In Kepler and Newton we studied the basics of gravitation. Now we will explore some analytical tools that simplify the calculations involved with gravitating systems. Most of these were developed in the nineteenth century when physicists sought to solve problems involving many objects interacting under complicated conditions. One of the most important concepts that came out of their mathematical analysis was the notion of energy conservation. The idea of energy conservation necessitated the concept of potential energy, which, unlike the kinetic energy that manifests itself in motion, was considered to be the latent ability of the system to produce useful work. As we shall see, because potential energy is a scalar and not a vector, it can greatly simplify the calculation of the potential inherent in a large number of bodies at any point--the problem just reduces to summing the contributions of all the individual bodies. The force can then be found by taking the negative of the spatial derivative in the usual way (see Newton's Second Law)
We will use the concept of the gravitational potential energy to prove Newton's Shell Theorem, which asserts that a spherical mass can be treated as if all its mass were concentrated at its center for the purposes of calculating the gravitational force on an object outside it, and that a massive, thin shell exerts no gravitational force on a mass inside itself. Furthermore, we will state the Principle of Equivalence, which states that inertial mass, appearing in Newton's Second Law, is the same as the gravitational mass appearing the Universal Law of Gravitation.