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. Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets, and visible stars.
The classical physical problem can be stated as follows:
The two-body problem has been completely solved and is discussed below. For three or more bodies the problem can only be solved completely in particular cases. In general, the problem is chaotic and can only be solved numerically.
The -body problem in general relativity is considerably more difficult to solve.
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 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 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 bywhere is the gravitational constant and is the magnitude of the distance between and .
Summing over all masses yields the -body equations of motion:where is the self-potential energy
Defining the momentum to be, Hamilton's equations of motion for the -body problem become
where the Hamiltonian function is
and is the kinetic energy
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
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
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.
The moment of inertia of an -body system is given by
and the virial is given by. Then the Lagrange–Jacobi formula states that
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, then the critical time for a system to settle down to a dynamic equilibrium is
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 are discussed on other Wikipedia pages. Here though, these subjects are discussed from the perspective of the -body problem.The two-body problem was first solved by Isaac Newton in 1687 using geometric methods, but a complete solution was given in 1710 by Johann Bernoulli by classical theory by assuming the main point-mass was fixed; this is outlined here. Consider then the motion of two bodies, say the Sun and the Earth, with the Sun fixed, then:
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:
Where
- is the vector position of relative to ;
- is the Eulerian acceleration ;
- .
It is incorrect to think of 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. 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. 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 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.
- 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.
- 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.
- 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 with its plot of the restricted three-body problem solution. An aside, see Meirovitch's book, pages 413–414 for his restricted three-body problem solution.
Moulton's solution may be easier to visualize if one considers the more massive body to be stationary in space, and the less massive body to orbit around it, with the equilibrium points maintaining the 60° spacing ahead of, and behind, the less massive body almost in its orbit. For sufficiently small mass ratio of the primaries, these triangular equilibrium points are stable, such that massless particles will orbit about these points as they orbit around the larger primary. The five equilibrium points of the circular problem are known as the Lagrangian points. See figure below:
In the restricted three-body problem math model figure above, the Lagrangian points L4 and L5 are where the Trojan planetoids resided ; is the Sun and 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. The -circles and closed loops echo the electromagnetic fluxes issued from the Sun and Jupiter. It is conjectured, contrary to Richard H. Batin's conjecture, the two 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 the mass of one of the bodies is negligible. 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.
The restricted problem 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. In the restricted problem, there exist five equilibrium points. Three are collinear with the masses 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.