Editing Three-body problem (section)

==Other problems involving three bodies==

The term "three-body problem" is sometimes used in the more general sense to refer to any physical problem involving the interaction of three bodies.

A quantum-mechanical analogue of the gravitational three-body problem in classical mechanics is the [[helium atom]],
in which a [[helium]] nucleus and two [[electrons]] interact according to the [[inverse-square]] [[Coulomb interaction]]. Like the
gravitational three-body problem, the helium atom cannot be solved exactly.<ref>{{cite book | author=Griffiths, David J. | author-link=David J. Griffiths | title=Introduction to Quantum Mechanics (2nd ed.) | publisher=Prentice Hall |year=2004 |isbn=978-0-13-111892-8 | oclc=40251748 |page=311}}</ref>

In both classical and quantum mechanics, however, there exist nontrivial interaction laws besides the inverse-square force that do lead to exact analytic three-body solutions. One such model consists of a combination of [[Harmonic oscillator|harmonic attraction]] and a repulsive inverse-cube force.<ref name="Crandall1984">{{cite journal |last1=Crandall |first1= R. |last2=Whitnell |first2=R. |last3=Bettega |first3=R. |title= Exactly soluble two-electron atomic model |journal=American Journal of Physics |volume=52 |issue=5 |pages=438–442 |year=1984 |doi=10.1119/1.13650 |bibcode=1984AmJPh..52..438C}}</ref> This model is considered nontrivial since it is associated with a set of nonlinear differential equations containing singularities (compared with, e.g., harmonic interactions alone, which lead to an easily solved system of linear differential equations). In these two respects it is analogous to (insoluble) models having Coulomb interactions, and as a result has been suggested as a tool for intuitively understanding physical systems like the helium atom.<ref name="Crandall1984"/><ref>{{cite journal |last=Calogero |first=F. |author-link=Francesco Calogero (physicist) |title=Solution of a Three-Body Problem in One Dimension |journal=Journal of Mathematical Physics |volume=10 |issue=12 |pages=2191–2196 |year=1969 |doi=10.1063/1.1664820 |bibcode= 1969JMP....10.2191C}}</ref>

Within the [[Two-dimensional point vortex gas|point vortex model]], the motion of [[Vortex|vortices]] in a two-dimensional ideal fluid is described by equations of motion that contain only first-order time derivatives. I.e. in contrast to Newtonian mechanics, it is the ''velocity'' and not the acceleration that is determined by their relative positions. As a consequence, the three-vortex problem is still [[Integrable system|integrable]],<ref>{{Cite journal |last=Aref |first=Hassan |author-link=Hassan Aref |date=1979-03-01 |title=Motion of three vortices |url=https://aip.scitation.org/doi/10.1063/1.862605 |journal=The Physics of Fluids |volume=22 |issue=3 |pages=393–400 |doi=10.1063/1.862605 |bibcode=1979PhFl...22..393A |issn=0031-9171}}</ref> while at least four vortices are required to obtain chaotic behavior.<ref>{{Cite journal |last1=Aref |first1=Hassan |author-link1=Hassan Aref |last2=Pomphrey |first2=Neil |date=1980-08-18 |title=Integrable and chaotic motions of four vortices |journal=Physics Letters A |language=en |volume=78 |issue=4 |pages=297–300 |doi=10.1016/0375-9601(80)90375-8 |bibcode=1980PhLA...78..297A |issn=0375-9601}}</ref> One can draw parallels between the motion of a passive tracer particle in the velocity field of three vortices and the restricted three-body problem of Newtonian mechanics.<ref>{{Cite journal |last1=Neufeld |first1=Z |last2=Tél |first2=T |date=1997-03-21 |title=The vortex dynamics analogue of the restricted three-body problem: advection in the field of three identical point vortices |url=https://iopscience.iop.org/article/10.1088/0305-4470/30/6/043 |journal=Journal of Physics A: Mathematical and General |volume=30 |issue=6 |pages=2263–2280 |doi=10.1088/0305-4470/30/6/043 |bibcode=1997JPhA...30.2263N |issn=0305-4470}}</ref>

The gravitational three-body problem has also been studied using [[general relativity]]. Physically, a relativistic treatment becomes necessary in systems with very strong gravitational fields, such as near the [[event horizon]] of a [[black hole]]. However, the relativistic problem is considerably more difficult than in Newtonian mechanics, and [[Numerical relativity|sophisticated numerical techniques]] are required.
Even the full [[Two-body problem in general relativity|two-body problem]] (i.e. for arbitrary ratio of masses) does not have a rigorous analytic solution in general relativity.<ref>{{cite journal | last1=Musielak | first1=Z. E. | last2=Quarles | first2=B. | title=The three-body problem | journal=Reports on Progress in Physics | volume=77 | issue=6 | page=065901 | year=2014 | issn=0034-4885 | doi=10.1088/0034-4885/77/6/065901 | pmid=24913140| arxiv=1508.02312 | bibcode=2014RPPh...77f5901M | s2cid=38140668 }}</ref>