N-body problem

In physics, the n-body problem is an ancient, classical problem^[1] of predicting the individual motions, and forces on same, of a group of celestial objects interacting with each other gravitationally. Solving this problem — from the time of the Greeks and on — has been motivated by the desire to understand the motions of the Sun, planets and the visible stars. In the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem too.^[2] The n-body problem in general relativity is considerably more difficult to solve.

The classical physical problem can be informally stated as: given the quasi-steady orbital properties (instantaneous position, velocity and time)^[3] of a group of celestial bodies, predict their interactive forces; and consequently, predict their true orbital motions for all future times.^[4]

To this purpose the two-body problem has been completely solved and is discussed below; as is the famous Restricted 3-Body Problem.^[5]

This page next develops a general, closed-form (statics) solution for determining the reactive forces for the general n-body problem case, owing to a concentrated applied body force (an external force and moment).

Rigid-body applications (i.e., where A 's = 1, see below) include distribution of useful loads in a finite element model (FEM); 3D rigid-body rivet analyses; and astronomy problems. Astrodynamics problems and the like are particular solutions that use this new, general n-body problem solution as an algorithm; and once the forces of the bodies, {m_ζ}, ζ = 1, 2, ... N, are known, velocities and accelerations, i.e. their motions, may be determined. Determining equations and numerical values for astronomy problems is beyond this page's scope. Structural analyses or soil analyses elastic solutions (A 's < 1) are possible too. Again, the latter applications are beyond the scope of this page. Note especially: rigid-body solutions for elastic structures are not valid.

An example problem is given.

Historical treatment

Knowing three orbital positions of a planet's orbit – positions obtained by Sir Isaac Newton (1643-1727) from Astronomer John Flamsteed^[6] – Newton was able to produce an equation by straightforward analytical geometry, to predict a planet's motion; i.e., to give its orbital properties: position, orbital diameter, period and orbital velocity.^[7] Having done so he and others soon discovered over the course of a few years, those equations of motion did not predict very well or even correctly some orbits^[8],;^[9] and Newton realized it was because gravitational interactive forces amongst all the planets was affecting all their orbits.

The above discovery goes right to the heart of the matter as to what exactly the n-body problem is physically: as Newton realized, it is not sufficient to just specify the initial position and velocity, or three orbital positions either, to determine a planet's true orbit: the gravitational interactive forces have to be known too. Thus came the awareness and rise of the n-body “problem” in the early 17th century. These gravitational attractive forces do conform to Newton's Laws of Motion and to his Law of Universal Gravitation, but the many multiple (n-body) interactions have historically made any exact solution intractable. Ironically, this conformity led to the wrong approach.

After Newton's time the n-body problem historically was not stated correctly because it did not include a reference to those gravitational interactive forces. Newton does not say it directly but implies in his Principia the n-body problem is unsolvable because of those gravitational interactive forces.^[10] Newton said (ref. Cohen's Scientific American essay referenced below) in his Principia, paragraph 21:

And hence it is that the attractive force is found in both bodies. The Sun attracts Jupiter and the other planets, Jupiter attracts its satellites and similarly the satellites act on one another. And although the actions of each of a pair of planets on the other can be distinguished from each other and can be considered as two actions by which each attracts the other, yet inasmuch as they are between the same, two bodies they are not two but a simple operation between two termini. Two bodies can be drawn to each other by the contraction of rope between them. The cause of the action is twofold, namely the disposition of each of the two bodies; the action is likewise twofold, insofar as it is upon two bodies; but insofar as it is between two bodies it is single and one ...

Newton concluded via his 3rd Law that "according to this Law all bodies must attract each other." This last statement, which implies the existence of gravitational interactive forces, is key.

As shown below, the problem also conforms to Jean Le Rond D'Alembert's non-Newtonian 1^st and 2^nd Principles and to the nonlinear n-body problem algorithm, the latter allowing for a closed form solution for calculating those interactive forces.

Two-body problem

Any discussion of planetary interactive forces has always started historically with the Two-body problem. The purpose of this Section is to relate the real complexity in calculating any planetary forces. Note in this Section also, several subjects, such as gravity, barycenter, Kepler's Laws, etc.; and in the following Section too (Three-body problem) — are discussed on other Wilkipedia pages. Here though, these subjects are discussed from the perspective of the n-body problem.

