General properties of the many-body problem
 
If the masses of the planets were vanishingly small compared to the Sun’s mass, then the orbit of any planet would be unchanging and the six elements would be constant. Indeed, Kepler’s three laws are the solution to the many-body problem in such a case. But the planetary masses are by no means negligible and, in the case of comets, near approaches to planets can occur so that, in general, the problem is much more complicated.
In the past three centuries, it has inspired (and frustrated!) many eminent astronomers and mathematicians. It is perhaps not obvious that even the three-body problem is of a much higher degree of complexity than the two-body problem. But if we consider that each body is subject to a complicated variable gravitational field due to its attraction by the other two, such that close encounters with either may be brought about, the result of each near-collision being an entirely new type of orbit, we see that it would require a general formula of unimaginable complexity to describe all the consequences of all such encounters.
In point of fact, several general and useful statements may be made concerning the many-body problem and these were proved quite early on in its history. They were known to Euler (1707–83) but since then no further overall properties have been discovered or are likely to be.
The statements follow from the only known integrals of the differential equations and refer to the centre of mass of the system, the total energy of the system and its total angular momentum. Without saying anything about the trajectories of the individual particles, the following statements can be made:
(a) The centre of mass of the system moves through space with constant velocity, i.e. it moves in a straight line at a fixed speed.
(b) The total energy of the system (the sum of all the kinetic energies and potential energy) is constant. Thus, although there is a continual trade-off among the members in kinetic energy and potential energy, the total energy is unaffected.
(c) The total angular momentum of the system is constant.
In addition to these properties, particular solutions of the three-body problem that exist when certain relationships hold among the velocities and mutual distances of the particles were found by Lagrange. He showed that if the three bodies occupy the vertices of an equilateral triangle, their speeds being equal in magnitude and inclined at the same angle to each mutual radius vector, they will remain in an equilateral triangle formation, though the triangle will rotate and may change its size. Lagrange also showed that if the three bodies are placed on a straight line at mutual distances depending upon
the ratios of their masses, they will remain on that line, though it will rotate. Although these equilateral triangle and collinear solutions of the three-body problem were thought to be of theoretical interest only at the time of their presentation, it was subsequently discovered that they occur in the Solar System. Two groups of asteroids, called the Trojans, revolve about the Sun in Jupiter’s orbit, so that their periods of revolution equal that of Jupiter. In their orbit about the Sun, they oscillate about one or other of the two points 60◦ ahead or behind Jupiter’s heliocentric position. Among Saturn’s moons, Telesto and Calypso remain 60◦ ahead or behind the more massive Tethys while Helene, in Dione’s orbit, keeps 60◦ ahead of Dione.