Lord of the Desert

Chapter 193
Just when Hong Fan was about to leave, he heard the sound of tables and chairs moving from the corner of the room.

It was the Bachelor Cheng who Wen Zhongguan mentioned before strode over.

"Mr. Zhuang, are you free?"

He asked, obviously not too concerned about the other party's availability.

"You said."

Zhuang Liren replied.

"We have solved the problem, it should be a cycloid."

Cheng Xueshi said, spreading out a piece of re-copied paper on the table.

Zhuang Liren looked at it carefully, but frowned.

"Your thinking is the same as mine, and the conclusion should be fine."

"But the process is not rigorous."

He took the charcoal pencil in the hand of the Process Scholar and circled it on the paper.

"With Mei Chengxue's temperament, this letter was sent to the supervisor of the Wuzhou University at the same time. He must have a complete proof method."

"We are sending this item to Hezhou now, and we will inevitably be ridiculed by him."

Hong Fan had heard of the name Mei Chengxue, who was supervised by Hezhou's Warehouse Supervisor.

Hearing this, Xueshi Cheng took a puff of air from his nostrils.

"Mr. Zhuang, just laugh, you won't lose a piece of meat."

"Those of us have thought about it for two days, and there is only this result, and there is nothing we can do."

He didn't feel good.

Zhuang Liren couldn't get into a fit, and was short of breath for a while.

At this time, Hong Fan saw the opportunity and spoke.

"Can I see it?"

He just glanced at the graphs and force analysis drawn on the paper, and felt very familiar.

Cheng Xueshi took a condescending glance.

"For those who don't understand, it's useless to read."

He curled his lips.

"This is a topic sent by Mr. Mei of Hezhou, please take a look."

Zhuang Liren thought for a while, then turned the paper.

"He is Jin Haihongfan, the one who wrote the three great essays "Taylor", "Bida", and "Langri"."

Faintly hearing the name "Hong Fan", the three bachelors who were still discussing quietly in the corner of the room immediately looked over here.

Cheng Xueshi was taken aback, and solemnly bowed his hand.

"Xijing Cheng Maode, I was rude just now."

Hong Fan returned a salute and read the title carefully.

This is an application question.

[The granary is full of grain.

Now Grain Officer A wants to design a slide so that the grain falls from the top of the slide.

Assume that the grain is only affected by the elemental magnetism in the process of movement, and the initial velocity is zero.

How to design the slide to make the food reach the ground in the shortest time? 】

After reading it, Hong Fan found that this was exactly an exercise he had done in his previous life study of functional analysis - finding the fastest descending line.

【Assume that A and B are two points on the vertical plane that are not on the same vertical line. Among all the plane curves connecting A and B, find a curve so that the mass point that is only affected by gravity and has zero initial velocity moves from point A to The time required to move along this curve to point B is the shortest. 】

The answer, as stated by Cheng Maode and Zhuang Liren, is the cycloid (x=r*(t-sint), y=r*(1-cost)).

(Of course, Dahua does not have Arabic numerals and English letters, but for the convenience of expression, the expressions involved in the symbol system in this book are all consistent with the reality, and everyone will think that I have translated them.)
The so-called cycloid is the trajectory formed by a certain point on the boundary of the circle when a circle moves along a straight line.

In Hong Fan's previous life, many mathematicians were attracted by its special properties, so this curve has another name, which is called "The Helen of Geometers".

Hong Fan continued to read the explanations of the four bachelors of science.

On the top is a concise particle force analysis diagram.

The solution process below is a bit complicated, but the general idea is to cut the curve into infinite layers and make each layer infinitely thin. Then the trajectory of the particle at each instant can be regarded as the tangent to the position of the curve.

Therefore, it can be deduced that an important property of the steepest drop line is that the ratio of the sine of the angle formed by the tangent line and the plumb line at any point to the square root of the falling height of the point is constant.

A curve with this property is a cycloid.

From the perspective of later generations, this answer is indeed not rigorous in theory, and it is no wonder that Zhuang Liren is dissatisfied.

"This solution is correct, but it is quite inferential."

After reading it, Hong Fan said.

"Do you have a better way?"

Cheng Xueshi asked directly, with a rather aggressive tone.

He didn't doubt Hong Fan's ability, but felt that this person was young after all, but it was too arrogant to make a conclusion after a hasty look.

"You can give it a try."

Hong Fan smiled at him, picked up the charcoal pen on the table, and started writing in the blank space.

Potential energy and kinetic energy theorems are ready-made, so there is the first equation.

[v=(2gy)^0.5]

Then the second equation can be easily obtained from the relation of particle motion.

【v=ds/dt=(1+y’^2)^0.5*dx/(2gy)^0.5】

Combining the two, the integral of dt naturally has a third equation.

【t=∫(1+y’^2)^0.5*dx/(2gy)^0.5】

(The formula editor can't send it out, and can't print the credit mark)

In this way, the time t of the whole movement of grain particles is a function of y(x), and the solution of the problem is to satisfy the boundary conditions

y(0)=0, y(p)=q
Among all continuous functions y(x) of , the function y that makes the above functional take the minimum value.

After Hong Fan finished writing the above sentence, he straightened up.

At this time, all four bachelors had gathered around the table.

"Then the question will be clear."

Hong Fan said with a relaxed face.

Cheng Maode frowned.

"Mr. Hong Fan, we have already listed your styles."

He was visibly disappointed.

"But there is no way to solve this thing."

Zhuang Liren also shook his head.

"Mr. Hong, your process is really clear and beautiful, but we don't have the tools at our fingertips to ask for this extremum function."

This is a situation that is often encountered in the instrumentation laboratory-deriving problems from typical physical phenomena and trying to find mathematical solutions, but there is no suitable mathematical tool.

But Hong Fan did not put down his pen.

"Everyone, since there are no tools, let's create tools."

These words were so arrogant that both Zhuang Liren and Cheng Maode were stunned when they heard them.

The charcoal pen leaves countless characters on the white paper in one go, as smooth as painting.

[For functional

S=∫L(f(x), f'(x), x)dx
Fix the two endpoints, and the function when the functional S reaches the extreme value is denoted as g(x),

Define a function "closer" to this function...]

In the quiet study room, there was only the rustle of writing for a while.

Hong Fan listened and deduced at the same time.

It seems that those memories that have been mottled and faded for a long time are flowing in the soul again.

After a while, he completed the whole process and wrote his name on the new theorem.

[Euler equation. 】

The Euler-Lagrange equation (Euler-Lagrange equation) is referred to as the EL equation, and it is often called the Lagrange equation in mechanics, which is the key theorem of the variational method.

"Now, we have the tools."

Hong Fan examined the theorem on paper, feeling a little ashamed.

But he quickly suppressed distracting thoughts and started to solve the functional of the steepest descent line with the EL equation.

The result is easily drawn.

【x=r*(t-sint), y=r*(1-cost)】

It is the cycloid.

Until Hong Fan gently put down the charcoal pen, no one in the room spoke.

Time has slipped by.

But that overwhelming sense of power still reverberates in Zhuang Liren's heart.

For example, as the otter saw, the stone dam blocked by the stream and hard to break by the wind and rain was run over by the dragon.

For example, the squirrel inhabits, the eternal tree towering in the forest is knocked away by the giant elephant.

Zhuang Liren didn't expect it.

After being in prison for decades, at this unexpected noon, he felt the insignificance of being a human being for the first time in a long time.

PS: Due to entering the write-now-publish mode, we cannot guarantee a fixed update time in the future, and it is possible at any time, please forgive me.

(End of this chapter)