N-body problem: Difference between revisions

This article is about an ancient, classical problem in classical mechanics. For the problem in quantum mechanics, see Many-body problem.

The n-body problem is an ancient, classical problem 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 need to understand the motions of the Sun, planets and the visible stars. In the 20th century, globular-cluster star-structures became an important n-body problem too.

Newton actually stated the solution to the n-body problem was unsolvable. Also see: Isaac Newton's Principia. (The n-body problem in general relativity is considerably more difficult).

Newton did prove in his Principia a spherically-symmetric body can be modeled as a point-mass. Since gravity is responsible for the motions of planets and stars, Newton expressed the gravitational interactions of point-masses in terms of his Law of Universal Gravitation—which is not a general solution to the n-body problem. An aside, Newton's Law of Universal Gravitation can be derived via the potential function V(r)  =  m/r, where m is mass and r is the spatial vector. Euler actually transformed Newton's 2nd Law equation—of changing momentum (mass subject to changing geometry), assumed equivalent to net force (is really a theorem, and not even the general theorem) -- into an ordinary, second order differential equation.

The physical problem can be informally stated as: given only the present positions and velocities of a group of celestial bodies, predict their motions for all future time and deduce them for all past time. The two-body problem has been completely solved. For n = 3, solutions exist for special cases, and for the so-called Restricted 3-body Problem discussed below. A general solution in terms of first integrals is known to be impossible. An exact theoretical solution for arbitrary n can be given in terms of Taylor series, but in practice such an infinite series must be truncated, giving an approximate solution. In addition, many solutions by numerical integration exist, but these too are approximate solutions.

This page develops a general (statics) solution for determining the reactive forces for the general n-body Problem case, owning to a concentrated body force (force and moment). Rigid body (A 's = 1, see below) applications include distribution of useful loads in a finite element model (FEM); 3D rigid body rivet analyses; and Astronomy problems. Spatial (Astrodynamic) problems and the like are particular solutions to the general n-body Problem and once the initial positions and forces on {m_ζ}, ζ = 1, 2, ... N, bodies are known, the velocities and accelerations, .i.e. their motions, may be determined. Determining equations and numerical values for the latter is beyond this page's scope (see Astrodynamics). Structural analyses or soil analyses elastic solutions (A 's < 1) are possible too. Again, the latter applications are beyond the scope of this page.

Informal version of the Newton n-body problem

More precise: consider $n$ point masses $m_{1}$, ... ,$m_{n}$ in three-dimensional (physical) space. Suppose the attractive force experienced between each pair of particles is Newtonian (i.e., conform to Newton's Laws of Motion). Then determine the spatial position of each particle at every future (or past) moment of time if initial positions and velocities in space are specified for every particle at some instant, $t_{0}$ of time. Note, the solution requires a definition of some coordinate system first.

In mathematical terms this means to find a global solution of the initial value problem for the differential equations describing the $n$-body problem.

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).

Classical Mathematical formulation

A general, closed-form, n-body problem solution (i.,e., n-formulas) can not be derived employing Newton's Law of Universal Gravitation equation as a solution approach—in any form—as outlined below (as has been proven time and again historically)! That said, most textbooks present, if they do at all, this general solution approach for the n-body Problem as shown below, anyway.

The general n-body problem of celestial mechanics is an initial-value problem for ordinary, second order, differential equations. Given initial values for the positions $\mathbf {q} _{j}(0)$ and velocities ${\dot {\mathbf {q} }}_{j}(0)$ of n particles (j = 1,...,n) with $\mathbf {q} _{j}(0)\neq \mathbf {q} _{k}(0)$ for all mutually distinct j and k , find the solution of the second order system

