Novel theory addresses centuries-old physics problems

The ‘three-body problem’, the term coined to predict the motion of three gravitational bodies in space, is essential for understanding a variety of astrophysical processes, as well as a large class of mechanical problems, and has some of the best occupied physicists around the world, astronomers and mathematicians for more than three centuries. Their efforts led to the discovery of several important fields of science; yet its solution remained a mystery.

At the end of the 17th century, Sir Isaac Newton succeeded in explaining the motion of the planets around the sun by a law of universal gravitation. He also tried to explain the motion of the moon. As both the earth and the sun determine the motion of the moon, Newton became interested in the problem of predicting the motion of three bodies moving in space under the influence of their mutual attraction, a problem that later became known. as’ the three-body problem. ‘

Unlike the two-body problem, however, Newton could not find a general mathematical solution to it. The problem with three body parts was indeed easy to define and yet difficult to solve.

New research, led by Professor Barak Kol at the Racah Institute of Physics of the Hebrew University of Jerusalem, adds a step to this scientific journey that began with Newton, touching on the limits of scientific prediction and the role of chaos in it.

The theoretical study presents a new and precise reduction of the problem, which is made possible by examining the basic concepts that underlie the previous theories. This allows for an accurate prediction of the probability that each of the three bodies can escape the system.

Following Newton and two centuries of fruitful research in the field, including by Euler, Lagrange, and Jacobi, the mathematical Poincare discovered by the late 19th century that the problem was extremely sensitive to the initial positions and velocities of the bodies. This sensitivity, later known as chaos, has far-reaching implications – it suggests that there is no deterministic solution in closed form to the three-body problem.

In the 20th century, the development of computers made it possible to re-examine the problem with the help of computerized simulations of the bodies’ movements. The simulations showed that under some general assumptions, a three-body system experiences periods of chaotic or random movement, alternating with periods of regular movement, until the system eventually disintegrates into several bodies orbiting away their common center of mass and a third one. move, or escape, from them.

The chaotic nature implies that a solution in closed form is not only impossible, but that computer simulations cannot provide specific and reliable long-term forecasts. However, the availability of large sets of simulations in 1976 led to the idea of ​​seeking a statistical prediction of the system, and in particular of predicting the probability of escape of each of the three bodies. In this sense, it was found that the original purpose, to find a deterministic solution, was wrong and that the real purpose is to find a statistical solution.

Determining the statistical solution is no easy task due to the three characteristics of this problem: the system offers chaotic motion that alternates with frequent motion; it is unlimited and susceptible to disintegration. A year ago, Dr. Nicholas Stone of Racah and his colleagues used a new calculation method and for the first time obtained a closed mathematical expression for the statistical solution. However, this method, like all of its predecessor’s statistical approaches, is based on certain assumptions. Inspired by these results, Col began a re-examination of these assumptions.

The infinite infinite range of the gravitational force indicates the occurrence of infinite probabilities by the so-called infinite phase-space volume. To avoid this pathology and for other reasons, all previous attempts have posited a somewhat arbitrary “strong area of ​​interaction” and only calculated configurations therein in the calculation of probabilities.

The new study, recently published in the scientific journal Celestial mechanics and dynamic astronomy, focuses on the outgoing current of phase volume, rather than the phase volume itself. Since the flood is finite, even if the volume is infinite, this flood-based approach avoids the artificial problem of infinite probabilities, without ever introducing the artificially strong interaction area.

The flood-based theory predicts the escape probability of each body, under a certain assumption. The predictions differ from all previous frameworks, and prof. Kol emphasizes that “tests by millions of computer simulations show strong similarity between theory and simulation.” The simulations were performed in collaboration with Viraj Manwadkar of the University of Chicago, Alessandro Trani of the Okinawa Institute in Japan, and Nathan Leigh of the University of Concepcion in Chile. This agreement proves that understanding the system requires a paradigm shift and that the new conceptual basis describes the system well. It then appears that even for the foundations of such an old problem innovation is possible.

The implications of this study are comprehensive and are expected to influence the solution of a variety of astrophysical problems and the understanding of a whole class of problems in mechanics. In astrophysics, it may apply to the mechanism that creates pairs of compact bodies that are the source of gravitational waves, as well as to deepen the understanding of the dynamics in stellar clusters. In mechanics, the three-body problem is a prototype for a variety of chaotic problems, so its progress is likely to reflect additional problems in this important class.