Hebrew University Researcher Introduces New Approach to Three-Body Problem, Predicts its Outcome Statistics.
The “three-body problem,” the term coined for predicting the motion of three gravitating bodies in space, is essential for understanding a variety of astrophysical processes as well as a large class of mechanical problems, and has occupied some of the world’s best physicists, astronomers and mathematicians for over three centuries. Their attempts have 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 through a law of universal gravitation. He also sought to explain the motion of the moon. Since 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 gravitational attraction (see illustration to the right), a problem that later became known as “the three-body problem.”
However, unlike the two-body problem, Newton was unable to obtain a general mathematical solution for it. Indeed, the three-body problem proved easy to define, yet difficult to solve.
New research, led by Professor Barak Kol at Hebrew University of Jerusalem’s Racah Institute of Physics, 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 novel and exact reduction of the problem, enabled by a re-examination of the basic concepts that underlie previous theories. It allows for a precise prediction of the probability for each of the three bodies to escape the system.
Following Newton and two centuries of fruitful research in the field including by Euler, Lagrange and Jacobi, by the late 19th century the mathematician Poincare discovered that the problem exhibits extreme sensitivity to the bodies’ initial positions and velocities. This sensitivity, which later became known as chaos, has far-reaching implications – it indicates 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’ motion. The simulations showed that under some general assumptions, a three-body system experiences periods of chaotic, or random, motion alternating with periods of regular motion, until finally the system disintegrates into a pair of bodies orbiting their common center of mass and a third one moving away, or escaping, from them.
The chaotic nature implies that not only is a closed-form solution impossible, but also computer simulations cannot provide specific and reliable long-term predictions. However, the availability of large sets of simulations led in 1976 to the idea of seeking a statistical prediction of the system, and in particular, predicting the escape probability of each of the three bodies. In this sense, the original goal, to find a deterministic solution, was found to be wrong, and it was recognized that the right goal is to find a statistical solution.
Determining the statistical solution has proven to be no easy task due to three features of this problem: the system presents chaotic motion that alternates with regular motion; it is unbounded and susceptible to disintegration. A year ago, Racah’s Dr. Nicholas Stone and his colleagues used a new method of calculation and, for the first time, achieved a closed mathematical expression for the statistical solution. However, this method, like all its predecessor statistical approaches, rests on certain assumptions. Inspired by these results, Kol initiated a re-examination of these assumptions.
The infinite unbounded range of the gravitational force suggests the appearance of infinite probabilities through the so-called infinite phase-space volume. To avoid this pathology, and for other reasons, all previous attempts postulated a somewhat arbitrary “strong interaction region”, and accounted only for configurations within it in the calculation of probabilities.
The new study, recently published in the scientific journal Celestial Mechanics and Dynamical Astronomy, focuses on the outgoing flux of phase-volume, rather than the phase-volume itself. Since the flux is finite even when the volume is infinite, this flux-based approach avoids the artificial problem of infinite probabilities, without ever introducing the artificial strong interaction region.
The flux-based theory predicts the escape probabilities of each body, under a certain assumption. The predictions are different from all previous frameworks, and Prof. Kol emphasizes that “tests by millions of computer simulations shows strong agreement between theory and simulation.” The simulations were carried out in collaboration with Viraj Manwadkar from the University of Chicago, Alessandro Trani from the Okinawa Institute in Japan, and Nathan Leigh from 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 turns out, then, that even for the foundations of such an old problem, innovation is possible.
The implications of this study are wide-ranging and is expected to influence both the solution of a variety of astrophysical problems and the understanding of an entire class of problems in mechanics. In astrophysics, it may have application 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 within star clusters. In mechanics, the three-body problem is a prototype for a variety of chaotic problems, so progress in it is likely to reflect on additional problems in this important class.
Reference: “Flux-based statistical prediction of three-body outcomes” by Barak Kol, 1 April 2021, Celestial Mechanics and Dynamical Astronomy.
… I was wrong, this problem doesn’t exist, it was explained to me, this is a lie…
…hey what are you doing, where is my comment…
… Torbjörn Larsson explained why I was wrong and now you are changing stuff… common, you can’t be serious, you don’t know better than Torbjörn Larsson…
The solution to the three body problem is straightforward.
Either add a fourth, or get rid of one of the three.
No more three body problem.
Magnets repel and attract… Then this undulation could maintain and order if a system, especially if there is a period of rest between opposing forces. It be other fields in combination of day photons interacting inside and outside.hmm
“Since both the earth and the sun determine the motion of the moon…”
That’s the problem …why do you feel the earth and sun determine the motion of the moon? Maybe the moon and sun determine the motion of the earth? Perhaps the earth and moon determine the motion of the sun?
