$ \vec F= q \vec E + q\vec v \times \vec B$
====== Lorentz Force Law ======
The Lorentz force law completes classical electromagnetic and describes the effect of electric and magnetic fields on a point charge.
In addition, the [[equations:maxwell_equations|Maxwell equations]] tells us how charges give rise to electric and magnetic fields.
**Derivation**
The [[formalisms:lagrangian_formalism|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 [[equations:euler_lagrange_equations|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.
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