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Part of a series on
Astrodynamics
Orbital mechanics
Orbital elements Apsis Argument of periapsis Eccentricity Inclination Mean anomaly Orbital nodes Semi-major axis True anomaly
Types oftwo-body orbits by eccentricity Circular orbit Elliptic orbit Transfer orbit (Hohmann transfer orbit Bi-elliptic transfer orbit) Parabolic orbit Hyperbolic orbit Radial orbit Decaying orbit
Equations Dynamical friction Escape velocity Kepler's equation Kepler's laws of planetary motion Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation
Celestial mechanics
Gravitational influences Barycenter Hill sphere Perturbations Sphere of influence
N-body orbits Lagrangian points (Halo orbits) Lissajous orbits Lyapunov orbits
Engineering and efficiency
Preflight engineering Mass ratio Payload fraction Propellant mass fraction Tsiolkovsky rocket equation
Efficiency measures Gravity assist Oberth effect
Propulsive maneuvers Orbital maneuver Orbit insertion
v t e

Inphysics, then-body problem is the problem of predicting the individual motions of a group ofcelestial objects interacting with each othergravitationally.^[1] Solving this problem has been motivated by the desire to understand the motions of theSun,Moon,planets, and visiblestars. In the 20th century, understanding thedynamics ofglobular cluster star systems became an importantn-body problem.^[2] Then-body problem ingeneral relativity is considerably more difficult to solve due to additional factors like time and space distortions.

The classical physical problem can be informally stated as the following:

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]

Thetwo-body problem has been completely solved and is discussed below, as well as the famousrestrictedthree-body problem.^[5]

Knowing three orbital positions of a planet's orbit – positions obtained by SirIsaac Newton from astronomerJohn 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 some orbits correctly or even very well.^[8] Newton realized that this was because gravitational interactive forces amongst all the planets were affecting all their orbits.

The aforementioned revelation strikes directly at the core of what the n-body issue physically is: as Newton understood, it is not enough to just provide the beginning location and velocity, or even three orbital positions, in order to establish a planet's actual orbit; one must also be aware of the gravitational interaction forces. Thus came the awareness and rise of then-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 then-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 hisPrincipia then-body problem is unsolvable because of those gravitational interactive forces.^[9] Newton said^[10] in hisPrincipia, 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 histhird law of motion 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 toJean Le Rond D'Alembert's non-Newtonian first and second Principles and to the nonlinearn-body problem algorithm, the latter allowing for a closed form solution for calculating those interactive forces.

The problem of finding the general solution of then-body problem was considered very important and challenging. Indeed, in the late 19th century KingOscar II of Sweden, advised byGösta 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 seriesconverges uniformly.

In case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was awarded toPoincaré, even though he did not solve the original problem. (The first version of his contribution even contained a serious error.^[11]) The version finally printed contained many important ideas which led to the development ofchaos theory. The problem as stated originally was finally solved byKarl Fritiof Sundman forn = 3 and generalized ton > 3 by L. K. Babadzanjanz^[12][13] andQiudong Wang.^[14]

Then-body problem considersn point massesm_i,i = 1, 2, …,n in aninertial reference frame in three dimensional spaceℝ³ moving under the influence of mutual gravitational attraction. Each massm_i has a position vectorq_i.Newton's second law says that mass times accelerationm_i⁠d²q_i/dt²⁠ is equal to the sum of the forces on the mass.Newton's law of gravity says that the gravitational force felt on massm_i by a single massm_j is given by^[15]$${\displaystyle {\mathbf{F}}_{ij}=\frac{G{m}_i{m}_j}{{\Vert {\mathbf{q}}_j-{\mathbf{q}}_i\Vert }^2}\cdot \frac{\left({\mathbf{q}}_j-{\mathbf{q}}_i\right)}{\Vert {\mathbf{q}}_j-{\mathbf{q}}_i\Vert }=\frac{G{m}_i{m}_j\left({\mathbf{q}}_j-{\mathbf{q}}_i\right)}{{\Vert {\mathbf{q}}_j-{\mathbf{q}}_i\Vert }^3},}$$whereG is thegravitational constant and‖q_j −q_i‖ is the magnitude of the distance betweenq_i andq_j (metric induced by thel₂ norm).

