$V_{potential}$ is the potential energy. It can also be written $V_{potential}(r_{radius})$ because the potential energy is a function with only one factor contributing to it's changes: that of distance.

radius Spherical polar coords tell us that the distance from a (charged, and/or massive for classical physics) particle is equal to the radius of a sphere with the origin at the particle and the spherical shell intersecting the point of interest.

So, the potential is the simple sum of all the little pieces of mass, denoted with the index $i$: $$V_{potential} = - \frac{GMm_i}{r_i}$$ The $G$ is the gravitational constant and $M$ is the mass of any other astronomical body, including Earth satellites. The problem at hand depends on the shape of the Earth.

Diagram of $r_i$ along with break down of center of mass. In rotations, and Earth is rotating in more ways than one, there are tools to calculate rates. Center-of-mass is one tool, and principle moments of inertia is another.

Law of cosines break down into trig identities & defs.

Legendre polynomials are a mathematical discovery which we use

From Section 5-8 of [Goldstein]: