Do divergence and curl of Lorentz force have some physical meaning? Time ago I started thinking about this: if we take the well known Lorentz Force expression, namely
$$\mathbf{F} = q\left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right)$$
and we operate $\nabla\cdot \mathbf{F}$ and $\nabla\times\mathbf{F}$, what do we obtain? I performed the calculations, but I have to say that
1) I don't know if they are exact (at least until where I stopped)
2) I ignore their physical meanings
The question is
Do they have some interpretation? 
Calculation of $\nabla\cdot\mathbf{F}$
$$
\begin{align*}
\nabla\cdot\mathbf{F} & = \nabla\cdot\left(q\left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right)\right)
\\\\
& = q\nabla\cdot\mathbf{E} + q\nabla\cdot (\mathbf{v}\times\mathbf{B})
\end{align*}
$$
Using Maxwell equations, and considering $\nabla\times\mathbf{v} = \mathbf{\Omega}$ as the vorticity (that should be the definition of the curl of a velocity) we gain
$$\boxed{\nabla\cdot\mathbf{F} = \frac{q\rho}{\epsilon_0} + q\mathbf{\Omega}\cdot\mathbf{B} - q\mu_0 \mathbf{v}\cdot \left(\mathbf{J} + \epsilon_0\frac{\partial \mathbf{E}}{\partial t}\right)}$$
Calculation of $\nabla\times\mathbf{F}$
$$
\begin{align*}
\nabla\times\mathbf{F} & = q\left[\nabla\times\mathbf{E} + \nabla\times(\mathbf{v}\times\mathbf{B}\right]
\\\\
& = q\nabla\times\mathbf{E} + q\left[\mathbf{v}(\nabla\cdot\mathbf{B}) - \mathbf{B}(\nabla\cdot \mathbf{v}) + (\mathbf{B}\cdot\nabla)\mathbf{v} - (\mathbf{v}\cdot\nabla)\mathbf{B}\right]
\end{align*}
$$
Again, using Maxwell equations, and defining (if it has any sense..) $\mathbf{a} = \nabla\cdot \mathbf{v}$, we get:
$$\boxed{\nabla\times\mathbf{F} = -q\frac{\partial\mathbf{B}}{\partial t} + q\left[-\mathbf{B}\mathbf{a} + (\mathbf{B}\cdot\nabla)\mathbf{v} - (\mathbf{v}\cdot\nabla)\mathbf{B}\right]}$$
Epilogue
So, if all of that is correct, what now?
EDIT
Thanks to a page linked in the comment I understood that (thanks Rob Jeffires):
From the solenoidal law $\nabla \cdot {\bf B}=0$ always, and $\nabla \cdot {\bf v} = \partial/\partial t(\nabla \cdot {\bf r})=0$. Furthermore, $({\bf B}\cdot \nabla){\bf v} = ({\bf B}\cdot \frac{\partial}{\partial t} \nabla){\bf r} = 0$, so
$$ \nabla \times {\bf F} =  - q\left[\frac{\partial {\bf B}}{\partial t} + \frac{\partial {\bf B}}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial {\bf B}}{\partial y} \frac{\partial y}{\partial t} + \frac{\partial {\bf B}}{\partial z} \frac{\partial z}{\partial t}\right] $$
$$\nabla \times {\bf F} =  - q\frac{d {\bf B}}{d t}$$
and the force is only conservative in the case of stationary magnetic (and hence electric) fields. 
Now again:


*

*Can we solve this equation to obtain a different form for $\mathbf{F}$?

*What about the divergence?
 A: Yes, both of these quantities have physical interpretations. The divergence of the Lorentz force, $\nabla \cdot \mathbf{F}$, represents the rate of change of the total energy of a charged particle system in an electromagnetic field. The first term on the right-hand side, $\frac{q\rho}{\epsilon_0}$, is proportional to the charge density of the system and represents the change in the electric field energy. The second term, $q\mathbf{\Omega} \cdot \mathbf{B}$, is proportional to the vorticity of the system and the magnitude of the magnetic field, and represents the change in the magnetic field energy due to the motion of the charges. The third term, $-q\mu_0 \mathbf{v} \cdot \left(\mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right)$, represents the rate of work being done on the charges due to the electromagnetic field.
The curl of the Lorentz force, $\nabla \times \mathbf{F}$, represents the force due to the change in the magnetic field. The first term on the right-hand side, $-q\frac{\partial \mathbf{B}}{\partial t}$, represents the rate of change of the magnetic field energy, which produces a force proportional to the rate of change of the magnetic field. The second term, $q\left[-\mathbf{B}\mathbf{a} + (\mathbf{B} \cdot \nabla)\mathbf{v} - (\mathbf{v} \cdot \nabla)\mathbf{B}\right]$, represents the force due to the interaction between the magnetic field and the motion of the charges. The first term, $-\mathbf{B}\mathbf{a}$, is proportional to the magnitude of the magnetic field and the acceleration of the charges. The second and third terms, $(\mathbf{B} \cdot \nabla)\mathbf{v}$ and $(\mathbf{v} \cdot \nabla)\mathbf{B}$, respectively, represent the change in the magnetic field due to the motion of the charges and the change in the velocity due to the magnetic field.
In summary, the Lorentz force, its divergence, and its curl have physical interpretations that can be understood in terms of the electric and magnetic fields, the motion of charges, and the energy of the system.