$m_{j}{\ddot {\mathbf {q} }}_{j}=G\sum \limits _{k\neq j}{\frac {m_{j}m_{k}(\mathbf {q} _{k}-\mathbf {q} _{j})}{|\mathbf {q} _{k}-\mathbf {q} _{j}|^{3}}},j=1,\ldots ,n\qquad \qquad \qquad (1)$

where $m_{1},m_{2},\ldots m_{n}$ are constants representing the masses of n point-masses, $\mathbf {q} _{1},\mathbf {q} _{2},\ldots ,\mathbf {q} _{n}$ are 3-dimensional vector functions of the time variable t, describing the positions of the point masses, and G is the gravitational constant. This equation is Newton's second law of motion (left side) combined with his Law of Universal Gravitation (right side). The left-hand side is the mass times acceleration for a particular j^th particle (F = ma in vector form); whereas the right-hand side is the sum of the forces on that particle relative to another particle. The forces are assumed here to be gravitational as given by Newton's law of universal gravitation; thus, they are proportional to the masses involved, and vary as the inverse square of the distance between the masses. The power in the denominator is three instead of two to balance the vector difference in the numerator, which is used to specify the direction of the force.

G, the constant of proportional, is the parameter making Newton's Law emperialistic (i.e., capable of being verified or disproved by observation or experiment). See Lord Cavendish's experiment measuring G. G allows mass (matter phenomenon) when combined with distance (spatial phenomenon) to be linked to force ("force" phenomenon). All three phenomena have different units of measurement. Then what is G physically? Newton's equation is empirical because G is empirical, but (m₁m₂/r³)/r is not a force and so G(m₁m₂/r³)/r is not equal to F_g (gravitational force) -- but only equivalent to it (and that only by assuming we know what "force" is). His equation establishes only an equivalence relationship via G: the equation maps the mass and spatial phenomena -to- force. As will be explain below, force is a functional.

G allows Newton's 2nd Law and all others too, to not require incorporating any constants of proportionality.

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.

For n = 2, the problem was completely solved by Johann Bernoulli, see below. (see Two-body problem below).

General considerations for the Classical Solution

In the physical literature about the $n$-body problem ($n$ ≥ 3), sometimes reference is made to the impossibility of solving the $n$-body problem (via employing the above approach). 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).

Employing Newton's generalized Law of Universal Gravitation formula above, the $n$-body problem contains 6$n$ variables, since each point particle is represented by three space (displacement) and three momentum components. First integrals (for ordinary differential equations) are functions that remain constant along any given solution of the system, the constant depending on the solution. In other words, integrals provide relations between the variables of the system, so each scalar integral would normally allow the reduction of the system's dimension by one unit. Of course, this reduction can take place only if the integral is an algebraic function not very complicated with respect to its variables. If the integral is transcendent the reduction cannot be performed.

The $n$-body problem has 10 independent algebraic integrals:

• three for the center of mass;
• three for the linear momentum;
• three for the angular momentum;
• one for the energy.

This allows the reduction of variables to 6$n$ − 10. The question at that time was whether there exist other integrals besides these 10. The answer was given in the negative in 1887 by Heinrich Bruns:

Theorem (First integrals of the $n$-body problem) The only linearly independent integrals of the $n$-body problem, which are algebraic with respect to $q$, $p$ and $t$ are the 10 described above.

(This theorem was later generalized by Poincaré). These results however do not imply there does not exist a general solution to the $n$-body problem, or that the perturbation series (Lindstedt series) diverges. Indeed Sundman provided such a solution by means of convergent series. (See Sundman's theorem for the 3-body problem).

Numerical integration

Main article: N-body simulation

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.

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(n²), but tree structured methods, such as Barnes-Hut simulation, can improve this to O(n log n), or even to O(n) such as with the fast multipole method.

Two-body problem

Main article: Two-body problem

The two-body problem solved by classical theory correctly, is outlined here Consider 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). This is the fundamental equation of the two-body problem, solved by Johann Bernoulli in 1734 (and not by Newton!). Notice, for this approach, forces have to be determined first, then the equation of motion resolved.

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 Sun's true barycenter, resulting mainly only from the combined masses of Jupiter and the Sun. Science Program stated "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." 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.

Newton fixed the Sun for his simple calculations, but all following after him have also made the same mistake analytically of fixing the Sun. The n-body problem realistically is much more complicated owing to the Sun's wobble than previously thought.

