# Gravity Differential Equations

I was just messing around with Newton's Law of gravitation, when I had the idea of converting Newton's Law into differential form (more or less like Maxwell's equations).

I did the following:

##### #1 Divergence of the field:

$$\iint_C {\mathbf g \cdot d\mathbf S} = -4\pi G M \rightarrow \ \iint_C {\mathbf g \cdot d\mathbf S} = \iiint_C {-4\pi G \rho \; d\mathbf V} \\ \iiint_C {\nabla \cdot \mathbf g \; dV} = \iiint_C {-4\pi G \rho \; dV} \\$$ $$\boxed{ \begin{array}{rcl} \nabla \cdot \mathbf g = -4\pi G \rho \end{array} }$$

##### #2 Curl of the field:

$$\mathbf g = -\nabla \phi \\ \nabla \times \mathbf g = \nabla \times (-\nabla \phi) = \mathbf 0$$ $$\boxed{ \begin{array}{rcl} \nabla \times \mathbf g = \mathbf 0 \end{array} }$$

Until now everithing is fine. Now I thinked if it was possible writting the equtions in terms of other field, like the velocity field $$\mathbf v$$, but I'm stucked.

It is well known that the velocity on an orbit obeys the following: $$v = \sqrt{\frac{GM}{r}} \\ v \propto \frac{1}{\sqrt{r}}$$

It is possible to express how this velocity field must behave to incorporates all we did knew about Newtonian Gravity using only vectors and vector calculus? In other words, it is possible to formulate gravity from its velocity field, using vectors? For example: $$\mathbf g = \frac{d \mathbf v}{dt} \\ \nabla \times \mathbf v = \gamma_0\mathbf L$$

Where $$\mathbf L$$, is the angular momentum, and then, if you rearange the equations, you could get the velocity which an object will have at a certain height, or find the escape velocity of a planet (maybe this is too idyllic). Also, would be possible to create a set of equations in which this Newtonian gravity has solutions of a wave equation (in a similar way that of Maxwell's Equations), resembling to the gravitational waves or GR?