The two-body problem, n = 2, was completely solved by Johann Bernoulli (1667-1748) by classical theory (and not by Newton) by assuming the main point-mass was fixed, is outlined here.^[11] Consider then the motion of two bodies, say Sun-Earth, with the Sun fixed, then:

m₁a₁ = (Gm₁m₂/r³₁₂)(r₂ - r₁)…..Sun-to-Earth m₂a₂ = (Gm₂m₁/r³₂₁)(r₁ - r₂)…..Earth-to-Sun

The equation describing the motion of mass m₂ relative to mass m₁ is readily obtained from the differences between these two equations and after canceling common terms gives: α + (η/r³)r = 0, where

• α is the Eulerian acceleration d²r/dt²;
• r = r₂ - r₁ is the vector position of m₂ relative to m₁;
• and η = G(m₁ + m₂).

α + (η/r³)r = 0 is the fundamental differential equation for the two-body problem Bernoulli solved in 1734. Notice for this approach forces have to be determined first, then the equation of motion resolved. This differential equation has elliptic, or parabolic or hyperbolic solutions^{[12], [13],}.^[14]

It is incorrect to think of m₁ (the Sun) as fixed in space when applying Newton's Law of Universal Gravitation, and to do so leads to erroneous results. Dr. Clarence Cleminshaw calculated the approximate position of the Solar System's true barycenter, a result achieved mainly by combining only the masses of Jupiter and the Sun. Science Program stated in reference to his work: "The Sun contains 98 per cent of the mass in the solar system, with the superior planets beyond Mars accounting for most of the rest. On the average, the center of the mass of the Sun-Jupiter system, when the two most massive objects are considered alone, lies 462,000 miles from the Sun's center, or some 30,000 miles above the solar surface! Other large planets also influence the center of mass of the solar system, however. In 1951, for example, the systems' center of mass was not far from the Sun's center because Jupiter was on the opposite side from Saturn, Uranus and Neptune. In the late 1950s, when all four of these planets were on the same side of the Sun, the system's center of mass was more than 330,000 miles form the solar surface, Dr. C. H. Cleminshaw of Griffith Observatory in Los Angeles has calculated."^[15] This means the Sun wobbles and Sun-spots are possibly caused via the movement of the barycenter, owing to Jupiter's 11-year cycles, producing Sun-spots every 22 years. It further needs to be pointed out the total mass orbiting the Sun is probably equal to the Sun's own mass^{[citation needed]}.

The Sun wobbles as it rotates around the galactic center, dragging the Solar System and Earth along with it. What mathematician Kepler did in arriving at his three famous equations was curve-fit the apparent motions of the planets using Tycho Brahe's data, and not curve-fitting their true circular motions about the Sun (see Figure). Both Robert Hooke and Newton were well aware Newton's Law of Universal Gravitation did not hold for the forces associated with elliptical orbits.^[16] In fact, Newton's Universal Law doesn't account for the orbit of Mercury, the Asteroid Belt's gravitational behavior, or Saturn's Rings.^[17] Newton stated (in the 11th Section of the Principia) the main reason however for failing to predict the forces for elliptical orbits was his math model was for a body confined to a situation that "hardly exist in the real world," namely, the "motions of bodies attracted toward an unmoving center." Some present physics and astronomy textbooks don't emphasize the negative significance of Newton's assumption and end up teaching that his math model is in effect reality. It is to be understood the classical two-body problem solution above is a mathematical toy. See also Kepler's first law of planetary motion.

Newton conveniently fixed the Sun so he could do simple calculations (in effect he cheated), but all following after him have also made the same mistake by analytically fixing the Sun too. They perpetuated the mathematical toy (see Truesdell's Essays in the History of Mechanics referenced below). An aside: Newtonian physics doesn't include (among other things) relative motion and may be the root of the reason Newton "fixed" the Sun.^[18][19] The Sun's wobbling means the n-body problem's solution realistically is much more complicated than maybe previously thought (see below).

