celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...

and the mathematics of the -body problem, a central configuration is a system of point masses with the property that each mass is pulled by the combined

gravitational force In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...

of the system directly towards the

center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...

, with acceleration proportional to its distance from the center. Central configurations may be studied in Euclidean spaces of any dimension, although only dimensions one, two, and three are directly relevant for celestial mechanics.

Examples

For equal masses, one possible central configuration places the masses at the vertices of a

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...

(forming a

Klemperer rosette A Klemperer rosette is a gravitational system of heavier and lighter bodies orbiting in a regular repeating pattern around a common barycenter. It was first described by W. B. Klemperer in 1962, and is a special case of a central configuration. ...

), a

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...

, or a

regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, f ...

in higher dimensions. The centrality of the configuration follows from its symmetry. It is also possible to place an additional point, of arbitrary mass, at the center of mass of the system without changing its centrality. Placing three masses in an equilateral triangle, four at the vertices of a regular

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

, or more generally masses at the vertices of a regular

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

produces a central configuration even when the masses are not equal. This is the only central configuration for these masses that does not lie in a lower-dimensional subspace.

Dynamics

Under

Newton's law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...

, bodies placed at rest in a central configuration will maintain the configuration as they collapse to a collision at their center of mass. Systems of bodies in a two-dimensional central configuration can orbit stably around their center of mass, maintaining their relative positions, with circular orbits around the center of mass or in elliptical orbits with the center of mass at a focus of the ellipse. These are the only possible stable orbits in three-dimensional space in which the system of particles always remains similar to its initial configuration. More generally, any system of particles moving under Newtonian gravitation that all collide at a single point in time and space will approximate a central configuration, in the limit as time tends to the collision time. Similarly, a system of particles that eventually all escape each other at exactly the escape velocity will approximate a central configuration in the limit as time tends to infinity. And any system of particles that move under Newtonian gravitation as if they are a rigid body must do so in a central configuration. Vortices in two-dimensional

fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...

, such as large storm systems on the earth's oceans, also tend to arrange themselves in central configurations.

Enumeration

Two central configurations are considered to be equivalent if they are similar, that is, they can be transformed into each other by some combination of rotation, translation, and scaling. With this definition of equivalence, there is only one configuration of one or two points, and it is always central. In the case of three bodies, there are three one-dimensional central configurations, found by

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

. The finiteness of the set of three-point central configurations was shown by

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiathree-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...

; Lagrange showed that there is only one non-collinear central configuration, in which the three points form the vertices of an

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...

. Four points in any dimension have only finitely many central configurations. The number of configurations in this case is at least 32 and at most 8472, depending on the masses of the points. The only convex central configuration of four equal masses is a square. The only central configuration of four masses that spans three dimensions is the configuration formed by the vertices of a regular

. For arbitrarily many points in one dimension, there are again only finitely many solutions, one for each of the linear orderings (up to reversal of the ordering) of the points on a line. For every set of point masses, and every dimension less than , there exists at least one central configuration of that dimension. For almost all -tuples of masses there are finitely many "Dziobek" configurations that span exactly dimensions. It is an unsolved problem, posed by and , whether there is always a bounded number of central configurations for five or more masses in two or more dimensions. In 1998,

Stephen Smale Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics facult ...

included this problem as the sixth in his list of "mathematical problems for the next century". As partial progress, for almost all 5-tuples of masses, there are only a bounded number of two-dimensional central configurations of five points.

Special classes of configurations

Stacked

A central configuration is said to be ''stacked'' if a subset of three or more of its masses also form a central configuration. For example this can be true for equal masses forming a

square pyramid In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral square pyramid, ...

, with the four masses at the base of the pyramid also forming a central configuration, or for masses forming a

triangular bipyramid In geometry, the triangular bipyramid (or dipyramid) is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces. As the name suggests, i ...

, with the three masses in the central triangle of the bipyramid also forming a central configuration.

