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 $$\begin{equation} \label{eq:Lagrangian-relativistic-EM} L(q,\dot{q}) = m|{\dot{q}}| + eA_i\dot{q}^i \end{equation}$$ so we can work out the Euler–Lagrange equations: $$\begin{align*} p_i = \frac{\partial L}{\partial \dot{q}^i} &= m\frac{\dot{q}_i}{|{\dot{q}}|} + eA_i\\ &= m v_i + e \,A_i \end{align*}$$ 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 $$\begin{align*} F_i = \frac{\partial L}{\partial q^i} &= \frac{\partial}{\partial q^i}\Bigl(e\,A_j\dot{q}^j\Bigr)\\ &= e\frac{\partial A_j}{\partial q^i} \dot{q}^j \end{align*}$$ So the Euler-Lagrange equations give us (using $A_i=A_j\Bigl(q(t)\Bigr)$: $$\begin{align*} \dot{p} &= F \\ \frac{d}{dt}\Bigl(mv_i+eA_i\Bigr) &= e\frac{\partial A_j}{\partial q^i}\dot{q}^j\\ m\frac{d v_i}{dt} &= e\frac{\partial A_j}{\partial q^i}\dot{q}^j - e\frac{d A_i}{dt}\\ m\frac{d v_i}{dt} &= e\frac{\partial A_j}{\partial q^i}\dot{q}^j - e\frac{\partial A_i}{\partial q^j}\dot{q}^j\\ &= e\left(\frac{\partial A_j}{\partial q^i} - \frac{\partial A_i}{\partial q^j}\right)\dot{q}^j . \end{align*}$$ Here, term in parentheses is $F_{ij}=$ the electromagnetic field, $F=dA$. Therefore, the equations of motion are

$$\begin{equation} m\frac{d v_i}{dt} = e F_{ij}\dot{q}^j, \end{equation}$$

which we call the Lorentz law.

Abstract

The classical mechanics of an electron propagating in an electromagnetic field on a spacetime X is all encoded in a differential 2-form on X, called the Faraday tensor F, which encodes the classical Lorentz force that the electromagnetic field exerts on the electron.https://arxiv.org/abs/1601.05956

For a derivation, using the Ehrenfest theorem, see http://www.physics.drexel.edu/~bob/PHYS517/Ehrenfest.pdf

Why is it interesting?