In the physical world, the phenomenon of stability is actually rare, and instability is more common, to paraphrase a modern word, it is called "chaos".
Xia Zhihong· Tenured Professor at Northwestern University
Gezhi Dao No. 44 | September 9, 2019 Beijing
Today I want to talk about the "three-body problem and celestial motion". Everyone probably knows a novel called "The Three-Body Problem", and many of the contents of the novel deal with some of the nature of the three-body movement. Today I want to talk about the three-body problem and some interesting questions related to it from a scientific point of view.
The origin of the three-body problem
Modern science began with Newton. Newton was a very remarkable scientist, perhaps the greatest scientist of mankind, who discovered Newtonian mechanics, invented calculus, discovered the law of gravitation.
This is a cartoon drawn by a famous American cartoonist about Newton's discovery of gravity. There is an apple tree in the comics, and under the apple tree sits Newton, next to which there is a fallen apple.
It is said that while Newton was taking a nap under an apple tree at Cambridge University, an apple fell and hit his head, triggering his inspiration to discover the law of gravity. Of course, this is just a legend.
In fact, the discovery of the law of universal gravitation was made through the common observation and hard work of many scientists hundreds of years before Newton, and it was summarized based on many observations of the motion of the planets of the solar system, the most famous of which should be Kepler.
Kepler proposed the "Three Laws of Planetary Motion". Where do these three laws come from? It was obtained from an astronomer named Tycho. Tycho is a very interesting person, if you are interested, you can check his relevant information.
Tycho was a Danish astronomer who had a short temper but had a good relationship with the emperor, who gave him an island for astronomical observations. Tycho was also the last astronomer to observe the movement of the planets with the naked eye. The observation mission was very difficult, but the emperor gave him a lot of resources, and even built a paper mill on the island for him to study the paper he needed to use.
Tycho was hot-tempered, fought with people when he was young, and his nose was cut off. After a period of astronomical research, the new emperor took the throne, but the new emperor did not like him, and Tycho had to go to the Czech Republic, because the Czech emperor liked him at that time, so he went to the Czech Republic to continue his astronomical research. Tycho frequented the Czech Imperial Palace, but four years after arriving in the Czech Republic, he died once after returning from the Imperial Palace. At that time, people were debating why Diacho died when he returned from the palace.
Although some people suspect that he may have been poisoned, it is more commonly believed that he drank too much alcohol in the palace, because he was embarrassed to go to the toilet, and as a result, he let the urine suffocate! He may be the only scientist to suffocate urine. Of course, this claim has been controversial. So in 1901, 300 years after Tycho's death, his body was dug up to determine if he was really poisoned. But it turned out that Tycho was indeed not poisoned, and he really let the urine suffocate.
What is particularly unfortunate is that after another 100 years, people are arguing about another thing about Tycho - Tycho had his nose cut because of a fight, so what material was the later fake nose made of? Some people argue that it is made of iron, and some people argue that it is made of copper. So 10 years ago, Tycho's body was dug up again. Upon examination, his prosthetic nose was made of iron. This man is really interesting and unlucky, but it is this person who laid the foundation of the law of gravity.
As just said, Newton discovered calculus, Newtonian mechanics, and the law of gravitation, and these three discoveries just turned an astronomical problem into a mathematical problem. Why do you say that? Because we can accurately calculate the trajectory of the planets according to the laws of physics.
I graduated from the astronomy department of Nanjing University, but I started doing mathematics after I came to the United States, in fact, some of the work I did was related to astronomy and mathematics. Astronomical problems become mathematical problems, that is, solutions to a set of differential equations. As you probably know, equations have algebraic equations as well as differential equations. To some extent, predicting the motion of celestial bodies becomes understanding a mathematical differential equation.
Of course, the simplest are two-body problems, such as predicting the orbits of the sun and a planet. At this time, the differential equation to be solved is relatively simple. Anyone with simple training can write a solution to the two-body problem.
But the three-body problem is more complicated, which is also the topic we are talking about today. Take an example of the three-body problem. For example, studying the trajectory of the sun and two planets constitutes a three-body problem. Of course, it is also possible that there is a three-body problem like two stars like the sun and a planet.
