt=e∂Aj∂qi˙qj−edAidtmdvidt=e∂Aj∂qi˙qj−e∂Ai∂qj˙qj=e(∂Aj∂qi−∂Ai∂qj)˙qj.
Here, term in parentheses is Fij= the electromagnetic field, F=dA. Therefore, the equations of motion are
mdvidt=eFij˙qj,
which we call the Lorentz law.