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The sunlight on Sunday morning was softer than usual, like a thin layer of golden gauze, gently covering the glass windows of the library.
When Xu Yang arrived ten minutes early, his old seat by the window had already been taken - Lu Jinyu was lying on the long table with his head buried in his arms. The hood of his light gray sweatshirt had slipped down, covering most of his face, leaving only a small section of his white neck and soft hair exposed.
A collection of real-life competition mechanics questions lay spread out on the table, with two steaming cups of American coffee beside it. Water droplets condensed on the walls of the cups were slowly dripping down.
Xu Yang walked over lightly and was about to sit down when he saw Lu Jinyu suddenly raise his head, his eyes still hazy from waking up, like a frightened deer.
"You're here?"
He rubbed his eyes, his voice a little hoarse from just waking up, and reached out to push one of the American coffees towards Xu Yang.
"I figured you'd arrive early today, so I took a detour and bought your usual drink, less ice and half sugar."
Xu Yang looked at the familiar cup of American coffee and felt a warm feeling in his heart.
When he sat down, his fingertips accidentally touched the back of Lu Jinyu's hand. They both paused, then quickly moved away, but the tips of their ears turned red at the same time.
"Thank you."
Xu Yang took a sip of the coffee, and found it was just the right temperature:
"Why are you here so early?"
"I was thinking about that Samsung system problem last night. The more I thought about it, the more I couldn't sleep, so I just came here early."
Lu Jinyu pointed to the red-circled question in the real exam book, his brow furrowed again. "Look at this. Three celestial bodies are performing circular motion under the influence of gravity, and we have to analyze their stability under small perturbations. I drew three force analysis diagrams alone, and they're getting messy the more I draw."
Xu Yang looked down at the question: Three celestial bodies with masses m, m, and m respectively, initially form an equilateral triangle and perform uniform circular motion around the center of mass. If one of the celestial bodies is subjected to a tiny radial disturbance, determine whether the system is stable.
This is indeed a difficult point in the competition. It not only involves the knowledge of celestial mechanics, but also requires the use of Lagrange equations in analytical mechanics, which requires a high level of mathematical deduction ability.
"Don't worry, let's start with the simplest model."
Xu Yang took a red pen from his pencil case and drew an equilateral triangle on a piece of scratch paper, labeling the three vertices A, B, and C. "First, we need to determine the center of mass of the system and use it as the origin of the coordinate system. This will eliminate the overall translation of the system and simplify the calculation."
He paused and added, "Remember what we said last time when we reviewed the law of center of mass motion? For a multi-body system, the motion state of the center of mass is determined only by external forces. However, the gravitational forces between the three celestial bodies here are all internal forces, so the center of mass is stationary."
Lu Jinyu nodded and immediately took out his wrong answer book and turned to the relevant chapter. There was a blue pen saying "Application skills of the center of mass coordinate system"
There's also a small diagram drawn next to it.
"I remember this," he said suddenly, "the last time you explained the two-body problem to me, you simplified it using the barycentric coordinate system, turning the motion of the two celestial bodies into the relative motion of one celestial body relative to the other."
"Yes, this idea can be extended to multi-body problems."
Xu Yang wrote down the formula for the center of mass coordinates on a piece of scratch paper. "Let's first calculate the distance from the center of mass O to each celestial body. Since we initially have an equilateral triangle with a side length of a, we can set the position vector of point A to r, point B to r, and point C to r. According to the definition of the center of mass, mr + mr + mr = 0. This condition can be used to eliminate a variable."
Lu Jinyu immediately picked up a pen and followed Xu Yang's idea to calculate on the draft paper: "Assuming m = m = m, m = 2m, then the center of mass should be closer to point C, right?"
As he calculated, he said, "The lengths of r and r should be equal. The length of r is half of r?"
"That's about it, but it requires precise calculation."
Xu Yang helped him sort out the calculation process. "Let R be the distance from the center to the vertex of the equilateral triangle. Then the distance from each star to the center of mass can be expressed as r = (m + m)R / (m + m + m), and so on. However, because the initial motion is uniform circular motion, we also need to use the condition that universal gravitation provides the centripetal force to calculate the initial angular velocity ω."
Work together on scratch paper to derive an expression for the initial angular velocity.
Lu Jinyu was responsible for calculating the magnitude of the gravitational force between the two celestial bodies, while Xu Yang was responsible for decomposing the gravitational force into radial and lateral components, which together provided centripetal force.