The Real Motion v.s. Kepler's Apparent Motion

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 famous equations was curve-fit the apparent motion of the planets using Tycho Brahe's data, and not curve-fitting their true circular motions about the Sun (see Figure). See also Kepler's first law of planetary motion.

If the common center of mass (i.e., the barycenter) of the two bodies is considered to be at rest, then each body travels along a conic section which has a focus at the barycenter of the system. In the case of a hyperbola it has the branch at the side of that focus. The two conics will be in the same plane. The type of conic (circle, ellipse, parabola or hyperbola) is determined by finding the sum of the combined kinetic energy of two bodies and the potential energy when the bodies are far apart. (This potential energy is always a negative value; energy of rotation of the bodies about their axes is not counted here).

• If the sum of the energies is negative, then they both trace out ellipses.
• If the sum of both energies is zero, then they both trace out parabolas. As the distance between the bodies tends to infinity, their relative speed tends to zero.
• If the sum of both energies is positive, then they both trace out hyperbolas. As the distance between the bodies tends to infinity, their relative speed tends to some positive number.
  

Note: The fact a parabolic orbit has zero energy arises from the assumption the gravitational potential energy goes to zero as the bodies get infinitely far apart. One could assign any value to the potential energy in the state of infinite separation. That state is assumed to have zero potential energy (i.e. 0-joules) by convention.

Three-body problem

Main article: Three-body problem

Not as much is known about the n-body problem for n ≥ 3 as for n = 2. The case n = 3 was most studied and for many results can be generalized to larger n. Many of the early 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 taken in the definition and study of the problem of the movements of three bodies subject to their mutual gravitational attractions. His descriptions were verbal and geometrical, 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.
• 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 ${\ddot {q}}=kq$ for some constant k>0 .

Motion of three particles under gravity, demonstrating chaotic behaviour

Specific solutions to the Three-body problem result in chaotic motion with no obvious sign of a repetitious path. 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.

The Restricted Three-body Problem assumes the mass of one of the bodies is negligible; 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). 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; for another stable system, see Lagrangian point.

The restricted problem (both circular and elliptical) was worked on extensively by many famous mathematicians and physicists, notably Poincaré at the end of the 19th century. Poincaré's work on the Restricted Three-body Problem was the foundation of deterministic chaos theory. 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. This may be easier to visualize 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 maintaining the 60 degree-spacing ahead of and behind the less massive body almost in its orbit (although in reality neither of the bodies is truly stationary; 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 Lagrange points. See figure below:

Restricted 3-Body Problem

In the Restricted 3-Body Problem math model figure above (ref. Moulton), Lagrangian points L₄ and L₅ are where the Trojan planets resided; 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 h-circles and closed loops echo the electromagnetic fluxes issued from the Sun and Jupiter.

Theoretical Solutions

This is the most elemental way of solving the n-body Problem, the theoretical expression. It 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:

${\underset {i}{\overset {2}{x_{j}}}}(t)=G\sum _{k=0,k\neq i}^{n}{\frac {{\underset {k}{m}}\left({\underset {k}{\overset {0}{x_{j}}}}(t)-{\underset {i}{\overset {0}{x_{j}}}}(t)\right)}{\left(\left({\underset {k}{\overset {0}{x_{1}}}}(t)-{\underset {i}{\overset {0}{x_{1}}}}(t)\right){}^{2}+\left({\underset {k}{\overset {0}{x_{2}}}}(t)-{\underset {i}{\overset {0}{x_{2}}}}(t)\right){}^{2}+\left({\underset {k}{\overset {0}{x_{3}}}}(t)-{\underset {i}{\overset {0}{x_{3}}}}(t)\right){}^{2}\right){}^{3/2}}}$,

where (in ${\underset {i}{\overset {2}{x_{j}}}}(t)$) the upper index $2$ indicates the second derivative with respect to time $t$, $i$ represents the number of each body and $j$ the coordinate.