Summing over all masses yields then-bodyequations of motion:

whereU is theself-potential energy

Defining the momentum to bep_i =m_i⁠dq_i/dt⁠,Hamilton's equations of motion for then-body problem become^[16]$${\displaystyle \frac{d{\mathbf{q}}_i}{dt}=\frac{\mathrm{\partial }H}{\mathrm{\partial }{\mathbf{p}}_i}\frac{d{\mathbf{p}}_i}{dt}=-\frac{\mathrm{\partial }H}{\mathrm{\partial }{\mathbf{q}}_i},}$$where theHamiltonian function is$${\displaystyle H=T+U}$$andT is thekinetic energy$${\displaystyle T=\sum _{i=1}^n\frac{{\Vert {\mathbf{p}}_i\Vert }^2}{2m_i}.}$$

Hamilton's equations show that then-body problem is a system of6n first-orderdifferential equations, with6ninitial conditions as3n initial position coordinates and3n initial momentum values.

Symmetries in then-body problem yield globalintegrals of motion that simplify the problem.^[17]Translational symmetry of the problem results in thecenter of mass$${\displaystyle \mathbf{C}=\frac{{\displaystyle \sum _{i=1}^n{m}_i{\mathbf{q}}_i}}{{\displaystyle \sum _{i=1}^n{m}_i}}}$$moving with constant velocity, so thatC =L₀t +C₀, whereL₀ is the linear velocity andC₀ is the initial position. The constants of motionL₀ andC₀ represent six integrals of the motion.Rotational symmetry results in the totalangular momentum being constant$${\displaystyle \mathbf{A}=\sum _{i=1}^n{\mathbf{q}}_i\times {\mathbf{p}}_i,}$$where × is thecross product. The three components of the total angular momentumA yield three more constants of the motion. The last general constant of the motion is given by theconservation of energyH. Hence, everyn-body problem has ten integrals of motion.

BecauseT andU arehomogeneous functions of degree 2 and −1, respectively, the equations of motion have ascaling invariance: ifq_i(t) is a solution, then so isλ^−2/3q_i(λt) for anyλ > 0.^[18]

Themoment of inertia of ann-body system is given by$${\displaystyle I=\sum _{i=1}^n{m}_i{\mathbf{q}}_i\cdot {\mathbf{q}}_i=\sum _{i=1}^n{m}_i{\Vert {\mathbf{q}}_i\Vert }^2}$$and thevirial is given byQ =⁠1/2⁠⁠dI/dt⁠. Then theLagrange–Jacobi formula states that^[19]$${\displaystyle \frac{d^{2}I}{d{t}^2}=2T-U.}$$

For systems indynamic equilibrium, the longterm time average of⟨⁠d²I/dt²⁠⟩ is zero. Then on average the total kinetic energy is half the total potential energy,⟨T⟩ =⁠1/2⁠⟨U⟩, which is an example of thevirial theorem for gravitational systems.^[20] IfM is the total mass andR a characteristic size of the system (for example, the radius containing half the mass of the system), then the critical time for a system to settle down to a dynamic equilibrium is^[21]$${\displaystyle t_{\mathrm{c}\mathrm{r}}={\left(\frac{GM}{R^3}\right)}^{-1/2}.}$$

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 asgravity,barycenter,Kepler's Laws, etc.; and in the following Section too (Three-body problem) are discussed on other Wikipedia pages. Here though, these subjects are discussed from the perspective of then-body problem.

The two-body problem (n = 2) was completely solved byJohann Bernoulli (1667–1748) by classical theory (and not by Newton) by assuming the main point-mass was fixed; this is outlined here.^[22] Consider then the motion of two bodies, say the Sun and the Earth, with the Sun fixed, then:

The equation describing the motion of massm₂ relative to massm₁ is readily obtained from the differences between these two equations and after canceling common terms gives:$${\displaystyle \alpha +\frac{\eta }{r^3}\mathbf{r}=\mathbf{0}}$$Where

• r =r₂ −r₁ is the vector position ofm₂ relative tom₁;
• α is theEulerian acceleration⁠d²r/dt²⁠;
• η =G(m₁ +m₂).

The equationα +⁠η/r³⁠r = 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.^[23][24][25]

It is incorrect to think ofm₁ (the Sun) as fixed in space when applying Newton's law of universal gravitation, and to do so leads to erroneous results. The fixed point for two isolated gravitationally interacting bodies is their mutualbarycenter, and thistwo-body problem can be solved exactly, such as usingJacobi coordinates relative to the barycenter.

Dr. Clarence Cleminshaw calculated the approximate position of the Solar System's 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 from the solar surface, Dr. C. H. Cleminshaw of Griffith Observatory in Los Angeles has calculated.^[26]

The Sun wobbles as it rotates around theGalactic Center, dragging the Solar System and Earth along with it. What mathematicianKepler did in arriving at his three famous equations was curve-fit the apparent motions of the planets usingTycho Brahe's data, and not curve-fitting their true circular motions about the Sun (see figure). BothRobert Hooke and Newton were well aware that Newton'sLaw of Universal Gravitation did not hold for the forces associated with elliptical orbits.^[10] In fact, Newton's Universal Law does not account for the orbit of Mercury, the asteroid belt's gravitational behavior, orSaturn's rings.^[27] Newton stated (in section 11 of thePrincipia) that the main reason, however, for failing to predict the forces for elliptical orbits was that his math model was for a body confined to a situation that hardly existed in the real world, namely, the motions of bodies attracted toward an unmoving center. Some present physics and astronomy textbooks do not emphasize the negative significance of Newton's assumption and end up teaching that his mathematical model is in effect reality. It is to be understood that the classical two-body problem solution above is a mathematicalidealization. See alsoKepler's first law of planetary motion.

This section relates a historically importantn-body problem solution after simplifying assumptions were made.

In the past not much was known about then-body problem forn ≥ 3.^[28] The casen = 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 thePrincipia 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 foundcollinear motions, in which three bodies of any masses move proportionately along a fixed straight line. TheEuler's three-body problem is the special case in which two of the bodies are fixed in space (this should not be confused with thecircular restricted three-body problem, in which the two massive bodies describe a circular orbit and are only fixed in a synodic reference frame).
• 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 ofcentral configurations, for whichq̈ =kq for some constantk > 0.
• A major study of the Earth–Moon–Sun system was undertaken byCharles-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" inperturbation theory.
• In 1917,Forest Ray Moulton published his now classic,An Introduction to Celestial Mechanics (see references) with its plot of therestricted three-body problem solution (see figure below).^[29] An aside, see Meirovitch's book, pages 413–414 for his restricted three-body problem solution.^[30]

Moulton's solution may be easier to visualize (and definitely easier to solve) if one considers the more massive body (such as theSun) to be stationary in space, and the less massive body (such asJupiter) to orbit around it, with the equilibrium points (Lagrangian points) maintaining the 60° 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 therestricted three-body problem math model figure above (after Moulton), the Lagrangian points L₄ and L₅ are where theTrojan planetoids resided (seeLagrangian point);m₁ is the Sun andm₂ is Jupiter. L₂ is a point within the asteroid belt. It has to be realized for this model, this whole Sun-Jupiter diagram is rotating about its barycenter. The restricted three-body problem solution predicted the Trojan planetoids before they were first seen. Theh-circles and closed loops echo the electromagnetic fluxes issued from the Sun and Jupiter. It is conjectured, contrary to Richard H. Batin's conjecture (see References), the twoh₁ are gravity sinks, in and where gravitational forces are zero, and the reason the Trojan planetoids are trapped there. The total amount of mass of the planetoids is unknown.

