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Recently, the TV series "three-body" is very popular. But what is trisomy? Is there a solution to the three-body problem? These problems are worth exploring.

In 1687, Isaac Newton published his principles, which included equations of motion and gravity, which turned our seemingly erratic universe into a predictable machine. In view of the current position and velocity of celestial bodies in the solar system, Newton equations can in principle be used to calculate their past and future. I add "in principle" here because it's not that simple.

Although Newtonian gravitational equations are beautiful, they provide a simple solution for planetary motion in only one case: if and only if two objects revolve around each other without the influence of other gravity. If you add another object, then in most cases, all the motion will become fundamentally chaotic. This is the three-body problem, which has a history of 300 years.

After the publication of the principles, many people are looking for simple analytical solutions for more complex systems, and the next step is naturally a three-body system. But even the influence of an extra object seems to make an accurate solution impossible. The three-body problem has become a problem for many great mathematicians. In the late 1980s, mathematicians Ernst Burrows and Henry Poincare convincingly asserted that there was no general analytical solution.

The actual situation of the three-body problem is that almost all the evolution of the initial configuration is dominated by chaotic dynamics, and the future state is highly dependent on the change of the initial small conditions. The orbit tends to be wild and unpredictable, and it is almost inevitable that a celestial body will eventually pop out of the system. Despite the obvious despair, it is good to learn to predict multibody exercise. For most of the three centuries since Newton, predicting the motion of the planets and the moon was crucial for navigation, and now it is crucial for space travel.

Approximate solution because most of the three-body problems do not have useful analytical solutions, we can try to find approximate solutions. For example, if the objects are far enough apart, then we can approximate the multi-body system to a series of two-body systems. This is like every planet in our solar system can be thought of as a two-body system with the sun, resulting in a series of simple elliptical orbits, as Kepler predicted. But these orbits will eventually change because of the interaction between the planets.

Another useful approximate solution is that when the mass of one of the three objects is very low compared with the other two objects, we can ignore the small gravitational influence of the smaller celestial body and assume that it moves in the fully solvable two-body orbit of its larger companion. We call it a simplified three-body motion, which is suitable for small objects such as artificial satellites around the earth. It can also be used to approximate the orbit of the moon relative to the earth and the sun, or the orbit of the earth relative to the sun and Jupiter. These approximate solutions are useful, but they still cannot be predicted perfectly. Even the smallest planetary bodies have a certain mass, and the entire solar system has many massive components. Before we joined Earth, the Sun, Jupiter and Saturn themselves automatically became a three-body system without analytical solutions.

However, the absence of an analytical solution does not mean that there is no solution. In order to obtain an accurate prediction of most three-body systems, we need to decompose the motion of the system into multiple parts, and a sufficiently small part of any gravitational trajectory can be approximated by an accurate analytical solution. If the problem is decomposed into sufficiently small paths or time steps, then the small motions of all objects in the system can be updated step by step. This method of solving differential equations one step at a time is called numerical integration. When applied to the motion of many bodies, it is an N-body simulation.

With the help of modern computers, N-body simulations can accurately predict the motion of planets in the distant future, or solve millions of objects to simulate the formation and evolution of the entire galaxy. These numerical solutions did not begin with the invention of the computer, and before that, these calculations had to be done by many people by hand.

The limitations of approximate solutions of some analytical solutions, the difficulty of computer numerical integration and the legendary status of the three-body problem have inspired generations of physicists and mathematicians to continue to seek accurate analytical solutions. In very special cases, Euler has found a series of solutions for three celestial bodies orbiting around a common center of mass, all of which remain in a straight line. Lagrange found a solution for three objects to form equilateral triangles.

In fact, for any two objects orbiting each other, the Euler and Lagrangian solutions define five additional orbitals of the third object, which can be described by a simple equation. These are the only perfect analytical solutions to the three-body problem. Put a low-mass object in any of these five orbits and it will stay there indefinitely. We now call these points Lagrange points, and they are useful places for us to park our spacecraft.

Long after Euler and Lagrange, the perfect solution of the three-body system has not been further discovered. In modern times, people use computers to search the vast space of possible orbits in order to find a three-body system with periodic motion. In the 1970s, Michel Henon and Roger Broucke found a series of solutions that involved two masses bouncing back and forth in the orbital center of a third celestial body. In the 1990s, Cris Moore discovered a stable "8" orbit in which three celestial bodies have equal masses. Then mathematicians proved the existence of figure 8 solution mathematically, which led to the upsurge of discovering new periodic three-body orbits.

Hundreds of stable three-body orbitals are known, but with the exception of Euler and Lagrangian solutions, none of these are likely to occur in nature.

This article comes from the official account of Wechat: Vientiane experience (ID:UR4351), author: Eugene Wang

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