Because ${\underset {i}{\overset {0}{x_{j}}}}(t_{0})$ and ${\underset {i}{\overset {1}{x_{j}}}}(t_{0})$ are given as initial conditions, then every ${\underset {i}{\overset {2}{x_{j}}}}(t_{0})$ are known. Doing implicit derivation over every ${\underset {i}{\overset {2}{x_{j}}}}(t)$ results in ${\underset {i}{\overset {3}{x_{j}}}}(t)$ which at $t_{0}$ are known because each depends on known ${\underset {i}{\overset {k<3}{x_{j}}}}(t_{0})$ precalculated and given constants and then the Taylor series are constructed theoretically in such way, performing this process infinitely.

Any solution to the n-body program requires using Jean Le Rond D'Alembert's 1st and 2nd Principles.

HOOKEAN Switches and Springs, Point-Masses and the Inertia Tensor

It is convenient to define the scalar operator, a functional, to be employed in the solution of the n-body Problem. Index notation is used in this section . Indexes and scalars are plain italic text; vectors are in bold, italic text.

HOOKEAN Switches: Let there be a scalar function, 0 ≤ A_iζ ≤ 1, where i := x, y, z or 1, 2, 3, are the component directions, ζ := 1...N, is the point-mass m_ζ coordinate number. The A_iζ's are functionally the feasible, translational reaction component's allowable degrees of freedom (DOF) at P_iζ⇒ (x_iζ, y_iζ, z_iζ), specified for any and all point-masses m_ζ, where the N point-masses m_ζ are defined in a right-handed, Euclidean coordinate system space with (x, y, z) coordinates.

Applying HOOKEAN switches A's at the supports or reaction points for every feasible translational reaction, i.e., for each P_xζ, P_yζ, P_zζ, would be the same as showing those feasible or allowable translational vectors visually on a free-body diagram; but instead of showing by default all three DOF possible translational vectors, say ( P _ζ )_i, with i = 1,2,3. There must be at least three point-masses for stability, having a total of six DOF , meaning there must be at least six feasible reactions for these three points; i.e., six A_iζ's. (This latter requirement is the same as balancing an aircraft's FEM on three points and then applying Jean Le Rond D'Alembert's two Principles.)

File:BOSCOVITCH Theorem.jpg

Radius Vectors and Massless Points

(Definition continues.) Let the radius vector be:

r _ζ := r_xζA_xζ i + r_yζA_yζ j + r_zζA_zζ k ,

where i, j, k are the unit directional vectors, and the A's are the HOOKEAN switches or springs defined above. For example, for m_i at P_iζ ⇒ (x_iζ, y_iζ, z_iζ), if all A_iζ = 0 then r _ζ = 0; if all A_iζ = 1 then there are three reaction components. These conditions describe HOOKEAN switches. All other values of A_iζ ≤ 1, but not zero, describe HOOKEAN springs.

(Definition continues.) Then for each point P_iζ there may be 0, or up to 3 DOF, meaning m_i may be, depending on A_iζ's components for any particular DOF in that direction, elastic or rigid body. If r _ζ has built in HOOKEAN Switches then r _ζ are said to be a HOOKEAN vector.

The concept is HOOKEAN switches allow point-masses to respond, or not respond, as reaction forces in selected directions by setting selected A_iζ's components to 0; and vice-verse. When all the feasible A_iζ = 1 the solution is a rigid-body solution. In this way HOOKEAN switches allows for the solution of the n-body Problem. HOOKEAN springs allows a type of finite element model (FEM) to be constructed (i.e., 3D rivet analyses).

Let M := m₁ + m₂ + m₃ + …+ m_N, for ζ = 1, 2, ... N point-masses. Of course for each m_ζ there is a spatial point P_iζ ⇒ (x_iζ, y_iζ, z_iζ).

