Editing Euler's three-body problem (section)

{{Short description|Solve for a particle's motion that is acted on by the gravitational field of two other point masses}}{{Primary sources|date=October 2024}}

In [[physics]] and [[astronomy]], '''Euler's three-body problem''' is to solve for the motion of a particle that is acted upon by the [[gravitational field]] of two other point masses that are fixed in space. It is a particular version of the [[three-body problem]]. This version of it is exactly solvable, and yields an approximate solution for particles moving in the gravitational fields of prolate and oblate [[Spheroid|spheroids]]. This problem is named after [[Leonhard Euler]], who discussed it in memoirs published in 1760.  Important extensions and analyses to the three body problem were contributed subsequently by [[Joseph-Louis Lagrange]], [[Joseph Liouville]], [[Pierre-Simon Laplace]], [[Carl Gustav Jacob Jacobi]], [[Urbain Le Verrier]], [[William Rowan Hamilton]], [[Henri Poincaré]] and [[George David Birkhoff]], among others.<ref name="Murray">{{cite book |title=Solar System Dynamics |author1=[[Carl D. Murray]] |author2=Stanley F. Dermott |page=Chapter 3 |url=https://books.google.com/books?id=aU6vcy5L8GAC&q=%22restricted+three-body+problem%22&pg=PA63
|isbn=978-0-521-57597-3 |year=2000 |publisher=Cambridge University Press |no-pp=true}}</ref>
The Euler three-body problem is known by a variety of names, such as the '''problem of two fixed centers''', the '''Euler–Jacobi problem''', and the '''two-center Kepler problem'''.  The exact solution, in the full three dimensional case, can be expressed in terms of [[Weierstrass's elliptic functions]]<ref>{{cite journal |author1=Francesco Biscani |author2=Dario Izzo |year=2015 |title=A complete and explicit solution to the three-dimensional problem of two fixed centres |journal=Monthly Notices of the Royal Astronomical Society |volume=455 |issue=4 |pages=3480–3493 |arxiv=1510.07959 |doi=10.1093/mnras/stv2512 |doi-access=free}}</ref>  For convenience, the problem may also be solved by numerical methods, such as [[Runge–Kutta methods|Runge–Kutta integration]] of the equations of motion.  The total energy of the moving particle is conserved, but its [[linear momentum|linear]] and [[angular momentum]] are not, since the two fixed centers can apply a net force and torque.  Nevertheless, the particle has a second conserved quantity that corresponds to the [[angular momentum]] or to the [[Laplace–Runge–Lenz vector]] as [[limiting case (mathematics)|limiting case]]s.

Euler's problem also covers the case when the particle is acted upon by other inverse-square [[central force]]s, such as the [[electrostatics|electrostatic interaction]] described by [[Coulomb's law]].  The classical solutions of the Euler problem have been used to study [[Chemical bond|chemical bonding]], using a [[Semiclassical physics|semiclassical approximation]] of the energy levels of a single electron moving in the field of two atomic nuclei, such as the diatomic ion HeH<sup>2+</sup>.  This was first done by [[Wolfgang Pauli]] in 1921 in his doctoral dissertation under [[Arnold Sommerfeld]], a study of the first ion of molecular hydrogen, namely the [[Dihydrogen cation|hydrogen molecular ion]] H<sub>2</sub><sup>+</sup>.<ref name="pauli_1922">{{cite journal | author = Pauli W | year = 1922 | title = Über das Modell des Wasserstoffmolekülions | journal = Annalen der Physik | volume = 68 | issue = 11 | pages = 177&ndash;240| doi = 10.1002/andp.19223731102 |bibcode = 1922AnP...373..177P | author-link = Wolfgang Pauli }}</ref>  These energy levels can be calculated with reasonable accuracy using the [[Einstein–Brillouin–Keller method]], which is also the basis of the [[Bohr model]] of atomic hydrogen.<ref name="knudson_2006">{{cite journal | author = Knudson SK | year = 2006 | title = The Old Quantum Theory for H<sub>2</sub><sup>+</sup>: Some Chemical Implications | journal = Journal of Chemical Education | volume = 83 | issue = 3 | pages = 464–472 | doi = 10.1021/ed083p464|bibcode = 2006JChEd..83..464K }}</ref><ref name="strand_1979">{{cite journal |vauthors=Strand MP, Reinhardt WP | year = 1979 | title = Semiclassical quantization of the low lying electronic states of H<sub>2</sub><sup>+</sup> | journal = Journal of Chemical Physics | volume = 70 | issue = 8 | pages = 3812–3827 | doi = 10.1063/1.437932|bibcode = 1979JChPh..70.3812S }}</ref> More recently, as explained further in the quantum-mechanical version, analytical solutions to the eigenvalues (energies) have been obtained: these are a ''generalization'' of the [[Lambert W function]]. 

Various generalizations of Euler's problem are known; these generalizations add linear and inverse cubic forces and up to five centers of force.  Special cases of these generalized problems include Darboux's problem<ref name="darboux">[[Jean Gaston Darboux|Darboux JG]], ''Archives Néerlandaises des Sciences'' (ser. 2), '''6''', 371–376</ref> and Velde's problem.<ref name="velde">Velde (1889) ''Programm der ersten Höheren Bürgerschule zu Berlin''</ref>