The sunlight slowly moved across the table, climbing from one end of the formula to the other. Occasionally, a breeze blew in from the half-open window, carrying the faint fragrance of cherry blossoms outside the window and falling on the pages of the two people's books.
"Should we use the law of cosines here to calculate the net force exerted on A by the gravitational forces of B and C?"
Lu Jinyu took a pen and pointed at the force analysis diagram on the draft paper and asked.
Point A is affected by the gravitational force F from B and the gravitational force F from C. The angle between the two forces is 60 degrees because it is initially an equilateral triangle.
"right."
Xu Yang nodded. "The magnitude of F is G mm/a. F is G mm/a. Then use the parallelogram rule to find the resultant force. Then decompose the resultant force into a radial component pointing toward the center of mass and a transverse component perpendicular to the radial direction. Since the initial motion is uniform circular motion, the transverse component should be zero, and the radial component provides the centripetal force mrω."
Lu Jinyu immediately began to calculate, his pen tip moving quickly on the draft paper.
After a while, he frowned and looked up: "That's not right. The radial component of the net force I calculated is greater than the centripetal force. Is there something wrong with the decomposition?"
Xu Yang leaned over to take a look at his calculation process and found that he had miscalculated the angle when decomposing the radial component of F.
"You see, point C is on the other side of the center of mass, so the angle between F and the radial is not 60 degrees, but 120 degrees, and the cosine value should be -0.5, not 0.5."
He circled the incorrect spot with a red pen. "You made a similar mistake last time when analyzing the forces on an inclined plane. Remember to first determine the direction of the force and the angle between the coordinate system."
Lu Jinyu suddenly realized and slapped his forehead:
"How could I make such a stupid mistake again!"
He immediately corrected himself and recalculated, finally getting the correct answer. "I figured it out! ω = G(m + m + m)/a. This result seems pretty neat."
"Very good, this is the initial equilibrium state."
Xu Yang nodded. "Next, we need to analyze the situation with tiny perturbations. Let's assume we give Star A a tiny radial perturbation, causing its distance to change from r to r + δr, where δr is a very small quantity. Then, we'll analyze its equation of motion to see if this perturbation increases over time. If it does, the system is unstable; if not, it's stable."
This step requires the use of the Lagrange equation, which Lu Jinyu is not very familiar with, and a look of embarrassment appeared on his face.
"I've only seen the Lagrange equation in textbooks and have never used it in practice. It seems so complicated."
He scratched his head. "L = T - V, where T is kinetic energy and V is potential energy, right? But if we add up the kinetic and potential energies of three stars, the expression is bound to be very long."
"Don't worry, we can write the general form of the Lagrangian first and then use symmetry to simplify it."
Xu Yang wrote down the expression for the Lagrangian on a piece of scratch paper. "T is the sum of the kinetic energies of the three celestial bodies. The kinetic energy of each celestial body is m + mrθ, because it is in polar coordinates. V is the gravitational potential energy between the three celestial bodies: V = -Gmm/a - Gmm/a - Gmm/a. However, after the perturbation, a will become a + δa, so V needs to be expanded at the equilibrium position to retain the quadratic term of δa."
Lu Jinyu listened carefully and took notes in his wrong answer book from time to time.
Xu Yang explained in great detail, from the derivation of the Lagrange equation to how to deal with small disturbances. He explained each step clearly and stopped from time to time to ask him if he understood.
The knowledge points that Lu Jinyu originally found obscure and difficult to understand gradually became clear after Xu Yang's explanation.
Before I knew it, two hours had passed. The sunlight outside the window had risen above my head, and more and more people were coming into the library.
With the help of Xu Yang, Lu Jinyu finally wrote the differential equation of motion after the disturbance.
"So, as long as the characteristic roots of this equation are purely imaginary numbers, the system is stable, right?"
He looked at the final result on the draft paper, his eyes full of anticipation.
"That's right."
Xu Yang smiled and nodded. "Look at this equation. The second derivative of δr plus a positive number multiplied by δr equals zero. This is an equation of simple harmonic motion. The solution is periodic, which means that the disturbance will not increase indefinitely and the system is stable."
Lu Jinyu cheered excitedly and almost jumped out of his chair: "Great! I finally figured it out!"
He looked at the densely packed formulas on the draft paper, then looked at Xu Yang, a grateful smile on his face. "Xu Yang, thank you so much. If it weren't for you, I would definitely still be staring at this problem in a daze."
"You're welcome."
Xu Yang felt warm at the sight of his happy face. "You've worked really hard yourself. You've been able to follow my train of thought and even raise questions. That's a huge improvement."
