Schematic diagram of a three-body.
Researchers at The Hebrew University of Jerusalem (Israel) have used a novel approach to make explicit statistical predictions about the classical three-body problem.
The three-body problem is the problem of the laws of motion of three celestial bodies under the action of mutual gravity, and scientists wanted to find a formula to describe the end of the last three bodies, which is important for a wide range of mechanical problems and laws of motion of celestial bodies.
However, scientists found that this problem seems to be simple, but it is difficult. For example, the result of the final motion of the system is very sensitive to the initial positions and velocities of the three celestial bodies. Scientists call this sensitivity chaos (chaos). This concept means that it is difficult to predict the final outcome of the three celestial bodies with a definite probability.
This puzzle has attracted a large number of outstanding physicists, astronomers and mathematicians for more than three centuries since it was proposed.
Now, researchers at Hebrew University have found a new way of thinking about the problem that goes beyond the framework of the original study and dramatically reduces the uncertainty of the problem by making precise predictions about the likelihood that each object in the system will eventually fly out of the system. The study was published April 1 in the journal Celestial Mechanics and Dynamical Astronomy.
Research History Review
At the end of the 17th century, the famous scientist Isaac Newton used the law of gravity to explain the law of planets orbiting the Sun. He also wanted to explain the laws of motion of the Moon, since both the Earth and the Sun influence the Moon’s orbit, so it is expected that the final motion of these bodies is complicated. Later such a problem developed into the classical three-body problem.
After Newton, mathematicians Leonhard Euler, Joseph-Louis Lagrange and Carl Gustav Jacob Jacobi, and then Henri Poincaré in the 19th century, all contributed to this problem.
In the 20th century, with the development of computers, scientists were able to use computer systems to simulate the evolutionary patterns of the three bodies. Some scientists found that, in the general case, the three-body system would undergo a transformation between chaotic and regular motion, eventually becoming two bodies orbiting a common center, while a third body would leave the system completely.
Scientists realized that even using computer simulations, it would be difficult to make predictions about the long-term end of the three-body system. However, a large number of simulations carried out in 1976 made new progress. These simulations showed that it was possible to make statistical predictions about the system. Specifically, it was possible to predict the likelihood of each object leaving the system. From this perspective, scientists believe that the original goal of going for deterministic endings was wrong and that the correct goal is to go for statistical predictions.
Another advance in this problem was made more than a year ago when Nicholas Stone of the Racah Institute of Physics at Hebrew University used a new method of computation for the first time to find a mathematical description of the deterministic ending of the three-body problem. However, this method again relies on some specific conditional assumptions.
He referred to the region of space under consideration as the phase-space volume. Since gravity has an infinite range of action, this also implies that there will be infinite possibilities in this space. In order to circumvent this problem, and for other reasons, this approach makes a somewhat arbitrary assumption of a “strong region of action”, taking into account only the situations that arise in the region of strong action.
A new research approach
In this latest study, Barak Kol, also of the Institute of Physics at Raqqa, takes these assumptions a step further – considering them in terms of outgoing flux in phase space, rather than in phase space itself. The study claims that this flux is finite even though the space is endless, so this approach does not require the use of arbitrary assumptions used in previous studies.
This theory makes predictions about the likelihood of each object leaving the system based on the outflow flux, and the results are different from all previous theories. “Millions of computer models have been tested showing that the theory and the models are in high agreement,” Cole said.
The computer models were done in collaboration with the University of Chicago, the Okinawa Institute in Japan, and the University of Concepcion in Chile.
The study claims that the match between the theory and the models proves that the new theory provides a good description of the three-body system. It also shows that it is entirely possible to gain innovative insights into even such an old and difficult problem if one can radically change the paradigm of thinking.
Implications of the Three-Body Problem
This research is useful for many astrophysical problems and mechanical fields. In astrophysics, it will improve scientists’ understanding of compact two-body systems (i.e., systems composed of two celestial bodies), and the internal mechanisms of stellar populations. A large number of gravitational waves now detected originate from two-body systems.
In mechanics, many problems involve chaotic systems, and the three-body problem is the prototype of chaotic systems, so a breakthrough in the three-body problem will have a profound impact on a range of mechanics problems.
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