Spiderweb

A ''spiderweb central configuration'' is a configuration in which the masses lie at the intersection points of a collection of

concentric circles In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center point ...

with another collection of lines, meeting at the center of the circles with equal angles. The intersection points of the lines with a single circle should all be occupied by points of equal mass, but the masses may vary from circle to circle. An additional mass (which may be zero) is placed at the center of the system. For any desired number of lines, number of circles, and profile of the masses on each concentric circle of a spiderweb central configuration, it is possible to find a spiderweb central configuration matching those parameters. One can similarly obtain central configurations for families of nested

s, or more generally group-theoretic orbits of any finite subgroup of the

orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...

James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...

suggested that a special case of these configurations with one circle, a massive central body, and much lighter bodies at equally spaced points on the circle could be used to understand the motion of the rings of Saturn. used stable orbits generated from spiderweb central configurations with known mass distribution to test the accuracy of classical estimation methods for the mass distribution of galaxies. His results showed that these methods could be quite inaccurate, potentially showing that less

dark matter Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not ab ...

is needed to predict galactic motion than standard theories predict.

References

{{reflist, refs= {{citation , last = Albouy , first = Alain , issue = 2 , journal = Comptes rendus de l'Académie des Sciences , mr = 1320359 , pages = 217–220 , title = Symétrie des configurations centrales de quatre corps , volume = 320 , year = 1995 {{citation , last = Albouy , first = Alain , contribution = The symmetric central configurations of four equal masses , doi = 10.1090/conm/198/02494 , mr = 1409157 , pages = 131–135 , publisher = American Mathematical Society , location = Providence, Rhode Island , series = Contemporary Mathematics , title = Hamiltonian dynamics and celestial mechanics (Seattle, WA, 1995) , volume = 198 , year = 1996 {{citation , last1 = Albouy , first1 = Alain , last2 = Fu , first2 = Yanning , arxiv = math-ph/0603075 , doi = 10.1134/S1560354707010042 , issue = 1 , journal = Regular and Chaotic Dynamics , mr = 2350295 , pages = 39–55 , title = Euler configurations and quasi-polynomial systems , volume = 12 , year = 2007, s2cid = 18065509 {{citation , last1 = Albouy , first1 = Alain , last2 = Kaloshin , first2 = Vadim , author2-link = Vadim Kaloshin , doi = 10.4007/annals.2012.176.1.10 , issue = 1 , journal =

Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the ...

, mr = 2925390 , pages = 535–588 , series = Second Series , title = Finiteness of central configurations of five bodies in the plane , volume = 176 , year = 2012, doi-access = free {{citation , last = Chazy , first = J. , authorlink = Jean Chazy , journal = Bulletin Astronomique , pages = 321–389 , title = Sur certaines trajectoires du problème des {{mvar, n corps , volume = 35 , year = 1918 {{citation , last = Hampton , first = Marshall , doi = 10.1088/0951-7715/18/5/021 , issue = 5 , journal =

Nonlinearity In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...

, mr = 2164743 , pages = 2299–2304 , title = Stacked central configurations: new examples in the planar five-body problem , volume = 18 , year = 2005 {{citation , last1 = Hampton , first1 = Marshall , last2 = Moeckel , first2 = Richard , doi = 10.1007/s00222-005-0461-0 , issue = 2 , journal =

Inventiones Mathematicae ''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors ...

