Editing Two-body problem (section)

=== Gravitation and other inverse-square examples ===

The two-body problem is interesting in astronomy because pairs of astronomical objects are often moving rapidly in arbitrary directions (so their motions become interesting), widely separated from one another (so they will not collide) and even more widely separated from other objects (so outside influences will be small enough to be ignored safely).

Under the force of [[gravity]], each member of a pair of such objects will orbit their mutual center of mass in an elliptical pattern, unless they are moving fast enough to escape one another entirely, in which case their paths will diverge along other planar [[conic section]]s. If one object is very much heavier than the other, it will move far less than the other with reference to the shared center of mass. The mutual center of mass may even be inside the larger object.

For the derivation of the solutions to the problem, see [[Classical central-force problem]] or [[Kepler problem]].

In principle, the same solutions apply to macroscopic problems involving objects interacting not only through gravity, but through any other attractive [[scalar potential|scalar force field]] obeying an [[inverse-square law]], with [[Coulomb's law|electrostatic attraction]] being the obvious physical example. In practice, such problems rarely arise. Except perhaps in experimental apparatus or other specialized equipment, we rarely encounter electrostatically interacting objects which are moving fast enough, and in such a direction, as to avoid colliding, and/or which are isolated enough from their surroundings.

The [[dynamical system]] of a two-body system under the influence of torque turns out to be a [[Sturm–Liouville theory|Sturm-Liouville equation]].<ref>{{cite journal |last1=Luo |first1=Siwei |title=The Sturm-Liouville problem of two-body system |journal=Journal of Physics Communications | date=22 June 2020 |volume=4 |issue=6 |page=061001 |doi=10.1088/2399-6528/ab9c30|bibcode=2020JPhCo...4f1001L |doi-access=free }}</ref>