The restricted three-body problem assumes themass 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, seeHill sphere; for binary systems, seeRoche 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 byPoincaré at the end of the 19th century. Poincaré's work on the restricted three-body problem was the foundation ofdeterministic chaos theory.^{[citation needed]} In the restricted problem, there exist fiveequilibrium 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.

Inspired by the circular restricted three-body problem, the four-body problem can be greatly simplified by considering a smaller body to have a small mass compared to the other three massive bodies, which in turn are approximated to describe circular orbits. This is known as the bicircular restricted four-body problem (also known as bicircular model) and it can be traced back to 1960 in a NASA report written by Su-Shu Huang.^[31] This formulation has been highly relevant in theastrodynamics, mainly to model spacecraft trajectories in the Earth-Moon system with the addition of the gravitational attraction of the Sun. The former formulation of the bicircular restricted four-body problem can be problematic when modelling other systems than the Earth-Moon-Sun, so the formulation was generalized by Negri and Prado^[32] to expand the application range and improve the accuracy without loss of simplicity.

Theplanetary problem is then-body problem in the case that one of the masses is much larger than all the others. A prototypical example of a planetary problem is the Sun–Jupiter–Saturn system, where the mass of the Sun is about 1000 times larger than the masses of Jupiter or Saturn.^[18] An approximate solution to the problem is to decompose it inton − 1 pairs of star–planetKepler problems, treating interactions among the planets as perturbations.Perturbative approximation works well as long as there are noorbital resonances in the system, that is none of the ratios of unperturbed Kepler frequencies is a rational number. Resonances appear as small denominators in the expansion.

The existence of resonances and small denominators led to the important question of stability in the planetary problem: do planets, in nearly circular orbits around a star, remain in stable or bounded orbits over time?^[18][33] In 1963,Vladimir Arnold proved usingKAM theory a kind of stability of the planetary problem: there exists a set of positive measure ofquasiperiodic orbits in the case of the planetary problem restricted to the plane.^[33] In the KAM theory, chaotic planetary orbits would be bounded by quasiperiodic KAM tori. Arnold's result was extended to a more general theorem by Féjoz and Herman in 2004.^[34]

Acentral configurationq₁(0), …,q_N(0) is an initial configuration such that if the particles were all released with zero velocity, they would all collapse toward the center of massC.^[33] Such a motion is calledhomothetic. Central configurations may also give rise tohomographic motions in which all masses moves along Keplerian trajectories (elliptical, circular, parabolic, or hyperbolic), with all trajectories having the same eccentricitye. For elliptical trajectories,e = 1 corresponds to homothetic motion ande = 0 gives arelative equilibrium motion in which the configuration remains an isometry of the initial configuration, as if the configuration was a rigid body.^[35] Central configurations have played an important role in understanding thetopology ofinvariant manifolds created by fixing the first integrals of a system.

Solutions in which all masses move on thesame curve without collisions are called choreographies.^[36] A choreography forn = 3 was discovered by Lagrange in 1772 in which three bodies are situated at the vertices of anequilateral triangle in the rotating frame. Afigure eight choreography forn = 3 was found numerically by C. Moore in 1993^[37] and generalized and proven by A. Chenciner and R. Montgomery in 2000.^[38] Since then, many other choreographies have been found forn ≥ 3.

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

In the physical literature about then-body problem (n ≥ 3), sometimes reference is made to "the impossibility of solving then-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 byAbel andGalois about the impossibility of solvingalgebraic equations of degree five or higher by means of formulas only involving roots).

One way of solving the classicaln-body problem is "then-body problem byTaylor series".