, mr = 2207019 , pages = 289–312 , title = Finiteness of relative equilibria of the four-body problem , volume = 163 , year = 2006, s2cid = 1293751 {{citation , last1 = Hénot , first1 = Olivier , last2 = Rousseau , first2 = Christiane , author2-link = Christiane Rousseau , doi = 10.1007/s12346-019-00330-y , issue = 3 , journal = Qualitative Theory of Dynamical Systems , mr = 4028598 , pages = 1135–1160 , title = Spiderweb central configurations , volume = 18 , year = 2019, doi-access = free {{citation , last = Maxwell , first = James Clerk , author-link = James Clerk Maxwell , bibcode = 1859osms.book.....M , doi = 10.3931/e-rara-244 , location = Cambridge , publisher = Macmillan , title = On the stability of the motion of Saturn's rings , url = https://archive.org/details/onstabilityofmot00maxw , year = 1859 {{citation , last = Moeckel , first = Richard , editor1-last = Llibre , editor1-first = Jaume , editor2-last = Moeckel , editor2-first = Richard , editor3-last = Simó , editor3-first = Carles , contribution = Central configurations , doi = 10.1007/978-3-0348-0933-7_2 , location = Basel , mr = 3469182 , pages = 105–167 , publisher = Springer , series = Advanced Courses in Mathematics - CRM Barcelona , title = Central Configurations, Periodic Orbits, and Hamiltonian Systems , year = 2015 {{citation , last = Montaldi , first = James , doi = 10.1007/s10569-015-9625-4 , issue = 4 , journal =

Celestial Mechanics and Dynamical Astronomy ''Celestial Mechanics and Dynamical Astronomy'' is a scientific journal covering the fields of astronomy and astrophysics. It was established as ''Celestial Mechanics'' in June 1969. The journal is published by Springer Science+Business Media and ...

, mr = 3368140 , pages = 405–418 , title = Existence of symmetric central configurations , volume = 122 , year = 2015, s2cid = 16906550 , url = http://eprints.maths.manchester.ac.uk/2269/1/centralConfigurations.pdf {{citation , last = Moulton , first = F. R. , authorlink = Forest Ray Moulton , doi = 10.2307/2007159 , issue = 1 , journal =

, jstor = 2007159 , mr = 1503509 , pages = 1–17 , series = Second Series , title = The straight line solutions of the problem of {{mvar, n bodies , volume = 12 , year = 1910 {{citation , last = Pizzetti , first = Paolo , authorlink = Paolo Pizzetti , journal = Rendiconti della Reale Accademia dei Lincei , pages = 17–26 , title = Casi particolari del problema dei tre corpi , volume = 13 , year = 1904 {{citation , last = Saari , first = Donald G. , authorlink = Donald G. Saari , editor1-last = Shubin , editor1-first = Tatiana , editor2-last = Hayes , editor2-first = David , editor3-last = Alexanderson , editor3-first = Gerald , editor3-link = Gerald L. Alexanderson , contribution = Central Configurations—A Problem for the Twenty-first Century , contribution-url = https://www.math.uci.edu/~dsaari/BAMA-pap.pdf , isbn = 978-0-88385-571-3 , location = Washington, DC , mr = 2849696 , pages = 283–297 , publisher = Mathematical Association of America , series = MAA Spectrum , title = Expeditions in mathematics , year = 2011 {{citation , last = Saari , first = Donald G. , authorlink = Donald G. Saari , date = April 2015 , doi = 10.1088/0004-6256/149/5/174 , issue = 5 , journal =

The Astronomical Journal ''The Astronomical Journal'' (often abbreviated ''AJ'' in scientific papers and references) is a peer-reviewed monthly scientific journal owned by the American Astronomical Society (AAS) and currently published by IOP Publishing. It is one of the ...

, page = 174 , title = {{mvar, N-body solutions and computing galactic masses , volume = 149 {{citation , last = Smale , first = Steve , authorlink = Stephen Smale , doi = 10.1007/BF03025291 , issue = 2 , journal =

The Mathematical Intelligencer ''The Mathematical Intelligencer'' is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals. Volumes are released quar ...

, mr = 1631413 , pages = 7–15 , title = Mathematical problems for the next century , volume = 20 , year = 1998, s2cid = 1331144 {{citation , last = Wintner , first = Aurel , authorlink = Aurel Wintner , location = Princeton, New Jersey , mr = 0005824 , publisher = Princeton University Press , series = Princeton Mathematical Series , title = The Analytical Foundations of Celestial Mechanics , volume = 5 , year = 1941 Classical mechanics Orbits