January 1889, King of Sweden II. Celebrated Oscar's 60th birthday. To commemorate this milestone, the monarch, who studied mathematics in his youth and even founded the journal Acta Mathematica (still considered one of the most prestigious in this field), decided to organize a scientific competition. He offered a prize to anyone who could solve the tricky three-body problem by taking into account the orbits of three-body systems.

When Isaac Newton published his "Principia" in 1687, he was the first to formulate mathematical principles that made it possible to accurately predict the motion of two very close celestial bodies. This achievement strengthened the idea of a functioning mechanical universe. as a giant clock. However, Newton soon discovered that he could not find a correct general solution when another object was added to the system.

## Who is Henri Poincare?

The "three-body problem" remained without a mathematical solution for nearly 200 years, despite the best efforts of scientists. This is where Oscar II brings this unsolvable problem to its conclusion. The French mathematician Henri Poincaré won the competition, who was awarded a gold medal and 2.500 Swedish crowns. His solution has been published in the Royal Mathematical Journal.

But then Poincare discovered a miscalculation. He hastened to purchase all editions of the magazine containing the error – which cost him 3.500 crowns – and published a revised version the following year. He proved that the interactions between the three bodies are fundamentally chaotic, and therefore no deterministic mathematical solution to the problem can be found, to the disappointment of the king and the proponents of the mechanical understanding of the universe (that is, Poincaré could not find a formula).

## What is Chaos Theory?

This proof is considered one of the foundations of chaos theory. The lack of a deterministic solution to the "three-body problem" means that scientists cannot predict what happens during the close interaction between two orbiting bodies, such as the Earth and the Moon, and a third object approaching them.

But now, 121 years after Poincaré's findings were published, Yonadav Barry Ginat, PhD student from Technion – Israel Institute of Technology, Haifa, and Prof. Hagai Perets claims to have found a complete statistical solution to the problem.

## Three Hull Systems

Computer simulations of three-body systems show that they develop in a two-stage process: in the first, chaotic stage, the three bodies are very close together and exert equally intense gravitational forces on each other, therefore constantly changing. We can call this the relative motion of three bodies. Eventually, a celestial body is removed from the system and the two are left to orbit each other in an elliptical, deterministic orbit. If the third object is in a bound orbit, it eventually comes back to the other two, whereupon the first phase begins again.

This three-way dance ends in the second stage when one of the corpses escapes in an untethered trajectory, never to return.

### A Drunk Man Walking

While a complete solution to the "three-body problem" is not possible due to the chaotic nature of the process, it is possible to calculate the probability that a triple interaction will end in a certain way – for example, which object will be launched, at what speed, etc. Over the years, solutions using different methods have been proposed to arrive at a calculation of this probability as accurately as possible.

Two researchers from Technion's physics department have used tools from a branch of mathematics known as random walk theory, sometimes referred to as the "drunk's walk," since mathematicians began studying how drunk people move. Since a drunkard apparently took every step at random, mathematicians understood it as a random process. However, it is possible to estimate, for example, the distance a drunk will travel after a few steps (this is a statistical solution that results in an average distance of about 10 steps from the starting position for every hundred steps taken).

The ternary system basically behaves similarly: like a drunkard's walk, after phase 1 has taken place, an object is thrown randomly, returns, etc. into a ditch).

Rather than predicting the true outcome of each three-body interaction, Ginat and Perets calculated the probability of each possible outcome at each stage of the interaction and then combined all the individual stages using random walk theory to calculate the final probability of each.

The two began considering the random walk model in 2017 when Ginat was an undergraduate in one of Perets' lectures and was writing an essay on the three-body problem. Their solution was recently published in the journal Physical Review X.

According to Perets, “This is a big challenge to understand any situation where there are high-density star clusters. There was no solution until the 1970s. However, with advances in computing power, numerical solutions have been tried” – that is, by throwing the data into the simulation and seeing what happens.

source: haaretz.com

📩 19/08/2021 18:57

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