n-body problem
[1] Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets, and visible stars.
[2] The n-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: 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.
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.
After Newton's time the n-body problem historically was not stated correctly because it did not include a reference to those gravitational interactive forces.
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 problem as stated originally was finally solved by Karl Fritiof Sundman for n = 3 and generalized to n > 3 by L. K. Babadzanjanz[12][13] and Qiudong Wang.
[14] The n-body problem considers n point masses mi, i = 1, 2, …, n in an inertial reference frame in three dimensional space ℝ3 moving under the influence of mutual gravitational attraction.
Because T and U are homogeneous functions of degree 2 and −1, respectively, the equations of motion have a scaling invariance: if qi(t) is a solution, then so is λ−2/3qi(λt) for any λ > 0.
The two-body problem (n = 2) 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.
[23][24][25] 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 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.
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.
[10] In fact, Newton's Universal Law does not account for the orbit of Mercury, the asteroid belt's gravitational behavior, or Saturn's rings.
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.
Many earlier attempts to understand the three-body problem were quantitative, aiming at finding explicit solutions for special situations.
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 by Poincaré at the end of the 19th century.
[31] This formulation has been highly relevant in the astrodynamics, mainly to model spacecraft trajectories in the Earth-Moon system with the addition of the gravitational attraction of the Sun.
Perturbative approximation works well as long as there are no orbital resonances in the system, that is none of the ratios of unperturbed Kepler frequencies is a rational number.
[35] Central configurations have played an important role in understanding the topology of invariant manifolds created by fixing the first integrals of a system.
[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 by Abel and Galois about the impossibility of solving algebraic equations of degree five or higher by means of formulas only involving roots).
[41] Donald G. Saari has shown that for 4 or fewer bodies, the set of initial data giving rise to singularities has measure zero.
Second, in general for n > 2, the n-body problem is chaotic,[43] which means that even small errors in integration may grow exponentially in time.
Variational methods and perturbation theory can yield approximate analytic trajectories upon which the numerical integration can be a correction.
The use of a symplectic 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 of 1/2n2 computations to evaluate the potential energy over all pairs of particles, and thus have a time complexity of O(n2).
Alternative optimizations to reduce the O(n2) time complexity to O(n) have been developed, such as dual tree algorithms, that have applicability to the gravitational n-body problem as well.
[48] In the context of particle-laden turbulent multiphase flows, determining an overall disturbance field generated by all particles is an n-body problem.
