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 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. A popularization of the historical events and bickering between those parties, but more importantly about the results they produced. – 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.See 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.Rudolf Kurth has an extensive discussion in his book (see References) on planetary perturbations. An aside: these mathematically undefined planetary perturbations (wobbles) still exist undefined even today and planetary orbits have to be constantly updated, usually yearly. See Astronomical Ephemeris and the American Ephemeris and Nautical Almanac, prepared jointly by the Nautical Almanac Offices of the United Kingdom and the United States of America. 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.See Principia, Book Three, System of the World, "General Scholium", page 372, last paragraph. Newton was well aware that his mathematical model did not reflect physical reality. This edition referenced is from the Great Books of the Western World, Volume 34, which was translated by Andrew Motte and revised by Florian Cajori. This same paragraph is on page 1160 in Stephen Hawking, On the Shoulders of Giants, 2002 edition; is a copy from Daniel Adee's 1848 addition. Cohen also has translated new editions: Introduction to Newton's Principia, 1970; and Isaac Newton's Principia, with Variant Readings, 1972. Cajori also wrote History of Science, which is online. 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.For details of the serious error in Poincare's first submission 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 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 byMeyer 2009, pp. 27–28
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
Defining the momentum to be , Hamilton's equations of motion for the -body problem becomeMeyer 2009, p. 28
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.Meyer 2009, pp. 28–29 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 .Chenciner 2007
The moment of inertia of an -body system is given by
and the virial is given by . Then the Lagrange–Jacobi formula states thatMeyer 2009, p. 34
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
Special cases
Two-body problem
Any discussion of planetary interactive forces has always started historically with the two-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 (1667–1748) by classical theory 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:
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 ; .
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.For the classical approach, 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.For this approach see Lindsay's Physical Mechanics, Chapter 3: "Curvilinear Motion in a Plane", and specifically paragraphs 3–9, "Planetary Motion"; pp. 83–96. Lindsay presentation goes a long way in explaining these latter comments for the fixed two-body problem; i.e., when the Sun is assumed fixed.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 by convention.
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: