Top student: All my teachers are academic whizzes!

Chapter 21: Two Solutions!

Upon hearing Han Chuan say he knew a little bit, a hint of surprise flashed in Fang Jianyu's eyes, and he handed over a piece of chalk.

Han Chuan took the chalk and began writing on the blackboard.

Instead of following the conventional solution steps to first establish the differential equation, he created a blank area next to that huge three-dimensional coordinate system.

In an electromagnetic field, the Hamiltonian of a charged particle is H = (1/(2m))*(P - qA)² + qφ + mgz.

Choose an appropriate specification. For a uniform magnetic field B along the z-axis, a symmetric specification can be adopted:

A = (-By/2, Bx/2, 0).

[The electric field E is along the -z direction, and the electric potential φ = -Ez. Further combining the gravitational potential energy mgz with the electric potential energy qφ: V(z) = mgz - qEz = (mg - qE)z = 4qEz.]

The Hamiltonian can be written as: H = (1/(2m))*[(Px + qBy/2)²+(Py - qBx/2)²+ Pz²]+ 4qEz.

[...Within one period, the relationship between Pz and velocity vz is Pz = m * vz. The velocity of the ball between the upper and lower rebound points is determined by V(z). Using constant acceleration kinematics, we can obtain: Jz = (2√(2m)/(3π)) * 4qE * (Δz/2)^(3/2)...]

...

On the podium, Han Chuan followed the Hamiltonian system, first writing out the Hamiltonian of the system, and then decomposing the variables and action step by step to derive the results.

The classroom was completely silent, with most of the students staring blankly at Han Chuan on the stage, wondering what he was writing.

Only one or two top students sitting in the front row, who had studied some university content and elective books in advance, or who had tried to participate in physics competitions, stared at the stage, frowning and thinking.

Of course, this is perfectly normal.

After all, Hamiltonian is not a regular topic in high school physics courses and is not among the compulsory subjects for high school students.

Using Hamiltonians to solve this kind of problem was something that Mr. Zhou Peiyuan only recently taught him.

While Han Chuan was writing out the solution steps, Fang Jianyu stared blankly at the equation on the blackboard, his eyes filled with astonishment and disbelief.

This question is far beyond the curriculum for an average high school student.

He originally called Han Chuan up to solve the problem just to see how this student, praised by Cao Wen, would react when faced with a real challenge.

Should I just scribble something down or give up altogether?

To be honest, he never thought Han Chuan could solve the problem.

But now, this student is not only solving the problem, but is also using Hamiltonian, a concept that ordinary high school students wouldn't learn, to solve it.

How is that possible?!

A student who scored only 51 points in the previous monthly exam, and even with his rapid improvement in the last monthly exam, only scored 91 points, was able to solve a problem that was good enough to be used in the provincial competition.

No way, is this learning talent really that terrifying?!

On the platform, Han Chuan continued, the blackboard already covered with densely packed equations.

[…By matching dimensions and substituting the known conditions mg = 5qE and qB = m√(g/R), the above variational equation can be simplified to: (4qE * Δz) = (qB)²/(2m) * R², which is completely equivalent to the core proportion in Solution 1.]

Finally, after substituting the data, we get: Δz = R/2, T = 2π√(R/g).

【...】

A blackboard, one half of which is covered with problems written by physics teacher Fang Jianyu, while the other half is covered with answers completed by Han Chuan.

After obtaining the final answer, he did not report to Fang Jianyu immediately. Instead, he took two steps back, holding the chalk, and looked at his solution.

"Teacher Fang."

The classroom was completely silent. Han Chuan called out softly to Fang Jianyu, who was standing there in a daze.

Coming to his senses, Fang Jianyu glanced at the answers on the blackboard, his eyes and face filled with barely concealed excitement and joy, and for a moment he didn't know what to say.

The blackboard was covered with densely packed terms such as Hamiltonian, canonical momentum, magnetic vector potential, action-angular variable, adiabatic invariant...

This competition problem is so difficult that even a physics teacher like him wouldn't be able to figure out the answer at a glance, let alone a second-year high school student; he would have to work out the solution on scratch paper.

And this student in front of me, who scored only 51 points in physics two months ago, wrote it out so casually.

"Okay, okay!"

After muttering something to himself, he took a deep breath, looked at Han Chuan, and asked, "The conventional solution to this problem, the Hamiltonian, is something you only learn in university physics departments. Have you studied that?"

Putting down the chalk, Han Chuan thought for a moment and replied simply, "I've studied some of it, but solving this problem using high school mechanics is too complicated."

"The ball is deflected by the Lorentz force in the magnetic field, and with the elastic collision at the boundary, the trajectory will be very complex, and the direction will be different after each collision."

"But if you put it into the Hamiltonian framework, write the system as a whole, and then look for the system's conserved quantities and adiabatic invariants, you won't be disturbed by those messy trajectories in the middle."

"Zhou... uh, my self-taught textbook says that for dynamic problems with electromagnetic fields and boundary constraints, analytical mechanics is much better than vector mechanics."

Upon hearing this, Fang Jianyu's face showed surprise, and he pressed, "So, you can solve this problem in other ways?"

Han Chuan gave a slightly shy smile and said, "Actually, I think there's another, even simpler way."

"Um?"

This time, Fang Jianyu was genuinely surprised: "A simpler method? Without Hamiltonian?"

Han Chuan nodded and said, "It's not necessary."

He paused for a moment, then turned around and continued writing on the blank area of ​​the blackboard.

"If we use analytical mechanics to solve this problem, although it is rigorous, the process will be too long."

"What it requires is a final stable amplitude and period, so we can actually take a more direct approach to this."

"That is, we don't track how the particles turn or collide in the magnetic field, but only look at the stable state of motion they eventually enter."

"Because in the context of energy, the Lorentz force only changes the direction of motion, not the magnitude of kinetic energy. The core force driving the ball to move back and forth in the vertical direction is the resultant force of gravity and electric field."

"The resultant force in the vertical direction is mg - qE. Substituting mg = 5qE into the given equation, we get Fz = 4qE, which is directed downwards. This force is constant throughout the entire motion."

"That is, the driving power is ∝(4qE·Δz)/T, and the magnetic confinement power is ∝[(qB)²/m]·R²/T."

"Then, when both are equal in a steady state, T is eliminated, and we can obtain 4qE·Δz = C·(qB)²/ m· R²....."

While writing the equations on the blackboard, Han Chuan briefly explained his problem-solving approach.

The chalk made a crisp tapping sound on the blackboard. Compared to the steps used to solve the problem using Hamiltonians, this method only requires less than a third of the steps.

In just a few lines, Han Chuan had already calculated that the period T is the cyclotron period in the magnetic field: [T = 2πm/(qB) = 2π√(R/g)]

......

P.S.: There will be another update tonight, please vote and follow!