Theorem: Let r _iζ at P_iζ, where ζ = 1,...N mass points, be a HOOKEAN vector for each m_ζ in all (x, y, z) space, for a right-hand, rectangular Cartesian coordinate system, then the six massless barycenters (six owning to using index notation) for {M} are, by simple statics:

x_xz = ΣA_zζx_ζ/ΣA_zζ, x_xy = ΣA_yζx_ζ/ΣA_yζ, y_yz = ΣA_zζy_ζ/ΣA_zζ;

y_xy = ΣA_xζy_ζ/ΣA_xζ, z_yz = ΣA_yζz_ζ/ΣA_yζ, z_xz = ΣA_xζz_ζ/ΣA_xζ.

File:Centroids.jpg

Continuing, the massless centroidal mass moments of inertia for N masses, m_i, again by simple statics are:

I_xx = ΣA_yζz_ζ² + ΣA_zζy_ζ² - y_yz²ΣA_zζ - z_yz²ΣA_yζ ;

I_yy = ΣA_xζz_ζ² + ΣAzζx_ζ² - x_xz²ΣAzζ - z_xz²ΣAx_ζ ;

I_zz = ΣA_xζy_ζ² + ΣA_zζx_ζ² - x_xy²ΣA_yζ - y_xy²ΣA_xζ ;

I_xy = - (ΣA_zζx_ζy_ζ + x_xzy_yzΣA_zζ) ;

I_yz = - (ΣA_xζy_ζz_ζ + y_xyz_xzΣA_xζ) ;

I_yz = - (ΣA_yζx_ζz_ζ + x_xyz_yzΣA_yζ) ;

where, say x_xy, etc, are the barycenters calculated above The reason these equations are presented as a theorem will be explained below.

But before that...qualification for a right-hand coordinate system arises because some USA Aerospace firms use the left-hand system, making their signs for the cross products positive. If the inertia tensor if symmetric and positive-definite (via the value of the its determinant), then the coordinate system is right-handed ― always check this, and if not then your signs are wrong.

Applied forces and moments may be HOOKEAN too. These HOOKEAN definitions (and concept) are missing from physics textbooks, but are somewhat similar in concept to grid-point releases in a finite elements analyses. HOOKEAN points are not Lagrangian coordinates.

The Force

Interestingly, if empirical force laws and Euclidean geometry are true, and the Principle of Equilibrium is true of course, then any logical combination therein is also true. In all that follows is naught but simple physics (statics).

If, for example, when the vector s_i = Є _ijk r_j θ _k (analogous to s = rθ) from simple geometry is combined with D'Alembert's two Principles R = 0 and M = 0 , then a formula called The Force, can be developed giving the reactive forces for a pattern of points subject to an external resultant force F _p and moment M₀ . r_j may be a HOOKEAN vector, s_i covers all space. It will be shown The Force can be used to completely solve any n-body Problem for any n (an aside, see Meirovitch, pages 414 and 413 for the Three-body Problem).

Let the set {r _ζ} contain at least three spatial reaction points: r := {(r _ζ)_i: ζ ≥ 3, i = 1, 2, 3} in Euclidean R³, react to an applied resultant force F _i and moment ( M ₀)_i, also in R³ at (x_p, y_p, z_p). Then reaction displacements (∆s_ζ)_i of any point (r _ζ)_i owning to angular rotations (∆θ_ζ)_i are:

Reaction Displacements: (∆ s _ζ)_i := Є _ijk r_j∆ θ _k.

Assume only infinitesimal angular displacements (∆θ_ζ)_i are associated with reaction point motions (∆ s _ζ)_i in any instant of time. Further, since this is an instant of time (time is fixed) the solution becomes one of statics. Now let:

( r bar _ζ)_j = ( r _ζ)_j - rbar _j,

where rbar _j is the centroid of the reaction set {r _ζ}; and ( r bar _ζ)_j is a particular reaction point, number ζ. The HOOKEAN (∆ s _ζ)_i equations in expanded scalar form are exactly analogous to M = r x F or M = Є _ijk r_j F _k.

