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.