In solving the three-body problem, researchers found "islands of regularity."

For several centuries, scientists have been puzzled by the problem from classical mechanics concerning the motion of three bodies in a system, more commonly known as the “three-body problem”. In such a system, objects begin to behave unpredictably. The goal of researchers is to calculate the future motion of these bodies, determining the coordinates and velocities of three material points at any given moment in time under specified initial conditions, while only considering gravitational interactions. There are thousands of particular solutions to the problem, but they are not universally accepted and merely suggest isolated conclusions on how it might be resolved.

The history of the problem begins with Isaac Newton, who first formulated it in the 17th century. He believed that while the motion of two interacting objects in space is predictable, the introduction of a third body disrupts the system. This introduces randomness into the motion of the bodies. Despite his efforts, Newton was unable to find a general solution and acknowledged that the problem was too complex for an analytical approach.

The reason for the complexity in finding a solution lies in the chaotic nature of the system. Even slight variations in the initial conditions of one of the bodies (in mass, velocity, or spatial position) can lead to vastly different outcomes. Minor changes in the movement parameters of one of these bodies can entirely alter the fate of the system.

Some researchers believe that even within chaos, there are hidden patterns, suggesting that chaos is not mere disorder. Many mathematicians are attempting to prove this. Supercomputers come to their aid.

An international team of mathematicians and physicists led by Alessandro Trani (Alessandro Trani) from the Niels Bohr Institute (Denmark) utilized a supercomputer for numerical modeling of the three-body problem and discovered gaps in the chaos of these objects' movements—“islands of order.” These “islands” indicated where the objects behaved predictably and adhered to specific rules: the trajectory of their orbits directly depends on how the three objects are positioned relative to each other at the moment of approach, as well as their speed and angle of approach. The study is detailed in a press release published on the institute's website.

Trani and his colleagues developed the Tsunami program, which sifted through millions of possible combinations of the interactions among the three bodies in the system, calculating their movement trajectories using Einstein's theory of general relativity and Newton's laws. This program employed a statistical method and required a special map where each point corresponded to a specific set of initial conditions.

Next, each point was assigned a color based on which of the three objects was ultimately “ejected” from the system as a result of the interactions (in most cases, the object with the least mass).

If the movements of the objects in the system were random, meaning the problem itself implied chaos, the colors on the map would have been mixed randomly. However, in their research, the scientists did not observe this pattern. They found areas on the map that were colored uniformly. According to the researchers, these areas are the “islands of order”: where the initial conditions predetermine the subsequent evolution of the system.

The authors of the study believe that on these “islands,” the motion of the objects follows strict mathematical laws, making it orderly and predictable. This, in turn, suggests the existence of a general solution to the three-body problem.

However, the “islands of order” discovered by Trani's team in solving the three-body problem may pose a challenge for researchers. The problem is that this approach effectively calculates chaotic trajectories of body movements, but struggles when it comes to “ordered” trajectories.

“When certain areas on the map, which suggest chaotic movement of the objects, suddenly transitioned into a phase of ordered movement, our statistical calculations for this phase were disrupted, leading to inaccurate predictions. Now our goal is to learn how to combine statistical methods with numerical ones to ensure high prediction accuracy when the system begins to become more ordered,” explained Trani.

The uncovered “islands of order” represent an important step towards a general solution to the three-body problem. Researchers still need to conduct a series of computer experiments to thoroughly investigate the properties of these “islands” and understand the mechanics of their formation, as well as assess their impact on the evolution of the system.

In space, the presence of systems consisting of three objects is not uncommon. Therefore, solving the three-body problem is not just a theoretical challenge but also an opportunity to uncover the mysteries of the universe.