Three-body problem

This Section relates an historically important n-body problem solution after simplifying assumptions were made.

In the past not much was known about the n-body problem for n equal to or greater than three.^[20] The case for n = 3 has been the most studied. Many earlier attempts to understand the Three-body problem were quantitative, aiming at finding explicit solutions for special situations.

• In 1687 Isaac Newton published in the Principia the first steps in the study of the problem of the movements of three bodies subject to their mutual gravitational attractions, but his efforts resulted in verbal descriptions and geometrical sketches; see especially Book 1, Proposition 66 and its corollaries (Newton, 1687 and 1999 (transl.), see also Tisserand, 1894).
• In 1767 Euler found collinear motions, in which three bodies of any masses move proportionately along a fixed straight line. The circular restricted three-body problem is the special case in which two of the bodies are in circular orbits (approximated by the Sun-Earth-Moon system and many others).
• In 1772 Lagrange discovered two classes of periodic solution, each for three bodies of any masses. In one class, the bodies lie on a rotating straight line. In the other class, the bodies lie at the vertices of a rotating equilateral triangle. In either case, the paths of the bodies will be conic sections. Those solutions led to the study of central configurations , for which for some constant k>0 .
• A major study of the Earth-Moon-Sun system was undertaken by Charles-Eugène Delaunay, who published two volumes on the topic, each of 900 pages in length, in 1860 and 1867. Among many other accomplishments, the work already hints at chaos, and clearly demonstrates the problem of so-called "small denominators" in perturbation theory.
• In 1917 Forest Ray Moulton published his now classic, An Introduction to Celestial Mechanics (see references) with its plot of the Restricted Three-body Problem solution (see figure below).^[21] An aside, see Meirovitch's book, pages 414 and 413 for his Restricted Three-body Problem solution.^[22]

Moulton's solution may be easier to visualize (and definitely easier to solve) if one considers the more massive body (e.g., Sun) to be "stationary" in space, and the less massive body (e.g., Jupiter) to orbit around it, with the equilibrium points (Lagrangian points) maintaining the 60 degree-spacing ahead of, and behind, the less massive body almost in its orbit (although in reality neither of the bodies are truly stationary, as they both orbit the center of mass of the whole system—about the barycenter). For sufficiently small mass ratio of the primaries, these triangular equilibrium points are stable, such that (nearly) massless particles will orbit about these points as they orbit around the larger primary (Sun). The five equilibrium points of the circular problem are known as the Lagrangian points. See figure below:

In the Restricted 3-Body Problem math model figure above (ref. Moulton), the Lagrangian points L₄ and L₅ are where the Trojan planetoids resided (see Lagrangian point); m₁ is the Sun and m₂ is Jupiter. L₂ is where the asteroid belt is. It has to be realized for this model, this whole Sun-Jupiter diagram is rotating about its barycenter. The Restricted 3-Body Problem solution predicted the Trojan planetoids before they were first seen. The h-circles and closed loops echo the electromagnetic fluxes issued from the Sun and Jupiter. It is conjectured, contrary to Richard H. Batin conjecture (see References), the two h₁'s are gravity sinks, in and where gravitational forces are zero, and the reason the Trojan planetoids are trap there. The total amount of mass of the planetoids is unknown.

The Restricted Three-body Problem assumes the mass of one of the bodies is negligible (???) ^{[citation needed]}. For a discussion of the case where the negligible body is a satellite of the body of lesser mass, see Hill sphere; for binary systems, see Roche lobe. Specific solutions to the Three-body problem result in chaotic motion with no obvious sign of a repetitious path.^{[citation needed]}

The restricted problem (both circular and elliptical) was worked on extensively by many famous mathematicians and physicists, most notably by Poincaré at the end of the 19th century. Poincaré's work on the Restricted Three-body Problem was the foundation of deterministic chaos theory.^{[citation needed]} In the restricted problem, there exist five equilibrium points. Three are collinear with the masses (in the rotating frame) and are unstable. The remaining two are located on the third vertex of both equilateral triangles of which the two bodies are the first and second vertices.

