Two orbiting bodies can be solved exactly, but adding a third breaks everything
Predicting how two stars or a planet and its moon move around each other is a solved problem with a neat, complete answer. Add just one more body and no general solution exists except in special cases. The gulf between two and three bodies is one of physics' starkest cliffs.
The classic setup treats two massive bodies as perfect spheres that never collide, with outside forces too weak to matter. Astronomy supplies ideal examples, because pairs of stars or planets tend to move quickly, sit far apart from each other and even farther from anything else. Under gravity, each body loops around their shared centre of mass along an ellipse, unless they are fast enough to escape, in which case their paths spread along other flat conic curves.
The trick that cracks it is splitting one hard problem into two easy ones. Adding Newton's equations for both bodies gives the motion of the centre of mass, which drifts steadily because total momentum is conserved. Subtracting them gives an equation for the separation between the bodies, equivalent to one particle moving in a fixed field of force. Solve that, and both original paths follow.
A cruder shortcut treats the heavier body as fixed, which works well for a light planet circling a massive star. The heavy partner really does move far less, and the shared balance point can even lie inside it. In principle the same maths covers any inverse-square force, such as electrostatic attraction, though charged objects rarely move fast and isolated enough for it to matter in practice.
Atoms are the tempting but misleading exception. Niels Bohr's early picture of electrons orbiting a nucleus gave us the word orbital, yet electrons do not truly orbit in any meaningful sense, and treating them classically yields little insight. Understanding them requires quantum mechanics.
Source: Two-body problem
