Planetary motion and few-body problems
In projectile motion discussed in Chapter 3, gravity is considered to be constant. This is an approximation, valid only for distances small compared to the size of Earth. Now imagine we stood atop Mt. Everest and fired a cannon horizontally. At low speed, the cannonball would travel just like projectile motion. With greater firing power and ever increasing speed, the cannonball would travel further and curve less. At some critical speed, it would fly all the way around Earth and come back from behind (ignoring drag, of course). This is exactly how the Moon moves around Earth, or the planets around the Sun. In a sense, planetary motion is akin to projectile motion in that a planet falls continuously toward the Sun. We can no longer treat gravity as a constant force, however.
The motion of planets, or the heavenly bodies as they were known, is a fascinating subject. Observations of planetary motion led to the discovery of Kepler's laws which, strictly speaking, are valid only for a two-body system (the planet and the Sun). The fact that these empirical laws could be successfully explained by Newton's theory of gravity confirmed the validity and universality of the classical theory of gravity. However, beyond two-body systems, no analytic solutions are known for the general configuration, even though simple three-body systems are still at the heart of physics at all scales, such as helium (two electrons plus the nucleus), water (one ...