We start by defining the system ofdifferential equations:^{[citation needed]}$${\displaystyle \frac{d^2{\mathbf{x}}_i(t)}{d{t}^2}=G\sum _{\genfrac{}{}{0ex}{}{k=1}{k\ne i}}^n\frac{m_k\left({\mathbf{x}}_k(t)-{\mathbf{x}}_i(t)\right)}{{\left|{\mathbf{x}}_k(t)-{\mathbf{x}}_i(t)\right|}^3},}$$

Asx_i(t₀) and⁠dx_i(t₀)/dt⁠ are given as initial conditions, every⁠d²x_i(t)/dt²⁠ is known. Differentiating⁠d²x_i(t)/dt²⁠ results in⁠d³x_i(t)/dt³⁠ which att₀ which is also known, and the Taylor series is constructed iteratively.^{[clarification needed]}

In order to generalize Sundman's result for the casen > 3 (orn = 3 andc = 0^{[clarification needed]}) one has to face two obstacles:

1. As has been shown by Siegel, collisions which involve more than two bodies cannot be regularized analytically, hence Sundman's regularization cannot be generalized.^{[citation needed]}
2. The structure of singularities is more complicated in this case: other types of singularities may occur (seebelow).

Lastly, Sundman's result was generalized to the case ofn > 3 bodies byQiudong Wang in the 1990s.^[39] 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[0,∞).

There can be two types of singularities of then-body problem:

• collisions of two or more bodies, but for whichq(t) (the bodies' positions) remains finite. (In this mathematical sense, a "collision" means that two pointlike bodies have identical positions in space.)
• singularities in which a collision does not occur, butq(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 forn > 3 byPainlevé (seePainlevé conjecture). Examples of this behavior forn = 5 have been constructed by Xia^[40] and a heuristic model forn = 4 by Gerver.^[41]Donald G. Saari has shown that for 4 or fewer bodies, the set of initial data giving rise to singularities hasmeasure zero.^[42]

While there are analytic solutions available for the classical (i.e. nonrelativistic) two-body problem and for selected configurations withn > 2, in generaln-body problems must be solved or simulated using numerical methods.^[21]

For a small number of bodies, ann-body problem can be solved usingdirect methods, also calledparticle–particle methods. These methods numerically integrate the differential equations of motion. Numerical integration for this problem can be a challenge for several reasons. First, thegravitational potential is singular; it goes to infinity as the distance between two particles goes to zero. The gravitational potential may be "softened" to remove the singularity at small distances:^[21]Second, in general forn > 2, then-body problem ischaotic,^[43] which means that even small errors in integration may grow exponentially in time. Third, a simulation may be over large stretches of model time (e.g. millions of years) and numerical errors accumulate as integration time increases.

There are a number of techniques to reduce errors in numerical integration.^[21] Local coordinate systems are used to deal with widely differing scales in some problems, for example an Earth–Moon coordinate system in the context of a solar system simulation. Variational methods and perturbation theory can yield approximate analytic trajectories upon which the numerical integration can be a correction. The use of asymplectic integrator ensures that the simulation obeys Hamilton's equations to a high degree of accuracy and in particular that energy is conserved.

Direct methods using numerical integration require on the order of⁠1/2⁠n² computations to evaluate the potential energy over all pairs of particles, and thus have atime complexity ofO(n²). For simulations with many particles, theO(n²) factor makes large-scale calculations especially time-consuming.^[21]

A number of approximate methods have been developed that reduce the time complexity relative to direct methods:^[21]

