One of the standard topics in classical mechanics, at any level, is that of Kepler's laws of planetary motion. This topic rests on Newton's law of universal gravitation,
F12 = −G m1 m2 r12/r123,
for the gravitational force on mass m1 due to mass m2, given a separation
r12 = |r1 − r2| = (r12 +
r22 − 2r1·r2)1/2,
and where G = 6.67 × 10−11Nm2/kg2. The force is attractive and lies
along the line separating the two masses.
Kepler's laws can be simply expressed as (1) planets move in ellipses with the
Sun at one focus (2) planets sweep out equal areas in equal times as it orbits
the Sun, and (3) the square of the period of a planet's orbit is proportional to
the cube of the semimajor axis, a, of its orbit: T2
= (4π2/GM)
a3.
Introductory treatments of Kepler's laws focus on the special cases of two-body
systems where the central object is much more massive than the orbiting body,
which itself usually orbits in circular motion. More advanced treatments
includes the general study of elliptical orbits and occasionally cases where the
objects are nearly the same mass. Less frequently, systems with more than
two objects are considered, and for good reason. These systems are, in
general, not exactly solvable. However, as we shall see, there are special
cases in which the three-, four-, and five-body problems can be solved.