(∆ s _ζ)₁ = ( r _ζ)₂(∆θ_ζ)₃ + ( r _ζ)₃(∆θ_ζ)₂ ;

(∆ s _ζ)₂ = ( r _ζ)₃(∆θ_ζ)₁ + ( r _ζ)₁(∆θ_ζ)₃ ;

(∆ s _ζ)₃ = ( r _ζ)₁(∆θ_ζ)₂ + ( r _ζ)₂(∆θ_ζ)₁ ;

If the reaction points' material is structural; i.e., elastic ( σ = E ε ), then it will be further assumed the structure will conform to a similar linear law, say HOOKE's Law, (F = k''x) of load-deformation proportionality, where k is the spring rate (this assumption is not entirely necessary as F = f(x) can be any functional without its exact definition given).

The load center is defined to be at (x_cg, y_cg, z_cg), who coordinates are basic input. (xbar_i, ybar_i, zbar_i) -- are use later to calculate the sum of the moments -- are equal to the {P_ζ} set for ζ = 1, ... N reaction points, and are the centers of gravity of those point-masses. Real masses may be reduced to point-masses the same way NEWTON did it. In this analysis physical real mass is not a variable; only forces, reactions, moments, and geometry are real variables. It is convenient to use HOOKE's Law for the purposes later of eliminating the ∆θ's: k x ≃ r ∆θ. (Other relationships may be employed.)

( F _ζ )_k = k(∆ s _ζ )_k = kЄ_klm(r_ζ)_l(∆θ_ζ)_m ,

where (F _ζ)_k are the symmetrical components of force at the reaction points. Taking moments at the centroid (the rest of this development is just simple but messy Algebra…):

( M _ζ)_i = kЄ_ijk( r bar _ζ)_j( F _ζ)_k = kЄ_ijk( r bar _ζ)_jЄ_klm( r bar _ζ)_l(∆θ_ζ)_m ,

where ( M _ζ)_i are the individual resultant moments about the centroid owning to each force ( F _ζ)k at each reaction point ζ. The last form with the two Є can be simplified.

Let k = 1 for the time being (k cancels out later anyway). Then in expanded scalar form:

( M _ζ)₁ = - [(r bar _ζ)₂² + (r bar _ζ)₃²] (∆θ_ζ)₁ + (r bar _ζ)₁(r bar _ζ)₂(∆θ_ζ)₂ + (r bar _ζ)₁(r bar _ζ)₃(∆θ_ζ)₃ ;

( M _ζ)₂ = (r bar _ζ)₁(r bar _ζ)₂(∆θ_ζ)₁ - [(r bar _ζ)₁² + (r bar _ζ)₃²](∆θ_ζ)₂+ (r bar _ζ)₂(r bar _ζ)₃(∆θ_ζ)₃ ;

( M _ζ)₃ = + (r bar _ζ)₁(r bar _ζ)₃(∆θ_ζ)₁ + (r bar _ζ)₂(r bar _ζ)₃(∆θ_ζ)₂2 - [(r bar _ζ)₁² + (r bar _ζ)₃²] (∆θ_ζ)₃ ;

(It can be easily seen how easy it would be to put these symmetrical equation in matrix form.) The resultant moments at each point owning then to all the reaction components at all the points are:

∑( M _ζ)₁ =∑(- [(r bar _ζ)₂² + (r bar _ζ)₃²] (∆θ_ζ)₁ + (r bar _ζ)₁(r bar _ζ)₂(∆θ_ζ)₂ + (r bar _ζ)₁(r bar _ζ)₃(∆θ_ζ)₃) ;

∑( M _ζ)₂ = ∑((r bar _ζ)₁(r bar _ζ)₂(∆θ_ζ)₁ - [(r bar _ζ)₁² + (r bar _ζ)₃²](∆θ_ζ)₂+ (r bar _ζ)₂(r bar _ζ)₃(∆θ_ζ)₃) ;

∑( M _ζ)₃ = ∑((+r bar _ζ)₁(r bar _ζ)₃(∆θ_ζ)₁ + (r bar _ζ)₂(r bar _ζ)₃(∆θ_ζ)₂2 - [(r bar _ζ)₁² + (r bar _ζ)₃²](∆θ_ζ)₃) ;

Of course the r terms above are immediately recognized as being part of the massless inertia above.

