Editing Euler's three-body problem (section)

==Overview and history==
Euler's three-body problem is to describe the motion of a particle under the influence of two centers that attract the particle with [[central force]]s that decrease with distance as an [[inverse-square law]], such as [[gravitation|Newtonian gravity]] or [[Coulomb's law]].  Examples of Euler's problem include an [[electron]] moving in the [[electric field]] of two [[atomic nucleus|nuclei]], such as the [[hydrogen molecule-ion]] {{chem2|H2+}}.  The strength of the two inverse-square forces need not be equal; for illustration, the two nuclei may have different charges, as in the molecular ion HeH<sup>2+</sup>.

In Euler's three-body problem we assume that the two centres of attraction are stationary. This is not strictly true in a case like {{chem2|H2+}}, but the protons experience much less acceleration than the electron. However, the Euler three-body problem does not apply to a [[planet]] moving in the gravitational field of two [[star]]s, because in that case at least one of the stars experiences acceleration similar to that experienced by the planet.

This problem was first considered by [[Leonhard Euler]], who showed that it had an exact solution in 1760.<ref name="euler_1760">[[Leonhard Euler|Euler L]], ''Nov. Comm. Acad. Imp. Petropolitanae'', '''10''', pp. 207&ndash;242, '''11''', pp. 152&ndash;184; ''Mémoires de l'Acad. de Berlin'', '''11''', 228&ndash;249.</ref>  [[Joseph Louis Lagrange]] solved a generalized problem in which the centers exert both linear and inverse-square forces.<ref name="lagrange" >[[Joseph Louis Lagrange|Lagrange JL]], ''Miscellanea Taurinensia'', '''4''', 118&ndash;243; ''Oeuvres'', '''2''', pp. 67&ndash;121; ''Mécanique Analytique'', 1st edition, pp. 262&ndash;286; 2nd edition, '''2''', pp. 108&ndash;121; ''Oeuvres'', '''12''', pp. 101&ndash;114.</ref>  [[Carl Gustav Jacob Jacobi]] showed that the rotation of the particle about the axis of the two fixed centers could be separated out, reducing the general three-dimensional problem to the planar problem.<ref name="jacobi">[[Carl Gustav Jacob Jacobi|Jacobi CGJ]], ''Vorlesungen ueber Dynamik'', no. 29.  ''Werke'', Supplement, pp. 221&ndash;231</ref>

In 2008, Diarmuid Ó Mathúna published a book entitled "Integrable Systems in Celestial Mechanics". In this book, he gives closed form solutions for both the planar two fixed centers problem and the three dimensional problem.<ref>{{Cite book |last=Ó'Mathúna |first=Diarmuid |url=https://books.google.com/books?id=oP1FBz3Y8_wC |title=Integrable Systems in Celestial Mechanics |date=2008-12-15 |publisher=Springer Science & Business Media |isbn=978-0-8176-4595-3 |language=en}}</ref>