King Oscar II Prize: historical perspective

The problem of finding the general solution of the n-body problem was considered very important and challenging. Indeed in the late 19th century King Oscar II of Sweden, advised by Gösta Mittag-Leffler, established a prize for anyone who could find the solution to the problem. The announcement was quite specific:

Given a system of arbitrarily many mass points that attract each according to Newton's law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series converges uniformly.

In case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prize-worthy.

The prize was finally awarded to Poincaré, even though he did not solve the original problem. (The first version of his contribution even contained a serious error; for details see the article by Diacu). The version finally printed contained many important ideas which led to the development of chaos theory. The problem as stated originally was finally solved by Karl Fritiof Sundman for n = 3 (see below).

More general considerations for the classical solution

For every solution of the problem, not only applying an isometry or a time shift but also a reversal of time (unlike in the case of friction) gives a solution as well.^{[citation needed]}

In the physical literature about the -body problem ( ≥ 3), sometimes reference is made to the impossibility of solving the -body problem (via employing the above approach)^{[citation needed]}. However, care must be taken when discussing the 'impossibility' of a solution, as this refers only to the method of first integrals (compare the theorems by Abel and Galois about the impossibility of solving algebraic equations of degree five or higher by means of formulas only involving roots).

(Classical) power series solution

This is the most elemental way of classically solving the n-body problem. The theoretical expression is often called "The n-body problem by Taylor series", which is an implementation of the Power series solution of differential equations.

We start by defining the differential equations system^{[citation needed]}:

,

where (in ) the upper index indicates the second derivative with respect to time , represents the number of each body and the coordinate.

Because and are given as initial conditions, then every are known. Doing implicit derivation over every results in which at are known because each depends on known precalculated and given constants and then the Taylor series are constructed theoretically in such way, performing this process infinitely.

Numerical integration

n-body problems can be solved by numerically integrating the differential equations of motion. Many different ways to do this to varying degrees of accuracy and speed exist.^[23]

The simplest integrator is the Euler method, but this is only first order. A second order method is leapfrog integration, but higher-order integration methods such as the Runge–Kutta methods can be employed. Symplectic integrators are often used for n-body problems.

Numerical integration has a time complexity of O(<var>n</var>²), but tree structured methods, such as Barnes-Hut simulation, can improve this to O(<var>n</var> log <var>n</var>), or even to O(<var>n</var>) such as with the fast multipole method.

Sundman's theorem for the 3-body problem

In 1912, the Finnish mathematician Karl Fritiof Sundman proved there existed a series solution in powers of for the 3-body Problem. This series is convergent for all real t, except initial data which correspond to zero angular momentum. However these initial data are not generic since they have Lebesgue measure zero.

An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Therefore it is necessary to study the possible singularities of the 3-body Problems. As it will be briefly discussed below, the only singularities in the 3-body Problem are:

1. binary collisions,
2. triple collisions.

Now collisions, whether binary or triple (in fact of arbitrary order), are somehow improbable since it has been shown they correspond to a set of initial data of measure zero. However there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution. So Sundman's strategy consisted of the following steps:

1. He first was able, using an appropriate change of variables, to continue analytically the solution beyond the binary collision, in a process known as regularization.
2. He then proved triple collisions only occur when the angular momentum L vanishes. By restricting the initial data to he removed all real singularities from the transformed equations for the 3-body problem.
3. The next step consisted in showing that if , then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by using Cauchy's existence theorem for differential equations, there are no complex singularities in a strip (depending on the value of L) in the complex plane centered around the real axis (shades of Kovalevskaya).
4. The last step is then to find a conformal transformation which maps this strip into the unit disc. For example if (the new variable after the regularization) and if then this map is given by:

This finishes the proof of Sundman's Theorem. Unfortunately the corresponding convergent series converges very slowly. That is, getting the value to any useful precision requires so many terms, that his solution is of little practical use.

A generalized Sundman global solution

In order to generalize Sundman's result for the case n > 3 (or n = 3 and c = 0) one has to face two obstacles:

1. As it has been shown by Siegel, collisions which involve more than two bodies cannot be regularized analytically, hence Sundman's regularization cannot be generalized.
2. The structure of singularities is more complicated in this case: other types of singularities may occur (see below).

