A few years ago, there was a "fishing" joke that China and American students took part in a mathematical modeling contest. After that, China students packed their things and left, while American students silently packed the models on the table and took them away before leaving.

Although it was an interesting story to catch a lot of fish in those days, today's focus is not here. What we need is the "model" established by mathematical modeling.

The so-called "modeling" means that when we face a complex problem, we can simplify it into some methods with clear logic, easy analysis and understanding, especially easy calculation through layer-by-layer analysis.

Give an example of mathematical modeling.

For example, the expected number of coronavirus infections this time is a typical mathematical modeling. Of course, we don't know the exact infection data yet, but everything is not completely untraceable. We have a lot of data, such as the growth rate of confirmed infection, the total population of Wuhan, the population flow data between Wuhan and the outside world, and the approximate proportion of infection after contact with patients. ...

Through these data, we can simply create a mathematical model. For example, the earliest data of possible infection is calculated according to the determined population of Wuhan going abroad. I can't find the specific data, so I can simulate it for the convenience of demonstration (that is to say, I made up the following data).

At that time, the idea of this media was that three passengers going abroad from Wuhan were diagnosed, and there were about 4,000 passengers flying abroad from Wuhan Airport every day these days. From the discovery of the epidemic to the day, there were 15 days, of which Wuhan natives accounted for about 50%, that is to say, 3 out of 3000×15× 50% = 30,000 Wuhan people were infected, and the infection rate was 0.065438. It is known that the population of Wuhan is about140,000, which means that about1400,000× 0.01%=1400 people are infected.

This is a typical example of analyzing a problem with a mathematical model. Although this figure is far from the real figure, it is more convincing than any other guess under the limited conditions at that time.

Newtonian mechanics is actually a mathematical model.

Now let's review the Newtonian mechanics learned in middle school physics. If we take it seriously, in fact, many conclusions are incorrect. Not to mention the problem of complete failure under low-beam motion, even in the low-speed world, there are many wrong places when you think about it carefully.

For example, in the physics problem of junior high school, the object is absolutely rigid, and the lever used to transmit force never has the concept of "deformation", even if the earth is tilted, it makes sense. There are only two kinds of friction, sliding and rolling, and the friction coefficient is a constant, no matter what speed, it will not change. But we know that friction will generate heat, and temperature will obviously change the physical properties of materials.

▲ An example in which friction completely denatures an object. This is a match head.

Such a world is called a "linear" world, and all data are predictable. If the friction coefficient of your sole is 10, and the pressure of 1 will bring 10 N static friction, then putting a moon on your instep will naturally get 10 times the static friction of the moon's gravity: 7.2×10 24 n.

But is it possible? Of course not! When the pressure on the sole is greater than a certain value, it will be crushed. This is the real world, and this is the so-called "nonlinearity". And after the emergence of quantum mechanics, we realized that the world was even discontinuous. In the macro world, the function that can accurately describe the motion of an object can only be changed into a rather fuzzy probability function.

▲ The seemingly smooth and straight space seems to be surging when it is enlarged enough.

So Newtonian mechanics is only a mechanical model, and it is a mechanical model with strict restrictions. All the concepts and definitions in it have nothing to do with truth (although Newton himself thought he had found the truth at that time because he thought the truth was hidden in mathematics). But it doesn't matter, we can still learn, because the force analysis of Newtonian mechanics can solve many simple physical problems for us and get relatively accurate results, and there is no doubt that these figures are of practical value.

▲ "Mathematical Principles of Natural Philosophy" shows Newton's admiration for mathematics.

All physical studies are models.

I believe you may have thought, is it true that Einstein's theory of relativity and quantum mechanics swept through the 20th century? Of course not. It can be said with great certainty that they are all physical models, but they are closer to reality than any previous models. The astronomical data calculated by relativity can be accurate to more than ten decimal places, which is consistent with the observation height. Therefore, we believe that relativity is correct, but we are not sure whether there will be a brand-new theory in the future, which can be more accurate and closer to the truth than relativity, just as relativity embraces Newtonian mechanics.

Finally, let's sum up the initial question-why is there only three elements of force? Because this is an element that cannot be simplified under Newton's mechanical model, it is impossible to carry out force analysis without it. If you want a very accurate calculation, you can get 100 elements, or even 1000 elements. For example, "fluid mechanics" may never have a "universal formula", because there are too many parameters involved, and the design of aircraft still depends on wind tunnel experiments.