Many-Body Problems

Since we have developed the equations for the two-body problem, the question naturally arises about having three or more masses in the system. Unfortunately, there are no general closed form solutions for the \(n\)-body problem when \(n > 2\). The system can still be solved numerically, but it is highly chaotic and complex.

Given this is the case, the next question that arises is, how useful is the two-body solution, really? It turns out that the easiest way to solve the many-body problem for systems of engineering interest is to solve the two-body problem and treat other gravitational influences as perturbations of the two-body problem.

For example, consider a satellite orbiting Earth. The dominant gravitational force on the satellite is from the Earth. There will also be influences from the Sun, Moon, and other planets. However, those effects can be treated as perturbations on the solution of the two-body problem involving the Earth and the satellite.