Lastly, Sundman's result was generalized to the case of n > 3 bodies by Q. Wang in the 1990s. Since the structure of singularities is more complicated, Wang had to leave out completely the questions of singularities. The central point of his approach is to transform, in an appropriate manner, the equations to a new system, such that the interval of existence for the solutions of this new system is .

Singularities of the n-body problem

There can be two types of singularities of the n-body problem:

• collisions of two or more bodies, but for which q(t) (the bodies' positions) remains finite. (In this mathematical sense, a "collision" means that two point-like bodies have identical positions in space.)
• singularities in which a collision does not occur, but q(t) does not remain finite. In this scenario, bodies diverge to infinity in a finite time, while at the same time tending towards zero separation (an imaginary collision occurs "at infinity").

The latter ones are called Painlevé's conjecture (no-collisions singularities). Their existence has been conjectured for n > 3 by Painlevé (see Painlevé's conjecture). Examples of this behavior have been constructed by Xia^[24] and Gerver.

See also

• Einstein–Infeld–Hoffmann equations
• Few-body systems
• Natural units
• n-body choreography
• Virial theorem

Notes

References

• Heggie, Douglas and Hut, Piet: The Gravitational Million-Body Problem, A Multidisciplinary Approach to Star Cluster Dynamics, Cambridge University Press, 357 pages, 2003.
• Heggie, Douglas C.: "Chaos in the N-body Problem of Stellar Dynamics," in Predictability, Stability and Chaos in N-Body Dynamical Systems, Ed. by Roy A. E., Plenum Press, 1991.
• Aarseth, Sverre J.: Gravitational N-body Simulations, Tools and Algorithms, Cambridge University Press, 2003.
• van Winter, Clasine: "The n-body problem on a Hilbert space of analytic functions," Paper 11-29, in Analytic Methods in Mathematical Physics, edited by Robert P. Gilbert and Roger G. Newton, pp. 569–578, Gordon and Breach, 1970.
• Leimanis, E., and Minorsky, N.: Dynamics and Nonlinear Mechanics, Part I: Some Recent Advances in the Dynamics of Rigid Bodies and Celestial Mechanics (Leimanis), Part II: The Theory of Oscillations (Minorsky), John Wiley & Sons, Inc., 1958.
• Moulton, Forest Ray: An Introduction to Celestial Mechanics, Dover, 1970.
• Kurth, Rudolf, Introduction to the Mechanics of the Solar System, Pergamon Press, 1959.
• Meirovitch, Leonard: Methods of Analytical Dynamics, McGraw-Hill Book Co., 1970.
• Brouwer, Dirk and Clemence, Gerald M.: Methods of Celestial Mechanics, Academic Press, 1961.
• Cohen, Bernard I.: "Newton's Discovery of Gravity," Scientific American, pp. 167–179, Vol. 244, No. 3, Mar. 1980.
• Cohen, Bernard I.: The Birth of a New Physics, Revised and Updated, W.W. Norton & Co., 1985.
• Science Programs “' The Nature of the Universe ,” booklet, published by Nelson Doubleday, Inc., in 1968:
• Bate, Roger R.; Mueller, Donald D.; and White, Jerry: Fundamentals of Astrodynamics, Dover, 1971.
• Battin, Richard H.: An Introduction to The Mathematics and Methods of Astrodynamics, AIAA, 1987. He employs energy methods rather than a Newtonian approach.
• Gallian, Dave A. and Wilson, Henry E.: "The Integration of NASTRAN Into Helicopter Airframe Design/Analysis," American Helicopter Society Pub., Reprint No. 780, May 1973. This is a paper, not a book.
• Lindsay, Robert Bruce: Physical Mechanics, 3rd Ed., D. Van Nostrand Co., Inc., 1961.
