A central force is a force between two bodies that is always directed along the line connecting the two bodies. It can be an attractive force or a repulsive force. The two most common forces in the universe are central forces, and the dynamics of a body or bodies under a central force is an important problem in classical mechanics.
The mathematics of classical central force dynamics are well known and understood. Unfortunately, there is only an analytic solution to the equations of motion for two bodies. For more than two bodies (the infamous three body problem), the problem must either be approximated or solved numerically. Often, one of the bodies in the problem is considerably more massive than the others, for example, the case of two satellites and the Earth, in which the movement of the earth under the gravitational attraction of the satellites can be neglected. In this case the central gravitational attraction of the earth can be treated as a centripetal force, just causing the satellites to orbit the Earth rather than travel in a straight line. The science of orbits is essentially central force dynamics.
The central force problem is also important in quantum mechanics. Many of the properties of electrons in atoms are direct consequences of the central nature of the force exerted by the nucleus. Here, the important feature of the central force is that it allows the angular component of the Schroedinger Equation to be free, similar to the angular component of the Laplace equation in electrostatics. This leads to the quantization of angular momentum, and the effect termed 'space quantization', both of which are important properties of atoms.
Due to the importance of gravity and the electrostatic force, central force problems appear very frequently in physics. Thus, they are well-characterised and are taught quite thoroughly in any undergraduate physics program.
This writeup is copyright 2003-2004 D.G. Roberge and is released under the Creative Commons Attribution-NoDerivs-NonCommercial licence. Details can be found at http://creativecommons.org/licenses/by-nd-nc/2.0/ .