“Drunken solution” to the problem of chaotic interaction of three bodies

(ORDO NEWS) — The three-body problem is one of the oldest problems in physics: it concerns the motion of a system of three bodies – for example, the Sun, Earth and the Moon – and the change in their orbits under the influence of mutual gravity. Scientists have been looking for a solution to this problem since the time of Newton.

With the gravitational interaction of two bodies, the resulting common trajectory becomes an ellipse, the shape of which can be predicted mathematically – that is, find a general solution to the problem. But under the influence of the third body, the system becomes chaotic and unpredictable – and this was shown by the French mathematician Jules Henri PoincarĂ© in 1889, who discovered that a general solution to the three-body problem could not be found, since such gravitational interactions are random in nature. In fact, this discovery gave rise to a new branch of science called chaos theory.

However, the numerical solution of the three-body problem is quite possible and successfully performed on modern computers. Such step-by-step calculations of the position of the bodies of a system of three bodies revealed two stages of “triple” interaction – at the first, “chaotic” stage, all the stars approach each other, and as a result of interactions one of the stars is ejected far into space, while the other two stars remain moving alone relative to the other in an elliptical orbit.

If the third star turns out to be thrown out not far enough to lose contact with the system, then it returns – and the first stage is repeated again. This “triple dance” ends when, in the second stage, one of the stars loses contact with the system and never returns.

In a new paper, Yonadav Barry Ginat of the Technion Israel Institute of Technology and colleagues proposed using the random nature of interactions to obtain a statistical solution for the entire two-step process. Instead of predicting the actual outcome, the scientists calculated the likelihood of any possible outcome of the first stage.

While chaos implies the impossibility of a complete solution, its random nature allows one to calculate the probabilities of each of the most common possible outcomes. In this way, a whole series of convergence of bodies can be modeled based on the theory of random motion, sometimes known as the “drunkard’s walk.” In this theory, each subsequent movement of the object is random,

Ginat and his colleagues in their new work calculated the probability of each possible configuration of the ternary system in the second stage (for example, the probability that the system will take a certain value of energy), and then connected all the individual stages using the theory of random walks to find the final probability of any possible states of the system after triple interaction. Now the problem of stage-by-stage gravitational interaction between bodies in a system of three objects can be considered completely statistically resolved, the authors note.

The work was published in the journal Physical Review X.

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