Just as Hong Fan was about to say goodbye, he heard the sound of tables and chairs moving in the corner of the room.

But it was the Cheng Master who had been mentioned by Wen Zhongguan before and strode over.

"Zhuang Gong, are you free?"

He asked, clearly not caring about whether the other party was available.

"You say."

Zhuang Liren replied.

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

Cheng said, spreading a recopied piece of paper on the table.

Zhuang Liren looked at it carefully, but frowned.

"You have the same way of thinking as I do, and the conclusion should be fine."

"But the process is not rigorous."

He took the charcoal from the Master of Process and circled the paper.

"With Mei Chengxue's temperament, this letter was sent to the chief supervisor of the five states at the same time, and he must have a complete proof method."

"We are sending this thing to Hezhou now, and it is indispensable to be laughed at by him."

The name Mei Chengxue, Hong Fan has heard of it, is the chief supervisor of Hezhou.

Hearing this, Master Cheng snorted from his nostrils.

"Zhuang Gong, just laugh, and there will be no less meat."

"We people have been thinking about it for two days, but we only have this result now, and we can't do anything about it."

He didn't have a good breath.

Zhuang Liren couldn't have a seizure, and he was depressed for a while.

At this time, Hong Fan saw the opportunity to speak.

"Can you show me?"

He had just glanced at the figures and force analysis drawn on the paper, and felt familiar.

Master Cheng glanced at him condescendingly.

"Those who don't understand it are useless to read it."

He pouted.

"This is the title sent by Mei Gong of Hezhou, take a look."

Zhuang Liren thought for a moment and turned the paper.

"He is Kim Hae Hong-beom, the one who wrote the three eloquent essays of "Taylor", "Bida", and "Langri".

Vaguely hearing the name "Hong Fan", the three bachelors who were still sitting in the corner of the room and discussing in a whisper immediately looked at this side.

Bachelor Cheng was taken aback and solemnly surrendered.

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

Hong Fan returned the salute and read the question carefully.

This is a practical question.

[The granary is full of grain.

Now the grain officer has to design a slide so that the grain falls from the top of the slide.

Suppose that the grain is only affected by the elemental magnetism during its motion, and the initial velocity is zero.

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

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

[Let A and B be two points on the straight plane of lead that are not on the same lead line, and find a curve among all the plane curves connecting A and B, so that the time required for a particle with zero initial velocity to move along this curve from point A to point B is the shortest.] 】

The answer, as described 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 sake of convenience, the expressions in the part of the book involving the symbol system are all consistent with reality, and you have translated them by me.) ）

The so-called cycloid is the trajectory formed by a fixed point on the boundary of a circle when it moves along a straight line.

In his previous life, many mathematicians were attracted by its special properties, so this curve is also nicknamed "The Helen of Geometers".

Hong Fan continued to look down at the solutions of the four Bachelor of Science students.

At the top is a concise diagram of the force analysis of the particles.

The following solution process is a little complicated, but the general idea is to cut the curve into infinite layers, so that each layer is infinitely thin, then the trajectory of the particle in each instantaneous motion can be regarded as the tangent of the position of the curve.

Thus, it can be deduced that an important property of the most rappelling line is that the ratio of the sine of the angle of the tangent and plumb line at any point to the square root of the height at which the point falls is constant.

A curve with this property is the 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."

Hong Fan finished reading it and said.

"Is there a better way?"

Cheng asked directly, his tone was quite impulsive.

He didn't doubt Hong Fan's ability, but he just felt that this person was young after all, but he made a conclusion after a hasty look, which was too arrogant.

"You can give it a try."

Hong Fan smiled at him, picked up the charcoal pen on the table, and began to write in the blank space.

Both the potential energy and the kinetic energy theorem are readily available, so there is the first equation.

【v=（2gy）^0.5】

Then the second equation is easily obtained from the particle motion relation.

【v=ds/dt=(1+y’^2)^0.5*dx/(2gy)^0.5】

The two are combined, and there is naturally a third equation for the DT integral.

【t=∫(1+y’^2)^0.5*dx/(2gy)^0.5】

(The formula editor can't send it, and it can't play the integration angle mark)

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

y(0)=0，y(p)=q

of all the continuous functions y(x) such that the above functional takes the minimum value of the function y.

Hong Fan finished writing the above sentence and straightened up.

By this time, all four scholars were already gathered around the table.

"Then the problem is clear."

Hong Fan said, his face full of light.

Cheng Maode frowned.

"Hong Fan Gongzi, we have already listed your formulas."

He was visibly disappointed.

"But there's no way to solve this thing."

Zhuang Liren also shook his head.

"Young Master Hong, your process is indeed clear and beautiful, but we don't have the tools to take advantage of this extreme value function yet."

This is a common situation in prison – drawing problems from typical physical phenomena and trying to find mathematical solutions, but without the right mathematical tools.

However, Hong Fan did not put down his pen.

"Guys, if you don't have a tool, create it."

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

The charcoal pen leaves countless characters on the white paper, which are as smooth as painting.

For functionals

S=∫L(f(x)，f’(x)，x)dx

Two endpoints are fixed, and the function when the functional S is taken to an extreme value is denoted as g(x),

Define a function that is "close" to this function ......]

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

Hong Fan listens and deduces at the same time.

It is as if those memories that are so long that they are mottled and faded are flowing in the soul again.

Half an hour later, he completed the whole process and wrote his name above the new theorem.

Euler's equation. 】

The Euler-Lagrange equation (E-L equation), often referred to as the Lagrangian equation in mechanics, is the key theorem of the variational method.

"Now, we have the tools."

Hong Fan examined the theorem on paper and felt a little ashamed in his heart.

But he quickly suppressed the distractions and began to solve the functional of the fastest descent line using the E-L equation.

The results are easily derived.

【x=r*(t-sint)，y=r*(1-cost)】

Exactly the cycloid.

Until Hong Fan gently put down the carbon pen, there was still no one in the room to speak.

Time has slipped away.

But the sense of power that destroys the withering and decaying still echoes in the hearts of Zhuang Liren.

For example, as the otter saw, the stone dam that was blocked in the stream and was indestructible by the wind and rain was crushed by the dragon.

For example, the squirrel inhabited the eternal tree that towers in the forest and is knocked away by a giant elephant.

Zhuang Liren didn't expect it.

Decades after entering the instrument as a prisoner, on this unforeseen afternoon, he actually felt the insignificance of being a human being for the first time in a long time.

PS: Due to entering the write-and-publish mode, there is no guarantee of a fixed update time in the future, and it is possible in each period, please forgive me.

(End of chapter)