Editing Euler's three-body problem (section)

==Generalizations==
An exhaustive analysis of the soluble generalizations of Euler's three-body problem was carried out by Adam Hiltebeitel in 1911.  The simplest generalization of Euler's three-body problem is to add a third center of force midway between the original two centers, that exerts only a [[Hooke's law|linear Hooke force]]. The next generalization is to augment the inverse-square force laws with a force that increases linearly with distance. The final set of generalizations is to add two fixed centers of force at positions that are [[imaginary number]]s, with forces that are both linear and [[inverse-square law]]s, together with a force parallel to the axis of imaginary centers and varying as the inverse cube of the distance to that axis.

The solution to the original Euler problem is an approximate solution for the motion of a particle in the gravitational field of a prolate body, i.e., a sphere that has been elongated in one direction, such as a cigar shape.  The corresponding approximate solution for a particle moving in the field of an oblate spheroid (a sphere squashed in one direction) is obtained by making the positions of the two centers of force into [[imaginary number]]s.  The oblate spheroid solution is astronomically more important, since most planets, stars and galaxies are approximately oblate spheroids; prolate spheroids are very rare.

The analogue of the oblate case in [[general relativity]] is a [[Kerr black hole]].<ref>Clifford M. Will, Phys. Rev. Lett. 102, 061101, 2009, https://doi.org/10.1103/PhysRevLett.102.061101</ref> The geodesics around this object are known to be integrable, owing to the existence of a fourth constant of motion (in addition to energy, angular momentum, and the magnitude of four-momentum), known as the [[Carter constant]]. Euler's oblate three body problem and a Kerr black hole share the same mass moments, and this is most apparent if the metric for the latter is written in [[Kerr–Schild coordinates]].

The analogue of the oblate case augmented with a linear Hooke term is a [[Kerr–de Sitter black hole]]. As in [[Hooke's law]], the [[cosmological constant]] term depends linearly on distance from the origin, and the Kerr–de Sitter spacetime also admits a Carter-type constant quadratic in the momenta.<ref>Charalampos Markakis, Constants of motion in stationary axisymmetric gravitational fields, MNRAS (July 11, 2014) 441 (4): 2974-2985. doi: 10.1093/mnras/stu715, https://arxiv.org/abs/1202.5228</ref>