Gravity

There are several formulations of this phenomena:

Newton's law of universal gravitation

Newton's law of universal gravitation states that the gravitational force $F$ between two point masses $m_{1}$ and $m_{2}$ is proportional to the product of their masses and inversely proportional to the square of the distance $r$ between them:

F = - G \frac{m_{1} m_{2}}{r^{2}} \hat{r}

where:

$G$ is the gravitational constant,
$r$ is the distance between the two masses,
$\hat{r}$ is a unit vector pointing from $m_{1}$ to $m_{2}$ .

Poisson formulation or Gauss's law for gravity

Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is equivalent to Newton's law of universal gravitation..
The differential form of Gauss's law for gravity states

\nabla \cdot g = - 4 π G ρ,

where $\nabla \cdot$ denotes divergence, $G$ is the universal gravitational constant, and $ρ$ is the mass density at each point.

Deriving Gauss's law from Newton's law

$g (r)$ , the gravitational field at $r$ , can be calculated by adding up the contribution to $g (r)$ due to every bit of mass in the universe (superposition principle). To do this, we integrate over every point $s$ in space, adding up the contribution to $g (r)$ associated with the mass (if any) at $s$ , where this contribution is calculated by Newton's law. The result is:

g (r) = - G \int ρ (s) \frac{r - s}{| r - s |^{3}} d^{3} s .

( $d^{3} s$ stands for $d s_{x} d s_{y} d s_{z}$ , each of which is integrated from $- \infty$ to $+ \infty$ .)

If we take the divergence of both sides of this equation with respect to $r$ , and use the known theorem:

\nabla \cdot (\frac{r}{| r |^{3}}) = 4 π δ (r),

where $δ (r)$ is the Dirac delta function, the result is:

\nabla \cdot g (r) = - 4 π G \int ρ (s) δ (r - s) d^{3} s .

Using the "sifting property" of the Dirac delta function, we arrive at:

\nabla \cdot g (r) = - 4 π G ρ (r),

which is the differential form of Gauss's law for gravity, as desired.

Deriving Newton's law from Gauss's law and irrotationality

Pending task. I have to read an adapt this Wikipedia entry...

Poisson's equation and gravitational potential

Since the gravity is a conservative force, it can be written as the gradient of a scalar potential, called the gravitational potential:

g = - \nabla ϕ .

Then the differential form of Gauss's law for gravity becomes Poisson's equation for gravity:

\begin{matrix} (*) & \nabla^{2} ϕ = Δ ϕ = 4 π G ρ, \end{matrix}

where $Δ$ is the Laplacian operator.

This provides an alternate means of calculating the gravitational potential and gravitational field. Although computing $g$ via Poisson's equation is mathematically equivalent to computing $g$ directly from Gauss's law, one or the other approach may be an easier computation in a given situation.

Newton--Cartan gravitation

See Newton-Cartan gravity.