This article is from the WeChat public account: SELF Gezhi On the Road pulpit (ID: SELFtalks) , author: Xia Zhihong (Northwestern University tenured professor)


“After so many years of observation and research, people increasingly realize that in the physical world, the phenomenon of stability is actually rare, and instability is more common.”

Today I am going to talk about “Three-Body Problem and Celestial Movement”. Everyone may know a novel called “Three-Body”. Many of the contents in the novel are related to some properties of the Three-Body Movement. Today I want to talk about the three bodies and related interesting issues from a scientific perspective.

The origin of the three-body problem

Modern science started with Newton. Newton was a very remarkable scientist, perhaps the greatest scientist of mankind. He discovered Newtonian mechanics, discovered calculus, and discovered the law of gravity.

This is a comic about Newton’s discovery of gravity drawn by a famous American cartoonist. There is an apple tree in the comics. Newton is sitting under the apple tree, and there is an apple falling down beside it.

It is said that when Newton took a nap under the apple tree at Cambridge University, an apple fell and hit him on the head, which triggered his inspiration and made him discover the law of gravity. Of course, this is just a legend.

In fact, the discovery of the law of gravitation has been the common observation and hard work of many scientists in the hundreds of years before Newton. It is based on the observations of the motion of solar system planets. The most famous scientist should Is Kepler .

Kepler proposed the “Three Laws of Planetary Motion.” Where do these three laws come from? It was obtained from an astronomer named Digu . Tycho is very interesting. If you are interested, you can check his relevant information.

Digu is a Danish astronomer. He has a bad temper, but he has a better relationship with the emperor. The emperor also gave him an island to facilitate astronomical observations on the island.

Digu is also the last astronomer to observe planetary motion with the naked eye . The observation mission at that time was very difficult, but the emperor gave him a lot of resources, and even built a paper mill on the island for his research paper.

Digu has a bad temper and fights with people when he is young. His nose is cut off. After working on astronomical research for a while, the new emperor came to power, but the new emperor did not like himTycho had to go to the Czech Republic, because the Czech emperor liked him so much, so he went to the Czech Republic to continue his astronomical research.

Tycho often enters and exits the Czech Royal Palace, but four years later in the Czech Republic, he once died after returning from the Royal Palace. At that time people were arguing why Tycho died after returning from the palace.

Although some people suspect that he may have been poisoned, it is more generally believed that he drank too much alcohol in the palace, because he was embarrassed to go to the toilet, and his urine was choked to death! He may be the only scientist who urinates to death.

Of course, this statement has always been controversial. So in 1901, 300 years after Tycho’s death, people dug up his body and wanted to determine if he was really poisonous. But it turned out that Tycho was indeed not poisoned. He really let the urine die.

What ’s particularly unfortunate is that another 100 years later, people are arguing about another thing about Tycho-Digu had a nose cut because of the fight. What was the fake nose that used to be? Made of materials?

Some people’s disputes are made of iron, and some people’s disputes are made of copper. So 10 years ago, Tycho’s body was dug up again. After examination, his fake nose was made of iron. This person is really interesting and unlucky, but it is this person that has laid a foundation for the law of gravity.

As I just said, Newton discovered calculus, Newtonian mechanics, and the law of universal gravitation. 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 planet according to the laws of physics.

I graduated from the Department of Astronomy, Nanjing University, but I started doing mathematics after I arrived in the United States. In fact, part of my work is related to astronomy and mathematics.

Astronomical problems become mathematical problems, that is, solving a set of differential equations. As you may know, there are algebraic equations and differential equations. To a certain extent, predicting the movement of celestial bodies becomes understanding a mathematical differential equation.

Of course, the simplest is the two-body problem , such as predicting the orbit of the sun and a planet. The differential equation to be solved at this time is relatively simple.

Solution to the two-body problem. People can write it, and people who have been trained simply can write a solution to the two-body problem.

So I can’t tell you the answer.

However, not writing does not mean that there is no solution. There are still solutions, but I cannot write a formula for it.

Of course, we can let the computer do the calculations, but this involves another problem- error . There is an error in letting the computer count. The short-term error is small, the longer the time, the larger the error.

So, what will happen after thousands of years, tens of thousands of years, and millions of years? It is still unreliable to use the solutions calculated by computers today.

This also means that we have no way to predict the future of planetary motion. Although it is impossible to predict, we still want to know the general situation of planetary motion.

For example, is the solar system stable? We cannot write a solution, but can we use other mathematical analysis methods to conclude that the solar system is stable? After all, this is important to us. If the solar system is unstable, the earth is too far from the sun, it is too cold; too close to the sun, it is too hot.

