N-Body Problem

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N-body problem Summary

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Encyclopedia of Nonlinear Science

N-BODY PROBLEM

The Newtonian N-body problem of celestial mechanics is a mathematical generalization of the solar system. In the Euclidean space consider N points of masses m_i>0, i=1,…, N, which attract each other by a force directly proportional to the product of the masses and inversely proportional to the square of the distance. The equations of motion are given by the 6N-dimensional system of differential equations

(1)

where the upper dot represents differentiation with respect to the time variable; q=(q₁,…,q_N) is the configuration of the particle system, with giving the coordinates of the point of mass m_i, is the momentum, where M is a 3N-dimensional square matrix having on the diagonal the elements m₁, m₁, m₁,…, m_N, m_N, m_N and zeros in rest; G is the gravitational constant; and is the potential function, −U(q) representing the potential energy. Standard re suits of the theory of differential equations ensure the existence and uniqueness of an analytic solution for any initial value problem as long as the initial data do not belong to the collision set where

Figure 1. The Eulerian solutions of the three-body problem.

Isaac Newton formulated this problem in his master work Principia, but Leonhard Euler was the first to write the equations as we know them today. The case N=2, also called the Kepler problem, is completely solved (see Albouy (2002) for a recent discussion of some early solutions). The relative motion of one body with respect to the other is planar and, depending on the initial conditions, can be a circle, an ellipse, a parabola, a branch of a hyperbola, or a line, in which case a collision takes place in the future or in the past. Kepler’s laws (See Celestial mechanics) can be recovered from Equation (1).

For N≥3, very little is known about the N-body problem in spite of thousands of research papers written over more than three centuries. The case N=3 was the most studied since many of the results obtained could be generalized to any larger N. The first attempts to understand the three-body problem were quantitative, aiming at finding explicit solutions. In 1767, Euler found the collinear periodic orbits, in which three bodies of any masses move such that they oscillate along a rotating line (Euler, 1767, Figure 1) and in 1772, Joseph-Louis Lagrange discovered some periodic solutions that lie at the vertices of a rotating equilateral triangle that shrinks and expands periodically (Lagrange, 1772, Figure 2). Those solutions led to the study of central configurations, for which q″=kq for some constant k>0. Each central configuration provides a class of periodic orbits.

Another idea was to reduce the order of the system with the help of first integrals. Ten linearly independent integrals are known: three for the center of mass, three for the momentum, three for the angular momentum, and one for the energy (see Wintner, 1947). Together with a certain symmetry, these integrals allow the reduction of the three-body problem from dimension 18 to 7. But unfortunately the dimension of the problem cannot be further reduced. In 1887, Heinrich Bruns proved that there exist no more linearly independent integrals, algebraic with respect to q, p and t (Bruns, 1887), thus showing the limitations of the quantitative methods. This led Henri Poincaré to

Figure 2. The Lagrangian solutions of the three-body problem.

attempt a qualitative approach. His first prolific ideas appeared in a memoir published in 1890 (Poincaré, 1890), for which he was awarded the King Oscar Prize (see Barrow-Green, 1997; Diacu & Holmes, 1996). There he laid the foundations of several branches of mathematics, including dynamical systems, chaos, KAM theory, and algebraic topology.

Poincaré aimed to understand the geometry of the phase space and the relative behavior of orbits and, thus, to answer questions regarding stability, asymptotic motion at infinity, existence of periodic orbits, etc. An important problem in this direction was that of singular solutions, that is, those that tend to the collision set A. It took mathematicians almost a century to prove that for N≥5, there exist singular solutions that do not end in collisions but in pseudocollisions, which are orbits that become unbounded in finite time (see Diacu & Holmes, 1996). For N=4, the problem is still open.

Recently, a lot of interest has been in finding choreographies, that is, periodic orbits for which all the bodies move on the same closed curve. For more than two centuries the only known example was the class of Lagrangian solutions in the particular case that all the masses are equal and the ellipses are circles. With the help of variational methods, in 2000, a spectacular new class was proved to exist: three bodies of equal mass chase each other along a curve resembling the figure eight (Chenciner & Montgomery, 2000; Montgomery, 2001, Figure 3). There is numerical evidence that this periodic orbit is KAM stable, that is, the solutions through most initial conditions in some sufficiently small neighborhood of the orbit stay close to it for all time, while the other solutions leave the neighborhood very slowly. Unfortunately, the stability region seems to be very small. Numerical experiments suggest that the probability of finding an eight in the universe is somewhere between one per galaxy and one per universe. Hundreds of other choreographies have been numerically put into the evidence.

Still far from fully understood are the questions regarding various restricted three-body problems. In the elliptic one, for example, it is asked to determine the motion of one body, assumed to have zero mass, while the other two move on ellipses as in the Kepler problem with negative energy.

Figure 3. The figure eight solution of the 3-body problem.

Numerical methods are also of help for getting insight into the problem. But due to the apparently chaotic character of the motion, they must be implemented with care. Recently, much progress has been made in successfully applying scientific computation to various aspects of the general and restricted three-body problem.

Many of the ideas of the classical N-body problem can be adapted to related problems for understanding the motion of particle systems given by other potentials, like those of Manev and Schwarzschild (also used in celestial mechanics), Coulomb (atomic and molecular theories), and Lennard-Jones (crystal formation). Based on the Coulomb potential, Niels Bohr’s model of the hydrogen atom led to the development of quantum mechanics. Several other branches of science have profited from the study of the N-body problem.

FLORIN DIACU

N-Body Problem

N-BODY PROBLEM

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