# Three-body problem: mathematicians find 12,000 new numerical solutions

A group of scientists from Bulgaria, Serbia, and Japan used numerical methods and a supercomputer to find 12 thousand new solutions to the well-known three-body problem. The preprint of the relevant article appeared in the arXiv database. This was written by Space.com.

The three-body problem is a special case of the N-body problem, a classical celestial mechanics problem in which it is necessary to find the trajectories of three bodies interacting according to the laws of gravity. The problem was first formulated by Isaac Newton in 1687 in his work “Mathematical Principles of Natural Philosophy” as a problem of the Moon’s motion in the gravitational field of the Sun and Earth.

In general, the problem does not have an exact solution due to the fundamental impossibility of solving a 6th-order differential equation with uncoupled variables. For some cases, an exact solution was found in the 18th century by Leonard Euler (for collinear arrangement of bodies) and Joseph-Louis Lagrange (for Lagrange’s triangular points).

In 1912, the Finnish mathematician Carl Sundman found an analytical solution to a general problem in the form of a convergent series. But it is not practical because of the enormous amount of computation that is unattainable for humanity, which must be done to apply it to astronomy.

Most of the existing partial solutions to the three-body problem have been found using numerical methods.

Three-body systems are quite common in the universe; there are many star systems with multiple stars or massive planets orbiting each other. Theoretically, the new solutions found by the group of mathematicians can be extremely valuable to astronomers. But they are only useful if the orbits of the system members are stable, meaning that the orbital patterns repeat over time without breaking up and throwing one of the components into space.

According to the study’s lead author, Ivan Hristov, a mathematician at Sofia University, the newly discovered orbits “have a very beautiful spatial and temporal structure.” Hristov and his colleagues calculated these orbits using a supercomputer, and the scientist is confident that with better technology they will be able to find “five times more solutions.”

“Their physical and astronomical relevance will be better known after the study of stability — it’s very important,” Hristov added.

But Juhan Frank, an astronomer at Louisiana State University, is skeptical that most of the orbits found will not be stable.

“They’re “probably never realized in nature,” says Frank. “After a complex and yet predictable orbital interaction, such three-body systems tend to break into a binary and an escaping third body, usually the least massive of the three.”

But despite skeptical comments, the array of solutions found is a mathematical miracle. According to Hristov, “whether they are stable or unstable, they are still of great theoretical interest.”