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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?

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Newtonian gravity does not have any forces that are not radially inward or outward, so if you are in pure Newtonian gravity, there is no generalization of magnetism.

Now, if you've seen a fancy derivation of magnetism using only electricity and special relativity, you would expect that something similar would be true in general relativity, and it does show up, in the form of the Lense-thirring effect, and works very similarly to magnetism.

You do, of course, also get gravitational waves, but figuring out exactly what waves, and the directions of the force, etc, are different in the details from electromagnetic waves, because gravitational radiation has spin 2, and couples to the quadrupole moment of the matter distribution, rather than having EM radiation spin 1 and coupling to the dipole moment of the charge distribution. So, ultimately, the equations you get aren't just perfect copies of the maxwell equations, the way that Gauss's Law and Newton's Law are.

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