Or how about the earth and sun and MILLIONS OF OTHER BODIES all determine each other in incalculable ways, but because all our maths are based on only what we can study, I fear we’ll always be ‘tweaking’ theories as they come about to match what we can observe.
If 3 bodies are unstable, how have the major bodies in our solar system been stable over billions of years?
Three boody system must have an easy solution.The origin(Sun) and primery(Planet) body have centre of mass calculated by usual method of calculation,confirmed.While substraction is made for secondary (Moon) body onè suitable method can be applied.
Motion of earth aroùnd sun is perpetual and calculated as before.If we àdd moon in the system then centre of mass of moon is determined and is shifted for the position of earth in previous situation.This new position is again modified by sustraction of a term.This term arises due to constrain motion of moon around the sun can have a suitable method of calculation.
Motion of earth aroùnd sun is perpetual and calculated as before.If we àdd moon in the system then centre of mass of moon is determined and is shifted for the position of earth in previous situation.This new position is again modified by sustraction of a term.This term arises due to constrain motion of moon around the sun can have a suitable method of calculation.Again moon will have a force or flux due stars or other massive matter having effective gravitation are present in this substracting term.
… when you look at the atoms and molecules they are all the same, jet somehow the stars, planets, etc are different, it not similar emerges from similar…
… anti symmetry emerges from symmetry, how wonderful and peculiar this thing it is, the symmetry gives rise to such a difference, but why…
… and if I may add, How?…
… since all bricks are the same, buildings are not, nice…
… I will look at desert lake floor that has dried and one boiling pot with awesome material in it…
That pattern is a birkeland current(electric field)
Check out The Thunderbolts Project on YouTube.
Thought of the day,
Why cant we look at these ‘large’ scale problems the way we look at atoms or molecular problems? Wouldn’t the same rules be present. Our solar system incased in a heliosphere is similar to cel structure, the the sun with planets is similar to a nucleous with protons, electrons etc.
Maybe the solution is really simple and people just can;t see the forest through the trees.
… when things scale nicely it is possible to look at those small problems, but it seems like the scientist are looking at the natures magic tricks and try to interpret it as it is not an illusion that we perceive…
The problem is due to the mistaken assumption of Sir Isaac Newton who assumed that matter attracts matter. The solution to the three body problem is solved by the correct assumption that matter and ‘space’ that is to say, the spacial field surrounding a body, are mutually repulsive.
A body will repulse a field of space proportional to the body’s mass. The expanded spacial field tends to collapse to a steady state field that would result if no body of mass was present. The two opposing forces of matter and space reach an equilibrium, and will behave according to how many bodies of mass are present, even three bodies!
You’re welcome. 🙂
This problem occurs because of a fundamental floor in how we view and use mathematics. The three bodies can’t exist independently of outside forces. Zero point therfore cannot be derived from any given point in time space for measurements. Zero point is rather a hypothetical ending of the relationships. So it exist within the calculus as a state which does not yet exist. You would get much better results if you used the number closest to zero that you could find.
I’m no physicist but they are talking about chaos. Which means very subtle changes that happen from even second to second and build of each other. Is that correct? Perhaps the problem is compounded with many other gravitational systems in the solar system. Such as different points in rotation alignment means a larger system to explain based on China’s of these interactions. A larger time line then the moon and the earth and the earth and the sun.
Keep in mind the Moon is slowly moving away from the Earth.
Would it be possible in a near future to develop a concept of some kind of engine capable of feeding (convertion) of electromagnetic waves from outer space;I have no idea how strong or weak are those EMW out there; what kind of systems do radio telescopes use?; I guess for me that one is an unknown highway
Dripping faucet, ticking clock, windshield wiper. Sometimes when they touch.
Excellent writing. Thank you to the writer.
The largest body always will dominate, largest to smallest in succession…
I wonder if calculating the center of gravity of the three body system as it is being calculated might give another perspective on a possible solution.
… what was bifurcation, after all? …
The moon is not a naturally occurring phenomenon, how could it be? That’s what blows a hole in any theory. Something that shouldn’t be there but it is and without it what would happen to life on earth? Theoretical sciences are mumbo jumbo. Even Hawking’s theory he built his whole career on was proven wrong when technology advanced enough to disprove it. Then we get “sorry guys, I was wrong but here’s my new theory”…spare me/us!