It is described in the novel “Three-body” that because the movement of the three-body is very irregular, sometimes the three suns appear at the same time, and the high temperature burns all people to death, and even burns into another form of life . Therefore, we are still very interested in such issues.

Newton considers planetary motion to be unstable. However, although Newton was a great scientist, he believed in God very much, and in the second half of his life he always wanted to try to prove the existence of God mathematically. He even thinks that the solar system is unstable, but if God helps, if God pushes the earth every once in a while, it can solve the problem.

People today are hard to believe that Newton actually took a long time to use mathematical formulas to deduce that God will come to push the earth someday.

Although Newton lived during the Renaissance, when everyone was more open-minded, this idea of ​​Newton was still criticized by many scientists.

In fact, at that time, basically all big scientists wanted to study the three-body problem, because it was a big problem that could not be solved. Every scientist has their own ideas, some think that planetary motion is stable for a long time, some think that they are unstable, and they all have their own ideas and methods of proof.

However, after so many years of observation and research, people increasingly realize that in the physical world, the phenomenon of stability is actually rare, and instability is more common. This unstable phenomenon is called “chaos” in a modern vocabulary.

What is chaos?

I want to tell you what chaos is. I hope that after listening, you can easily tell others what chaos is.

When it comes to “chaos,” you have to tell an interesting history. This is a portrait of Oscar II, who was also the emperor of Sweden and Norway at the same time.

Oscar II is a very interesting person. He likes art and science very much. He reads a lot of math books. He often invites some scientists to give lectures for him.

In the first two years of his 70th birthday, a mathematician named Mitag-Lefler suggested that he establish a science award, which will be presented at the emperor’s 70th birthday two years later. ThisIndividual awards are set for who can solve the three-body problem. Of course, we now know that the three bodies are inextricable, so this award is actually for nothing.

Many people wonder why the Nobel Prize does not set up a mathematics prize. It is said that Mitag-lefler stole Nobel’s wife. Of course, this is also a legend.

Oscar II was particularly fond of science. One day he invited a mathematician from the University of Paris to teach mathematics to the court. The mathematician was called Pan Levy . Pan Levy was the 84th and 92nd Prime Minister of France, and he was also a mathematician.

While giving a lecture for Oscar II, he put forward Pan Levy’s conjecture (In the case of several stars interacting through gravitation, one of them Individual stars may be thrown to infinity by other stars within a limited time) . Nearly 100 years after Pan Levi’s speculation, in the end I finally solved this problem in my doctoral dissertation.

Why can I solve it? In fact, because we now have a further understanding of the three-body or multi-body system, we know a structure called “chaos”, and I used the chaos mechanism to solve Panlev’s guess.

Return to the award set by Oscar II. Along with Pan Levy, there was another scholar, Poincaré . Poincaré also had a great influence on mathematics.

At that time, Poincaré wrote an article claiming that he had solved the trisomy problem, so the jury awarded the Oscar II award. But we know that the three-body problem is unsolvable.

In fact, a student in Poincaré soon discovered a fatal error in his article. This is troublesome. The prize went to Poincaré, who posted the wrong article. Poincaré began to realize the complexity of the three-body problem, so he re-written an article that first mentioned chaos.

Finally, the chairman of the jury, Weierstrass, decided that although Poincaré did not solve the trisomy problem, but because the rewritten new article was very important, he decided to give him the grand prize.

Interestingly, the prize amount was about two months’ salary for Poincaré, but because he wrote a wrong article, he had to rewrite, reprint, reissue the issue of the magazine that printed the article, and ended up spending After four months of salary, he still lost two months of salary.

Chaos and Instability

What is chaos? Let’s start with this simple comic. The story told in this comic may have been heard.

In the picture is an Indian mathematician kneeling on the ground. He is holding a chess board in his hand. The Indian emperor is sitting in the picture. The mathematician invented chess, and the emperor decided to give him a reward.

The mathematician said it is very simple. My reward is: you put 1 wheat on the first grid of the chessboard, 2 wheat on the second grid, and 4 wheat on the third grid. , Put 8 wheat on the fourth grid … and so on, you just need to fill the grid of this chessboard.

When the emperor heard this, he thought it was very simple, but it was just a few wheat.

But let ’s take a look. If you want to meet the requirements, how many wheats will you need? There are a total of 64 squares on the chessboard, which requires 264-1 wheat! Let’s convert and see how many liters of wheat are needed. It’s 140 trillion liters of wheat!