• Tree code methods, such as aBarnes–Hut simulation, are spatially-hierarchical methods used when distant particle contributions do not need to be computed to high accuracy. The potential of a distant group of particles is computed using amultipole expansion or other approximation of the potential. This allows for a reduction in complexity toO(n logn).
• Fast multipole methods take advantage of the fact that the multipole-expanded forces from distant particles are similar for particles close to each other, and uses local expansions of far-field forces to reduce computational effort. It is claimed that this further approximation reduces the complexity toO(n).^[21]
• Particle mesh methods divide up simulation space into a three dimensional grid onto which the mass density of the particles is interpolated. Then calculating the potential becomes a matter of solving aPoisson equation on the grid, which can be computed inO(n logn) time usingfast Fourier transform orO(n) time usingmultigrid techniques. This can provide fast solutions at the cost of higher error for short-range forces.Adaptive mesh refinement can be used to increase accuracy in regions with large numbers of particles.
• P³M andPM-tree methods are hybrid methods that use the particle mesh approximation for distant particles, but use more accurate methods for close particles (within a few grid intervals). P³M stands forparticle–particle, particle–mesh and uses direct methods with softened potentials at close range. PM-tree methods instead use tree codes at close range. As with particle mesh methods, adaptive meshes can increase computational efficiency.
• Mean field methods approximate the system of particles with a time-dependentBoltzmann equation representing the mass density that is coupled to a self-consistent Poisson equation representing the potential. It is a type ofsmoothed-particle hydrodynamics approximation suitable for large systems.

In astrophysical systems with strong gravitational fields, such as those near theevent horizon of ablack hole,n-body simulations must take into accountgeneral relativity; such simulations are the domain ofnumerical relativity. Numerically simulating theEinstein field equations is extremely challenging^[21] and aparameterized post-Newtonian formalism (PPN), such as theEinstein–Infeld–Hoffmann equations, is used if possible. Thetwo-body problem in general relativity is analytically solvable only for the Kepler problem, in which one mass is assumed to be much larger than the other.^[44]

Most work done on then-body problem has been on the gravitational problem. But there exist other systems for whichn-body mathematics and simulation techniques have proven useful.

In large scaleelectrostatics problems, such as the simulation ofproteins and cellular assemblies instructural biology, theCoulomb potential has the same form as the gravitational potential, except that charges may be positive or negative, leading to repulsive as well as attractive forces.^[45]Fast Coulomb solvers are the electrostatic counterpart to fast multipole method simulators. These are often used withperiodic boundary conditions on the region simulated andEwald summation techniques are used to speed up computations.^[46]

Instatistics andmachine learning, some models haveloss functions of a form similar to that of the gravitational potential: a sum of kernel functions over all pairs of objects, where the kernel function depends on the distance between the objects in parameter space.^[47] Example problems that fit into this form includeall-nearest-neighbors inmanifold learning,kernel density estimation, andkernel machines. Alternative optimizations to reduce theO(n²) time complexity toO(n) have been developed, such asdual tree algorithms, that have applicability to the gravitationaln-body problem as well.

A technique inComputational fluid dynamics called Vortex Methods sees thevorticity in a fluid domain discretized onto particles which are then advected with the velocity at their centers. Because the fluid velocity and vorticity are related via aPoisson's equation, the velocity can be solved in the same manner as gravitation and electrostatics: as ann-body summation over all vorticity-containing particles. The summation uses theBiot-Savart law, with vorticity taking the place of electrical current.^[48] In the context of particle-laden turbulent multiphase flows, determining an overall disturbance field generated by all particles is ann-body problem. If the particles translating within the flow are much smaller than the flow's Kolmogorov scale, their linear Stokes disturbance fields can be superposed, yielding a system of 3n equations for 3 components of disturbance velocities at the location ofn particles.^[49][50]

• Celestial mechanics
• Gravitational two-body problem
• Jacobi integral
• Lunar theory
• Natural units
• Numerical model of the Solar System
• Stability of the Solar System
• Few-body systems
• N-body simulation, a method for numerically obtaining trajectories of bodies in an N-body system.

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• Concepts in astrophysics
• Classical mechanics
• Computational problems
• Computational physics
• Gravity
• Orbits

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