▲ Complex three-body problem
The solution of a simple two-body problem formed by the Sun and a planet is relatively normative because of the relative rules of motion of stars. I draw you a trajectory of the three-body problem, and you will find that the trajectory of these three fulcrums in space is a very complex shape, and the trajectory it describes is irregular, which is also a very basic nature of the three-body problem - the motion of the three celestial bodies is irregular.
▲ Eight planets
In addition to the sun, there are eight planets in the solar system, dwarf planets such as Pluto, millions of asteroids, some planets and moons, and other large planets that have not been discovered now...
Therefore, the set of differential equations of the solar system alone is very large and complex, far more than the three-body problem, which is a many-body problem. It is difficult to solve even the three-body problem now, and it is even more difficult to solve the multi-body problem.
Whether the three-body problem is solvable
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Is the three-body problem solvable? That is, is there a formula that can be solved?
Unfortunately, there is no formula for solving general differential equations, because we have too limited a limited grasp of functions to write a solution using elementary methods.
As you probably know, algebraic equations are much simpler than differential equations. A quadratic equation can be solved by anyone, and people who have read cubic equations a little can also solve it, and quartic equations may be more complicated, but they can still be solved.
After the quintic equation, there are no more elementary solutions. That is, it is impossible to write out the solution of a quintic equation with a formula. Of course, this is not to say that the quintic equation is not solved, the quintic equation must have five roots, it must have a solution, but we have no way to write its solution in the form of a formula.
The famous Galois theory and Abel theorem both say that there is no elementary form of solution to a quintic equation. But in Newton's time, there were still many people trying to solve differential equations, and what they wanted to do most was to find the first integral, also called the classical solution.
To solve the equation, you need to find the first integral. Energy integration, angular momentum integration, momentum integration, these are all the first integrations. People have spent hundreds of years trying to find other first integrals for the three-body problem, but unfortunately, until now, modern mathematics has proved that there are no other first integrals.
That is to say, it is impossible to solve the three-body problem in this classical way, and in the classical sense, the three-body problem is unsolvable. What does it mean to react inexplicably practically? We can't write a formula, and we can't tell you an exact time.
For example, you want to know what the solar system will look like in a million years, but because there is no formula for the three-body problem, because I can't write the formula, I can't tell you the answer.
However, not being able to write it does not mean that there is no solution, there is still a solution, but I can't write its formula. We can let the computer do the calculations, but there is another problem involved in this - error. There is an error in letting the computer do the calculation. The error is small in the short term, and the longer the time, the greater the error.
Therefore, what will happen in thousands of years, tens of thousands, millions of years, with the solution calculated by the current computer, is still not credible. This means that there is no way to predict the future of planetary motion.
Although it is impossible to predict, we still want to know the general situation of the planetary motion. For example, the solar system is not stable. We can't write a solution, but can we use other mathematical analysis methods to conclude that the solar system is stable? After all, this is still quite important to us. If the solar system is unstable and the Earth is too far from the sun, it is too cold; Too close to the sun and too hot.
The novel "Three-Body Problem" describes that because the three-body movement is very irregular, sometimes three suns appear at the same time, and the excessive temperature burns all people, and even burns into another form of life. So, we're still interested in these kinds of issues.
Newton believed that planetary motion was unstable. However, although Newton was a great scientist, he believed in God very much, and he spent the rest of his life trying to prove the existence of God mathematically. He even believes that the solar system is unstable, but if God helps, if God comes to push the earth every once in a while, it can solve the problem.
It's hard to believe that Newton spent a long time using mathematical formulas to deduce that God would one day push the earth. Although Newton lived during the Renaissance, when everyone was more open-minded, Newton's ideas were still criticized by many scientists.
In fact, at that time, basically all the big scientists wanted to study the three-body problem, because it was a big problem that could not be solved. Every scientist has his own ideas, some believe that planetary motion is long-term stable, others believe that it is unstable, and they all have their own ideas and methods of proof.