• Gelman, Harry: Part I: The second orthogonality conditions in the theory of proper and improper rotations: Derivation of the conditions and of their main consequences, J. Res. NBS 72B (Math. Sci.)No. 3, 1968. Part II: The intrinsic vector; Part III: The Conjugacy Theorem,J. Res. NBS 72B (Math. Sci.) No. 2, 1969. A Note on the time dependence of the effective axis and angle of a rotation, J. Res. NBS 72B (Math. Sci.)No. 3&4, Oct. 1971. These papers are on the Internet.
• Meriam, J. L.: Engineering Mechanics, Volume 1 Statics, Volume 2 Dynamics, John Wiley & Sons, 1978.
• Quadling, Henley: "Gravitational N-Body Simulation: 16 bit DOS version," June 1994. nbody*.zip is available at the http://www.ftp.cica.indiana.edu: see external links.
• Korenev, G. V.: The Mechanics of Guided Bodies, CRC Press, 1967.
• Eisele, John A. and Mason, Robert M.: Applied Matrix and Tensor Analysis, John Wiley & Sons, 1970.
• Murray, Carl D. and Dermott, Stanley F.: Solar System Dynamics, Cambridge University Press, 606 pages, 2000.
• Bronowski, Jacob and Mazlish, Bruce: The Western Intellectual Tradition, from Leonardo to Hegel, Dorsey Press. 1986.
• Truesdell, Clifford: Essays in the History of Mechanics, Springer-Verlay, 1968.
• Hufbauer, Dr. Karl C. (History of Science): Exploring the Sun, Solar Science since Galileo, The Johns Hopkins University Press, 1991. This book was sponsored by the NASA History Office.
• Crandall, Richard E.: Topics in Advanced Scientific Computation, Chapter 5, "Nonlinear & Complex Systems," paragraph 5.1, "N-body problems & chaos," pp. 215–221, Springer-Verlag, 1996.
• Crandall, Richard E.: Projects in Scientific Computation, Chapter 2, "Exploratory Computation," Project 2.4.1, "Classical Physics," pp. 93–97, corrected 3rd printing, Springer-Verlag, 1996.
• Szebehely, Victor: Theory of Orbits, Academic Press, 1967.
• Rosenberg, Reinhardt M.: Analytical Dynamics, of Discrete Systems, Chapter 19, About Celestial Problems, paragraph 19.5, The n-body Problem, pp. 364–371, Plenum Press, 1977. Like Battin above, Rosenberg employs energy methods too, and to the solution of the general n-body problem but doesn't actually solve anything.
• Diacu, F.: The solution of the n-body problem, The Mathematical Intelligencer,1996,18,p. 66–70
• Mittag-Leffler, G.: The n-body problem (Price Announcement), Acta Matematica, 1885/1886,7
• Saari, D.: A visit to the Newtonian n-body problem via Elementary Complex Variables, American Mathematical Monthly, 1990, 89, 105–119
• Newton, I.: Philosophiae Naturalis Principia Mathematica, London, 1687: also English translation of 3rd (1726) edition by I. Bernard Cohen and Anne Whitman (Berkeley, CA, 1999).
• Wang, Qiudong (1991). "The global solution of the n-body problem". Celestial Mechanics and Dynamical Astronomy 50 (1): 73–88. Bibcode:1991CeMDA..50...73W. doi:10.1007/BF00048987. ISSN 0923-2958.  |accessdate= requires |url= (help)
• Sundman, K. F.: Memoire sur le probleme de trois corps, Acta Mathematica 36 (1912): 105–179.
• Tisserand, F-F.: Mecanique Celeste, tome III (Paris, 1894), ch.III, at p. 27.
• Hagihara, Y: Celestial Mechanics. (Vol I and Vol II pt 1 and Vol II pt 2.) MIT Press, 1970.
• Boccaletti, D. and Pucacco, G.: Theory of Orbits (two volumes). Springer-Verlag, 1998.
• Havel, Karel. N-Body Gravitational Problem: Unrestricted Solution (ISBN 978-09689120-5-8). Brampton: Grevyt Press, 2008. http://www.grevytpress.com
• Saari, D. G.; Hulkower, N. D. (1981). "On the Manifolds of Total Collapse Orbits and of Completely Parabolic Orbits for the n-Body Problem". Journal of Differential Equations 41 (1): 27–43. doi:10.1016/0022-0396(81)90051-6. 

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