He paused, then pulled out a sticky note from his pencil case. He drew a picture of a man holding a trophy, with the words "Mechanics Expert Lu Jinyu" written next to it. He stuck it on a corner of his draft paper. "This is a reward for you. To celebrate your success in solving this difficult problem."
Lu Jinyu looked at the little man on the sticky note and couldn't help laughing. He carefully peeled off the sticky note and stuck it in his wrong answer book, next to the strawberry little man.
"I'm going to keep this reward safe," he said. "Whenever I encounter difficulties in the future, I'll take it out and look at it to cheer myself up."
At this moment, Lu Jinyu suddenly slapped his head:
"Oops! I almost forgot something!"
He pulled out a small sealed bag filled with cat food from his canvas bag. "Yesterday I saw a ginger cat in the small garden behind the library. It was so skinny it was just a pile of bones. I bought some cat food. I had planned to feed it at noon, but I forgot about it when I started doing my homework."
Xu Yang looked at the cat food in his hand and couldn't help but smile: "You're quite caring. We've been reviewing all morning and are tired. Why don't you go feed the cat and come back for lunch?"
"Okay, okay!"
Lu Jinyu immediately packed up his things, grabbed Xu Yang's wrist and walked out. "It's hiding behind the hedge covered with roses. It's very afraid of people. I had to squat for a long time yesterday before it was willing to come out to eat something."
The two walked through the side door of the library and came to the small garden at the back.
In the garden in April, roses and roses are in full bloom. Pink, red and yellow flowers cover the fences and the air is filled with rich floral fragrance.
Sunlight streamed through the gaps in the leaves, casting mottled shadows on the ground. Lu Jinyu tiptoed to the edge of the fence, poured some cat food from the sealed bag into his palm, then slowly squatted down and whispered to the inside of the fence:
"Mimi? Xiaoju? Come out to eat, I brought you some delicious food."
Xu Yang stood not far behind him, looking at his cautious look, and couldn't help but raise the corners of his mouth slightly.
Lu Jinyu usually looks careless, but he is extremely gentle and careful when treating small animals.
After a while, an orange kitten the size of a palm poked its head out from the gap in the fence. Its eyes were sparkling as it looked at them warily, sniffed the cat food in the air, hesitated for a long time, and then slowly moved out.
"Don't be afraid, don't be afraid. We won't hurt you."
Lu Jinyu softened his voice and handed the cat food in his palm forward.
The kitten looked at him again, and after making sure there was no danger, it lowered its head and began to eat in small bites.
It ate very quickly, as if it had been hungry for a long time, and its small body trembled slightly from swallowing so hard.
Lu Jinyu looked at the kitten wolfing down its food and said distressedly, "Look how thin it is. It must have been hungry for several days."
He poured some cat food from the sealed bag onto the stone next to him and said, "Eat more. If it's not enough, I have plenty here."
The kitten finished eating the cat food in its palm and ran to the stone to continue eating.
Lu Jinyu quietly reached out his hand, wanting to touch its head, but as soon as he reached out, the kitten cautiously shrank back, making a "woooo" sound in its throat. Lu Jinyu smiled, withdrew his hand, and watched it eat patiently.
"It's quite timid," Xu Yang also came over, squatted beside Lu Jinyu, and whispered, "Maybe it's been bullied before."
"Yeah," Lu Jinyu nodded. "When I saw it yesterday, it was looking for something to eat next to the trash can. It had several wounds on its body and looked so pitiful."
He paused, then added, "From now on, I'll come and feed it every day. Maybe after a while, it won't be afraid of us anymore."
Xu Yang looked at his serious expression, and ripples appeared in his heart.
He suddenly felt that although the boy in front of him was a little confused sometimes, he had an extremely kind and soft heart.
"Okay," Xu Yang said with a smile, "When it's no longer afraid of us, let's give it a name, right?"
"Good idea!" Lu Jinyu's eyes lit up. "What should we call it? It's orange, how about 'Xiaoju'? It's simple and easy to remember."
"Okay, that's quite appropriate." Xu Yang agreed.
After eating the cat food, the kitten drank a few sips of water from the pool next to it, and then called to them twice in a small voice, as if to say thank you.
Lu Jinyu said happily: "Look, it seems to be thanking us!"
"Perhaps."
Xu Yang stood up and dusted off his pants. "We should go eat. We still have to continue reviewing in the afternoon."
"Um!"
Lu Jinyu also stood up and put away the remaining cat food. "I'll bring a small plate tomorrow so it will be easier for it to eat without getting its paws dirty."