The resultant moments ∑( M _ζ)_i for each point are equal to the resultant moments at the centroid of the reaction pattern per NEWTON's Third Law and by equilibrium considerations (i.e, D'Alembert's Second Principle):

∑( M _ζ)_i - ( M _R)_i = 0.

File:Moment Transfer.jpg

Transferring a Force Produces a Moment

Some explanation is in order before preceding: First, referring to the cantilever beam with the off-set load, if the load F₁ is moved to the cantilever's tip's end along the shaft's axis, then (to calculate loads at the root) if the original force, F₁, is reduplicated and transferred to the tip as F ₂, then it has to be balanced by an up-load, F₃, for the two to remain in equilibrium. This up-load F₃, combined with the original down-load F₁, is a couple and equivalent to a moment M (torque). Second, at the root of the cantilever the reaction forces at the attachment points (not shown) not only react with the vertical force F₂ in pure upward vertical shear divided by the number of attachment points, but by horizontal reaction couples in- and out- of the wall too. The new couple (moment M ) is reacted by only reaction couples at the root parallel to the wall. In this way (symbolically) the external moment M_external = M _internal = rF_reaction; i.e., the external moment(s) equal the in-plane and out-of-plane static reaction couples at the attachments in the wall (D'Alembert's Second Principle). Looking at the big picture, the total external forces and moments are equal to the sum of all the internal coupling reactions (moments) plus the direct shear reactions (translational reactions) divided by the number of attachments. D'Alembert's two Principles combined.

(Continuing.) The reaction forces owning to moment imbalances are added to the symmetric reaction force components to give the total reaction forces at each reaction point (explanation above). Let this resultant moment be ( M _R)_i = kЄ_ijk( r bar _p)_j R _k (notice the "k" in the equation), where

(r bar _p)_j = (r _p )_j - r bar _j ,

and (r _p)_j is the position of the applied force and moment. Then in expanded scalar form:

( M _R)₁ = (r bar _p)₂R₃ - (r bar _p)₃R₂ + ( M _o)₁ ;

( M _R)₂ = (r bar _p)₃R₁ - (r bar _p)₁R₃ + ( M _o)₂ ;

( M _R)₃ = (r bar _p)₁R₂ - (r bar _p)₂R₁ + ( M _o)₃ ;

When the (∆θ_ζ)_i terms are isolated in ∑( M _ζ)_i - ( M _R)_i = 0, they are substituted back into ( F _ζ)_k, resulting in their elimination. Putting all the above together results in an equation call The Force (whose proof is by calculating equilibrium again once the reactions have been found, see below):

( P _ζ)_i = [{( F _L)_i/(Σ(A_ζ)_i)}^T_1,3 + [x_s]_i[ I _s]⁻¹[ [x_L]_i{ F _L}_i^T + { M _o}_i^T]_3,3]{(A_ζ)_i}_1,3^T.

In the above equation :

• i = 1, 2, 3, ζ = 1, 2, 3, ... N point-mass m_i coordinate number; brackets [ ] and { } represent matrices where T is for transpose and -1 is for inverse; and subscript L stands for Load, s stands for support;
• ( P _ζ )_i are the vector reactions at each support point P_i⇒ (x_i, y_i, z_i);
• ( F _L)_i are the three components of the applied force, transposed; ( M ₀)_i are the the three components of the applied moment, transposed; F and M _o are basic inputs;
• The load centers at (x_cg, y_cg, z_cg is basic input.
• [ I _s]⁻¹ is the inverse of the inertia tensor of the reaction pattern, is a symmetrical 3x3 matrix;
• [x_L]_i is the position of the applied force and moment, 3x3 matrix; [x_L]_i( F _L) are the resulting couples in the support pattern caused by moving the applied force over to the pattern's centroids.
• [x_s]_i are the centroids of the reaction pattern, 3x3 matrix;
• [x_s]_i[ I _s]⁻¹[[x_L]_i{ F _L}_i^T + { M _o}_i^T]_3,3] are the unbalanced moments reacted as couples, 3x3 matrix;
• {( F _L)_i/(Σ(A_ζ)_i)}^T_1,3 are the symmetrical (translational) reactions.

