# Three body problem

The chaotic movements of three bodies

The three-body problem of celestial mechanics consists in finding a solution (prediction) for the course of the orbit of three bodies under the influence of their mutual attraction ( Newton's law of gravitation ). In order to obtain quantitative results, it has to be solved numerically in the general case .

## Mathematical-historical

Since the discoveries of Johannes Kepler and Nikolaus Copernicus, the three-body problem has been considered one of the most difficult mathematical problems with which many well-known mathematicians such as Alexis-Claude Clairaut , Leonhard Euler , Joseph-Louis Lagrange , Thorvald Nicolai Thiele and George William Hill have dealt with over the centuries and Henri Poincaré employed. In the general case, the movement is chaotic and can only be calculated numerically.

## Special case

The special case that one of the three bodies has an infinitesimally small mass and its effect on the other two can be neglected is called a restricted three-body problem. It plays an important role in astronomy (e.g. in research satellites such as the Planetary Grand Tour ), which leads to the problem of Lagrange points .

## General statements

The two-body problem can be solved analytically using Kepler's laws . In contrast, in the case of more than two celestial bodies , the integrals are no longer algebraic integrals and can no longer be solved with elementary functions . At the beginning of the 20th century, Karl Frithiof Sundman was the first to provide an analytical solution to the three-body problem in the form of a convergent power series , on the assumption that the total angular momentum of the system does not vanish and therefore a three-way collision does not occur in which the distance between all three Body is zero. Sundman's solution cannot be used for practical calculations, however, since at least 10 to the power of 8,000,000 terms would have to be taken into account for the sum in order to achieve sufficient accuracy.

The stability of a three-body system is described by the Kolmogorow-Arnold-Moser theorem .

Approximate or exact solutions are possible in some cases:

• If the mass of one of the heavenly bodies is small, then the three-body problem is solved iteratively , nowadays with computers , or calculates orbit disturbances that the smallest (lightest) body suffers from the larger (heavier) ones.
• The already mentioned special case of the equilibrium of the force of attraction between two large (heavy) bodies on a vanishingly small (light) body (taking into account the apparent forces occurring in the rotating reference system ) in the Lagrange points L 1 to L 5 can be solved exactly . The inner point L 1 is used, for example, in space travel for solar research . The SOHO solar observatory is located there.
• In the case of three equal masses, there is a solution in which the objects follow one another on a common path that has the shape of an infinity sign ( ).${\ displaystyle \ infty}$

## generalization

The generalization of the three-body problem is the multi-body problem. General multi-body problems are treated with multi-body simulations .

## Trivia

In the science fiction novel The Three Suns by the Chinese author Cixin Liu , the three-body problem plays a crucial role in communicating with an extraterrestrial civilization .

## literature

• Richard Montgomery: The Three Body Problem. In: Spectrum of Science . 2020, issue 3, pp. 12-19.

Wikibooks: The Multibody Problem in Astronomy  - Learning and Teaching Materials

## Remarks

1. ↑ Based on a theorem by Poincaré , which generalizes a theorem by Bruns .
2. June Barrow-Green: The dramatic episode of Sundman. In: Section 9. The reception of Sundman's work.

"In 1930 David Beloriszky [...] calculated that if Sundman's series were going to be used for astronomical observations then the computations would involve at least 10 8,000,000 terms!"