The two walked out of the small garden side by side. The sunlight stretched their shadows very long and pressed them close together.
When passing by a dessert shop, Lu Jinyu suddenly stopped and said, "Wait a minute, I'm going to buy something." He walked quickly into the store and came out a few minutes later with two strawberry ice creams. He handed one to Xu Yang and said, "I just made it. It's strawberry flavored. Try it."
Xu Yang took the ice cream and took a bite. The sweet strawberry flavor spread on the tip of his tongue, and the cool feeling instantly dispelled the fatigue of the morning.
"It's delicious," he said with a smile. "Thank you."
"You're welcome," Lu Jinyu also took a bite of his ice cream. "This is a reward for us solving the difficult problems this morning, and also a reward for us getting to know Xiaoju."
The two of them walked towards the cafeteria while eating ice cream.
On the way, Lu Jinyu talked non-stop about how he would take care of Xiaoju in the future, buying cat litter and cat toys for it, and taking it to the veterinary hospital for a checkup. Xu Yang listened quietly, nodding occasionally, feeling particularly warm in his heart.
When they arrived at the cafeteria, Lu Jinyu skillfully ordered Xu Yang's favorite tuna salad and sweet and sour pork ribs, and ordered a serving of Kung Pao Chicken and a bowl of rice for himself.
The two found a seat by the window and continued to discuss their afternoon review plan while eating.
Lu Jinyu said he wanted to review fluid mechanics because he made several mistakes in the last question on Bernoulli's principle. Xu Yang agreed and said he could help him compile a list of common mistakes so that he could review more specifically.
After lunch, the two returned to the library and continued their intense review. Following Xu Yang's advice, Lu Jinyu first reviewed the basic concepts of fluid mechanics and then worked on related examples and exercises.
Whenever he encountered something he didn't understand, he would ask Xu Yang for help. Xu Yang was sorting out a list of common mistakes, and from time to time he would look up at Lu Jinyu and notice that when he encountered a difficult problem, he would frown and bite the pen, looking particularly serious and cute.
Around four o'clock in the afternoon, Lu Jinyu suddenly said, "I'm a little sleepy. Let's go for a walk outside to wake up."
Xu Yang nodded, put down the pen in his hand, and walked out of the library with him.
The two walked slowly along the cherry blossom path in front of the library, pink cherry blossom petals drifting in the wind, landing on their shoulders and hair.
"Look, this cherry tree is in full bloom," Lu Jinyu said, pointing to a lush cherry tree. "Last time we took photos here, you were standing under the tree with petals falling from your hair. It was really beautiful."
Xu Yang's heart suddenly skipped a beat. He remembered the photo Lu Jinyu took of him last time, with the words "Xu Yang, a hundred times more beautiful than cherry blossoms" written on the back.
"Yeah?"
He pretended to be nonchalant and said, "I think it's just so-so."
"That's not the case," Lu Jinyu said seriously:
"In my heart, you are more beautiful than any flower."
After saying that, his face turned red instantly and he lowered his head embarrassedly.
Xu Yang looked at his flushed cheeks, and his heart seemed to be filled with something warm.
He plucked up his courage and gently held Lu Jinyu's hand.
Lu Jinyu was stunned for a moment, then took his hand, interlocking their fingers. The two of them didn't speak, just walking quietly along the cherry blossom path. The sunlight shone through the branches of the cherry trees, falling on their tightly clasped hands, warm and beautiful.
After returning to the library, Lu Jinyu's review efficiency improved significantly, and he became more and more adept at solving fluid mechanics problems.
Xu Yang watched his progress and felt genuinely happy for him. He handed Lu Jinyu a list of common mistakes he had compiled, detailing common mistakes and precautions for knowledge points like Bernoulli's principle, the continuity equation, and viscous fluids, along with corresponding example questions.
"Thank you so much, Xu Yang!"
Lu Jinyu took the list and said excitedly, "With this, it will be much easier for me to review."
He immediately reviewed the contents on the list, and discussed with Xu Yang when he encountered something he didn't understand. The two of them cooperated more and more tacitly.
In the evening, when the library closing bell rang, the two had already finished reviewing the key points of fluid mechanics.
On the way home, Xu Yang held Lu Jinyu's face and kissed him forcefully: "Lu Jinyu, do you love me?"
Lu Jinyu nodded: "I love you, but verbal love doesn't count. I want to prove my wholehearted love for you with actual actions."
Xu Yang smiled: "Okay, then let's work hard together and stay together forever."
Together forever.
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