N-body Problem

In physics, the -body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally.Leimanis and Minorsky: Our interest is with Leimanis, who first discusses some history about the -body problem, especially Ms. Kovalevskaya's 1868–1888 twenty-year complex-variables approach, failure; Section 1: "The Dynamics of Rigid Bodies and Mathematical Exterior Ballistics" (Chapter 1, "The motion of a rigid body about a fixed point (Euler and Poisson equations)"; Chapter 2, "Mathematical Exterior Ballistics"), good precursor background to the -body problem; Section 2: "Celestial Mechanics" (Chapter 1, "The Uniformization of the Three-body Problem (Restricted Three-body Problem)"; Chapter 2, "Capture in the Three-Body Problem"; Chapter 3, "Generalized -body Problem"). Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets, and visible stars. In the 20th century, understanding the dynamics of globular cluster star systems became an important -body problem.See references cited for Heggie and Hut. The -body problem in general 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:

The two-body problem has been completely solved and is discussed below, as well as the famous ''restricted'' three-body problem.

History

Knowing three orbital positions of a planet's orbit – positions obtained by Sir Isaac Newton from astronomer John Flamsteed – 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. 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. 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 the -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 (-body) interactions have historically made any exact solution intractable. Ironically, this conformity led to the wrong approach.

After Newton's time the -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 -body problem is unsolvable because of those gravitational interactive forces. Newton said in his ''Principia'', paragraph 21:

Newton concluded via his third 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 to Jean Le Rond D'Alembert's non-Newtonian first and second Principles and to the nonlinear -body problem algorithm, the latter allowing for a closed form solution for calculating those interactive forces.

The problem of finding the general solution of the -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:

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 to Poincaré, even though he did not solve the original problem. (The first version of his contribution even contained a serious error.) 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 and generalized to by L. K. Babadzanjanz and Qiudong Wang.

General formulation

The -body problem considers point masses in an inertial reference frame in three dimensional space moving under the influence of mutual gravitational attraction. Each mass has a position vector . Newton's second law says that mass times acceleration is equal to the sum of the forces on the mass. Newton's law of gravity says that the gravitational force felt on mass by a single mass is given by
$\mathbf_ = \frac \cdot \frac = \frac,$
where is the gravitational constant and is the magnitude of the distance between and ( metric induced by the norm).

Summing over all masses yields the -body equations of motion:where is the ''self-potential'' energy
$U = -\sum_ \frac.$

Defining the momentum to be , Hamilton's equations of motion for the -body problem become
$\frac = \frac \qquad \frac = -\frac,$
where the Hamiltonian function is
$H = T + U$
and is the kinetic energy
$T = \sum_^n \frac.$

Hamilton's equations show that the -body problem is a system of first-order differential equations, with initial conditions as initial position coordinates and initial momentum values.

Symmetries in the -body problem yield global integrals of motion that simplify the problem. Translational symmetry of the problem results in the center of mass
$\mathbf = \frac$
moving with constant velocity, so that , where is the linear velocity and is the initial position. The constants of motion and represent six integrals of the motion. Rotational symmetry results in the total angular momentum being constant
$\mathbf = \sum_^n \mathbf_i \times \mathbf_i,$
where × is the cross product. The three components of the total angular momentum yield three more constants of the motion. The last general constant of the motion is given by the conservation of energy . Hence, every -body problem has ten integrals of motion.

Because and are homogeneous functions of degree 2 and −1, respectively, the equations of motion have a scaling invariance: if is a solution, then so is for any .Chenciner 2007

The moment of inertia of an -body system is given by
$I = \sum_^n m_i \mathbf_i \cdot \mathbf_i = \sum_^n m_i \left\, \mathbf_i\right\, ^2$
and the ''virial'' is given by . Then the ''Lagrange–Jacobi formula'' states that
$\frac = 2T - U.$

For systems in ''dynamic equilibrium'', the longterm time average of is zero. Then on average the total kinetic energy is half the total potential energy, , which is an example of the virial theorem for gravitational systems. If is the total mass and 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 isTrenti 2008
$t_\mathrm = \left(\frac\right)^.$

Special cases

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 Wikipedia pages. Here though, these subjects are discussed from the perspective of the -body problem.

The two-body problem () was completely solved by Johann Bernoulli (1667–1748) by classical theory (and not by Newton) by assuming the main point-mass was fixed; this is outlined here.See Bate, Mueller, and White, Chapter 1: "Two-Body Orbital Mechanics", pp. 1–49. These authors were from the Department of Astronautics and Computer Science, United States Air Force Academy. Their textbook is not filled with advanced mathematics. Consider then the motion of two bodies, say the Sun and the Earth, with the Sun fixed, then:
$\begin m_1 \mathbf_1 &= \frac(\mathbf_2-\mathbf_1) &&\quad\text \\ m_2 \mathbf_2 &= \frac(\mathbf_1-\mathbf_2) &&\quad\text \end$

The equation describing the motion of mass relative to mass is readily obtained from the differences between these two equations and after canceling common terms gives:
$\mathbf + \frac \mathbf = \mathbf$
Where

* is the vector position of relative to ;
* is the ''Eulerian'' acceleration ;
*.

The equation 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.

It is incorrect to think of (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 mutual barycenter, and this two-body problem can be solved exactly, such as using Jacobi 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 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 that Newton's ''Law of Universal Gravitation'' did not hold for the forces associated with elliptical orbits.See I. Bernard Cohen's ''Scientific American'' article. In fact, Newton's Universal Law does not account for the orbit of Mercury, the asteroid belt's gravitational behavior, or Saturn's rings. Newton stated (in section 11 of the ''Principia'') 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 mathematical idealization. See also Kepler's first law of planetary motion.

Three-body problem

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

In the past not much was known about the -body problem for . The case 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 Euler's three-body problem is the special case in which two of the bodies are fixed in space (this should not be confused with the circular 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

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