From humans to wheat, not so much wheat is produced worldwide. According to the current output, it is estimated that it will not be possible to produce so much wheat after 2000.

This example shows that after doubling once and 63 times, this number will become an astronomical number. Therefore, no data can be doubled at a time.

For example, if you want GDP to double every 7 years, if you really calculate at this rate, it will be an astronomical figure. Therefore, the growth rate of geometric series is particularly fast.

What does this have to do with our physical system? for example. If I put a few air molecules in a box, I first measure the initial position and initial velocity of these molecules with small errors.

By observing the movement of these molecules, you will find that because the movement of molecules is very unstable, the error will double in less than a second. After another second, the error doubles again. When I say a second, the error doubles in less than a second.

In other words, after 60 seconds, the original error value may become the astronomical figure you just saw.

This shows that a physical system, if the small error in the micro state has been doubled, then this error will have a very large impact on the system.

Of course, although the value is large, the size of the box restricts the movement of the molecules. After the molecule moves to the edge of the box, it will bounce back, so as a whole, its error will not reach that astronomical number. But from a local and micro perspective, its error can make the original system completely different from the predicted system, which is why I will give this example.

I would like to show that for a chaotic dynamic system, small deviations 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 the error increases exponentially in a small area and in a microscopic state. In mathematics, this is called the positive Lyapunov exponent . This is a mathematical vocabulary, and it is the only mathematical vocabulary today.

What does chaos mean? Explain that the future is unpredictable.

Why is it unpredictable in the future? Because the accuracy of the initial test was not much accurate, the system after one minute was completely irrelevant to the original system. This is the unpredictable principle of a chaotic dynamic system in the future.

Application of Chaos System

What kind of system is a chaotic system? For example, meteorological systems. You may have heard of the “butterfly effect”. The weather forecast originally said that there was a storm in Beijing today, but it didn’t actually rain. Why?

It turned out that two weeks ago, in Chicago, on the other side of the earth, a butterfly suddenly shook its wings and disturbed the air.

It is such a small fluctuation, it may become a double-sized fluctuation after one second, and a second, it will become a four-fold fluctuation … Two weeks later, the “butterfly effect” affect Arrived in Beijing, so today Beijing is clear and sunny, there is no rain.

So, to accurately predict the weather, you must know what each butterfly in Chicago did two weeks ago. However, there are many objects larger than butterflies, such as airplanes and trains, which are very large.

In addition, to accurately predict the weather two weeks later, the movement of all things in Chicago must be figured out. Of course not only Chicago, but also New York. So, do n’t expect to watch the weather forecast, you can calmly arrange to go hiking on the weekend, maybe it ’s going to rain suddenly on the weekend.

But don’t blame the Meteorological Bureau. This has little to do with the Meteorological Bureau. If you blame it, blame the chaotic dynamic system. The meteorological system is a chaotic system.

There are many chaotic systems. The three-body problem has now proven to be a chaotic system, which is why the three-body is a very complex motion. Meteorological systems and turbulent mechanics systems are chaotic systems.

Also I said earlier, why can I prove Pan Levy’s guess? It was because I proved that there is a special chaotic dynamic system in the motion of celestial bodies.

Because of time, I can’t explain to you exactly what I prove. If you are interested, you can read a book called “Tian Yu”. It was an English popular science book, which introduced my related work. There is now a Chinese translation.

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 reached the sky, researchers found that the fuel was insufficient to reach the moon’s orbit.

So, Japan asked NASA for help, and NASA sent a mathematician named Belbruno to help the Japanese.

Belbruno redesigned the orbit, and finally returned the probe to the lunar orbit. Belbruno used limited fuel to send the detector to a chaotic area.

Isn’t the chaotic region unpredictable, so if you push the detector with a little fuel,The movement has a particularly large impact.

So it’s good to just place the detector in a suitable place; if it’s not suitable, let it jitter slightly.

One day, Belbruno suddenly called me. He said that an article I wrote proved in theory which region is most prone to chaos.

He said that it took him a month to design a new orbit. If I knew that article at that time, it might only take a few days to redesign the orbit, and then I could put the lunar probe Saved.

A few years later, the US Hughes company encountered the same problem after launching a satellite: the satellite was not fuel-rich enough to reach its intended orbit after it was launched.

At this time, Belbruno made light use of the road, redesigned the orbit, and successfully sent the satellite to the intended orbit. So, this is a very interesting application example for chaotic systems.

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This article comes from , author: Xia Zhihong (Northwestern University tenured professor)