However, through so many years of observation and research, people have increasingly realized that in the physical world, stable phenomena are actually rare, and instability is more common. This phenomenon of instability, to use a modern term, is called "chaos."
What is Chaos?
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Now, I'm going to tell you what chaos is, and I hope that after listening to it, you can easily tell others what chaos is.
When it comes to "chaos", we have to say an interesting history. This is a portrait of Oscar II, who was also the emperor of Sweden and Norway, which at that time were the same country.
Oscar II was a very interesting person, he liked art and science very much, read a lot of mathematics books, and often invited some scientists to give lectures for him.
Two years before his seventieth birthday, a mathematician named Mitag-Lefler suggested that he set up a science prize that would be awarded two years later at the emperor's seventieth birthday banquet. This prize is for those who can solve the three-body problem.
Of course, we now know that the three-body problem is insolvable, so this prize is actually for nothing. Many people wonder why the Nobel Prize does not have a mathematics prize, supposedly because Mitag-lefler snatched Nobel's wife away. Of course, this is also a legend.
Oscar II was particularly fond of science, and one day he invited a mathematician from the University of Paris, named Panlevie, to the court to lecture on mathematics. Panlevi was the 84th and 92nd Prime Minister of France and a mathematician.
In his lecture for Oscar II, he proposed the Panlevy conjecture: in the case of several stars interacting through gravity, one of them may be thrown to infinity by other stars for a limited time.
Nearly 100 years after the Penlevi guess was proposed, I have finally solved this problem in my doctoral thesis. Why can I fix it? In fact, because we now have a better understanding of three-body or many-body systems, we know a structure called "chaos", and I used the mechanism of chaos to solve the Panlevi guess.
Back to the grand prize set by Oscar II just mentioned. Panlevy was joined by another mathematician, Poincaré, who also had a great influence on mathematics.
At that time, Poincaré wrote an article claiming that he had solved the three-body problem, and the jury awarded him the Oscar II. But we know that the three-body problem is unsolvable. In fact, one of Poincaré's students soon discovered a fatal error in his article.
Things got into trouble, and the prize was awarded to Poincaré, who published the wrong article. Poincaré became aware of the complexity of the three-body problem, so he rewrote an article in which chaos was mentioned for the first time. Finally, the president of the jury, Karl Weierstrass, believes that although Poincaré did not solve the three-body problem, it was decided to award the prize to him because of the importance of the new article rewritten.
Interestingly, the prize amount was about two months' salary for Poincaré, but because he wrote the wrong article, he had to rewrite, reprint, and reissue the issue of the magazine in which the article was printed, which cost him four months' salary, and he lost two months' salary.
Chaos and instability
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So what exactly is chaos?
Let's start with this simple cartoon, and the story it tells may have been heard. Kneeling on the ground is an Indian mathematician clutching a chess board in his hand, and the emperor of India is seated. The mathematician invented chess, and the emperor decided to give him a reward.
The mathematician says: it's simple, you put 1 wheat on the first grid of the chessboard, 2 wheat on the second, 4 wheat on the third square, 8 wheat on the fourth compartment ... And so on. The reward I want is, you just have to fill the grid of this chessboard.
When the emperor heard this, he thought that this was very simple, just a few wheats. But let's see, how many wheat do we need to meet the requirements? There are 64 squares on the board, so you need 1+2+22+23
+...+263=264-1=18,446,744,073,709,551,615 wheat! Let's convert it and see how many liters of wheat we need. That's 140 trillion liters of wheat!
From the time humans grew wheat to the present, not so much wheat has been produced worldwide. According to current production, it is estimated that it will take 2000 years before so much wheat can be produced. This example shows that after doubling again and again, after doubling to 63 times, this number becomes an astronomical number.
Therefore, no data can be doubled at a time. For example, if you want to double GDP every 7 years, it would be astronomical if you really calculate at this rate. Therefore, the geometric progression grows particularly fast.
How does this relate to our physical systems? Take an example. If I put a few air molecules in a box, I measure the initial position and initial velocity of these molecules first, with a small error. By looking at the movement of these molecules, you will find that because the movement of molecules is very unstable, the error doubles in less than 1 second.