This solution takes an external force and moment and moves it over to the centroids of the reactions or supporting pattern (system); and by incorporating the massless inertial properties of that pattern, beams the applied loads to the support loads. Although somewhat messy algebraically, it's that simple. It's analogous to a 3D rigid body, rivet analysis.

The [x_s]_i[ I _s]⁻¹[x_L] term is a congruent transformation (Meirovitch) and for all physical entities having like congruent transformations, like stress, strain, they are always positive definite if the right-hand rule is used: a fundamental property of all matter. {A_ζ}_i are the HOOKEAN Switches defined above. If a particular motion is planar then out-of-plane reaction forces (P _ζ)_i will be zero by default. ( P _ζ)_i has been computerized, see below.

Equilibrium of Forces and Moments

The sum of the forces for the entire support or reaction points are:

The load center at (x_cg, y_cg, z_cg) is basic input. (xbar_i, ybar_i, zbar_i), equal to the {P_ζ} set for ζ = 1, ... N reaction points, are the center of gravity of the point-masses. Then the sum of the moments for the entire supports or reactions are:

Σ M _xx = Σ[(ybar_i - y_cg) P _zi - (zbar_i-z_cg) P _yi] ;

Σ M _yy = Σ[(zbar_i - z_cg) P _xi - (xbar_i-x_cg) P _zi] ;

Σ M _zz = Σ[(xbar_i - y_cg) P _yi - (ybar_i-y_cg) P _xi] ;

Solar System Motions

n-body particle simulations can be effected via computer

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 $t^{1/3}$ 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 $\mathbf {L} \neq 0$ he removed all real singularities from the transformed equations for the 3-body problem.
   
3. The next step consisted in showing that if $\mathbf {L} \neq 0$, then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by using the Cauchy 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 $s=t^{1/3}$ (the new variable after the regularization) and if $|\mathop {\text{Im}} \,s|\leq \beta$ then this map is given by:
   

$\sigma ={\frac {e^{\pi s/(2\beta )}-1}{e^{\pi s/(2\beta )}+1}}.$

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 $[0,\infty )$.

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 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 and Gerver.

See also

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

References

• Heggie, Douglas and Hut, Piet: The Gravitational Million-Body Problem, A Multidisciplinary Approach to Star Cluster Dynamics, Cambridge University Press, 357 pages, 2003.
• 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.
• 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.
• Science Program's “ 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.
• Batin, Richard H.: An Introduction to The Mathematics and Methods of Astrodynamics, AIAA, 1987.
• Gallian, Dave A. and Wilson, Henry E.: "The Integration of NASTRAN Into Helicopter Airframe Design/Analysis," American Helicopter Society Pub., May 1973.
• 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.

• 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. ISSN 0923-2958. {{cite journal}}: |access-date= 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.

External links

• Three-Body Problem at Scholarpedia
• More detailed information on the three-body problem
• Regular Keplerian motions in classical many-body systems
• Applet demonstrating chaos in restricted three-body problem
• Applets demonstrating many different three-body motions
• On the integration of the n-body equations
• Java applet simulating Solar System
• Java applet simulating a ring of bodies orbiting a large central mass
• Java applet simulating dust in the Solar System
• Java applet simulating a stable solution to the equi-mass 3-body problem
• Java applet simulating choreographies and other interesting n-body solutions
• A java applet to simulate the 3-d movement of set of particles under gravitational interaction
• Javascript Simulation of our Solar System
• The Lagrange Points - with links to the original papers of Euler and Lagrange, and to translations, with discussion
• http://ftp.math.utah.edu/pub/tex/bib/toc/sciam1980.html#242%281%29:January:1980
• http://www.ftp.cica.indiana.edu