Chapter 12 Gravitation
Newton discovered the law of gravitation through his studies of planetary motion. At the time of Newton, planetary motion was understood in terms of Kepler’s three laws, which were based on the astronomical observations of Tycho Brahe. Newton was able to explain the origin of Kepler’s laws from a more fundamental perspective by solving the equation of motion subject to a gravitational force given by his law of gravitation. Newton’s successful explanation of the planetary laws of Kepler was a major achievement and led to the acceptance of his laws of mechanics and ushered in the new era of physics.
The Newtonian explanation of the planetary motion is based on the solution of the equations for the dynamics of one planet revolving around the Sun. You may recall that these problems are called two-body problems and we use the techniques of multi-particle systems to study them. It turns out that the motion of two bodies interacting with the gravitational force is an important problem that can be solved exactly and therefore serves as prototype for may other problems in physics. The dynamics of a three-body system, such as a system consisting of the Moon, the Earth and the Sun system, are too difficult to solve analytically, and the solutions of the equations of motion for these and more complicated systems have not been found yet. Much of the dynamical behaviors of the three-body system and more complicated systems are known through observations and numerical solutions.