After another second, the error doubles again. It says 1 second, but in fact, less than 1 second, the error will be doubled. In other words, after 60 seconds, the original error value may become the astronomical amount you just saw.
This shows that in a physical system, if the small error in the microstate has been doubling, then this error will have a very large impact on the system. Of course, although the value is large, the size of the box limits the movement of molecules, and molecules will bounce back after moving to the edge of the box, so on the whole, its error will not reach that astronomical figure. But locally, micrographically, its error can make the original system completely different from the predicted system, which is why I want to give this example.
I want to show that in a chaotic dynamical system, small shifts or deviations can cause the error to increase exponentially, but the overall error is still within the limits of the box. So, what is chaos? Chaos is that in a small range, in a microscopic state, the error increases exponentially. In mathematics, this is called the positive Lyapunov index, which is a mathematical vocabulary and the only mathematical word today.
What does that chaos say? Indicates that the future is unpredictable. Why is the future unpredictable? Because the accuracy of the initial test is useless, the system after one minute has nothing to do with the original system. This is the principle that a chaotic dynamical system is unpredictable in the future.
Application of chaotic systems
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What kind of system is a chaotic system? For example, meteorological systems. You may have heard of the "butterfly effect". Originally, the weather forecast said that there would be a storm in Beijing today, but it was not actually raining, why?
▲ The weather system is chaotic
It turned out that two weeks ago, in Chicago, on the other side of the world, a butterfly suddenly shook its wings, disturbing the air. It's such a small fluctuation that can become a fluctuation twice the size after one second, and then wait another second and become a fluctuation of four times the size... Two weeks later, the "butterfly effect" affected Beijing, so today Beijing is clear and there is no rain.
So, to accurately predict the weather, you have to know what every butterfly in Chicago did two weeks ago. However, there are many objects that are much larger than butterflies, such as planes and trains, which are very large. In addition, to accurately predict the weather in two weeks' time, you must also figure out the movement of everything in Chicago.
Of course, not only Chicago, but also New York. So, don't expect to read the weather forecast, you can calmly arrange to go hiking on the weekend, maybe it will suddenly rain heavily on the weekend. But you don't blame the Meteorological Bureau, this has little to do with the Meteorological Bureau, if you want to blame it, blame the chaotic dynamic system, the meteorological system is a chaotic system. The three-body problem has also now proven to be a chaotic system, which is why the three-body problem is a very complex motion. Meteorological systems, turbulence mechanics systems are chaotic systems.
Also, I just mentioned, why can I prove the Panlevi guess? It is because I proved that there is a special chaotic dynamic system in celestial motion. Because of time constraints, I can't explain to you what I have proved, if you are interested, you can read a book called "Heavenly Encounter". It was an English popular science book that introduced my work and now has Chinese translations.
Finally, I will talk about an example of the application of chaotic systems. In April 1991, Japan launched a lunar probe called Hiten, but after the probe went to heaven, researchers found that there was not enough fuel to reach lunar orbit. So, Japan asked NASA for help, and NASA sent a mathematician named Belbruno to help the Japanese. Belbruno redesigned its orbit and finally put the probe back into lunar orbit.
Belbruno used limited fuel to send the probe to a chaotic region. Isn't the chaotic region unpredictable, so spending a little fuel to push the detector will have a particularly large impact on the movement of the detector. Therefore, it is advantageous to put the detector in a suitable place; If the place doesn't fit, let it shake a little.
One day, Belbruno suddenly called me. He said that an article I wrote theoretically proved which regions are most prone to chaos effects. He said he spent a month designing the new track. If you had known my article then, it would have taken a few days to redesign the orbit and save the lunar probe.
A few years later, the American company Hughes encountered the same problem after launching a satellite: the satellite did not have enough fuel to reach its intended orbit after it went into the sky. At this time, Belbruno redesigned the orbit and successfully sent the satellite to the intended orbit. So, this is a very interesting application example of chaotic systems.
That's all I have for today's sharing, thank you!
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The article is reproduced from the "Lecture on the Taoist Forum"
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