Chapter 4: The Success of Celestial Mechanics

Despite the absence of exact general solutions, the magicians who practice celestial mechanics have obtained remarkable results using perturbation theory. In this setting, when the differential equations are expanded in series, a particular class of terms is singled out as particularly problematic: the secular terms. They have very little effect over the short term but as time evolves, their contributions grow continually and, in the long run, these terms make preponderant contributions. This property means it is impossible to make long term predictions of planetary trajectories. Worse, these trajectories are extremely sensitive to initial conditions. These are the two most important properties of chaotic motion.