Upon hearing Qiao Ze's words, Peter Schultz couldn't help but stand up and step closer to Qiao Ze's side to look at the calculations he was working on his draft paper.
"Is this an elliptical model?"
"Yes, we first assu a three-body problem and represent the position of each object in the three-body system as a solution of elliptical functions."
Upon finishing, Qiao Ze wrote down three more formulas on the manuscript.
[ x_i(t) = a_i \\cos(\\oga_i t \\phi_i),]
[ y_i(t) = b_i \\sin(\\oga_i t \\phi_i),]
[ z_i(t) = c_i \\cos(\\oga_i t \\phi_i),]
He then began to explain, "(a_i, b_i, c_i) are the semi-major axis, semi-minor axis, and semi-interdiate axis of the ellipse, respectively, (\\oga_i) is the angular frequency of the ellipse, and (\\phi_i) is the initial phase."
Roth Dugan's face first showed a look of realization and then frowned as he asked, "But how does this affect the calculation of the interaction forces?"
"By approximating the interaction forces through a series expansion. For example, consider the gravitational force between object (i) and (j), then define the interaction force as..."
As he spoke, Qiao Ze wrote down a series of formulas on the manuscript.
[\\mathbf{F}_{ij}=-G \\frac{m_i m_j}{|\\mathbf{r}_i -\\mathbf{r}_j|^2}\\hat{\\mathbf{r}}_{ij},].
He then said, "(G) is the gravitational constant, which needs no explanation, (m_i, m_j) are the masses of objects (i) and (j) respectively, and (\\hat{\\mathbf{r}}_{ij}=(\\mathbf{r}_j -\\mathbf{r}_i)/|\\mathbf{r}_j -\\mathbf{r}_i|) is the unit vector."
"How do you perform a series expansion? Does transcendental geotry also involve calculations in chanics?"
"In transcendental geotry, we're allowed to use term-by-term approaching techniques, which can be used for series expansions, specifically we can get..."
[\\frac{1}{|\\mathbf{r}_i -\\mathbf{r}_j|} = \\sum_{k=0}^{\\infty}\\frac{\\Psi_k}{r^{k 1}},]
"Right, (\\Psi_k) are the coefficients."
Looking at the formulas on the manuscript that were becoming more comprehensive, Peter Schultz suddenly felt unwell, furrowed his brows and said, "This isn't right, there will be truncation errors in the series expansion, and these errors are uncontrollable, right?"
"There's a way, did you forget how I solved it during today's paper discussion? Set a truncation parater N, and only consider the first N terms of the series expansion. As long as the value of N is large enough, the model will mathematically approach the exact solution.
Of course, when it cos to calculating celestial positions, we don't need such high accuracy. We just need to take into account the performance of the supercomputer and the required accuracy to set the truncation parater. It will certainly require less computation than solving higher-order differential equations."
Roth Dugan subconsciously gestured with his hand and asked, unable to help himself, "In this iterative solving process, there will be issues of nurical instability, how do you resolve that?"
"Hmm?"
Qiao Ze wrote down the last stroke, reviewed his derivation process carefully, and then handed the pen back to Roth Dugan, saying, "I rember there's a self-adaptive step size nurical integration algorithm, and with the advantages of transcendental geotry in solving complex problems, we should be able to ensure the stability of the nurical solutions even when the distances between objects are small.
Of course, you can also use nurical stability analysis to adjust the paraters of the algorithm. In short, there should be many thods available, but if I were to solve this problem, I would definitely choose this approach. Moreover, if we're calculating the relativistic N-body problem, we use Einstein's field equations instead of the traditional Newtonian gravitational laws, and the general approach doesn't change much."
After speaking, Qiao Ze returned the manuscript and pen to Roth Dugan.
Roth Dugan foolishly took the manuscript handed over by Qiao Ze and, looking at its contents, his facial expression rapidly shifted.
Truly, after studying the N-body problem for so many years, he had never felt his thoughts so clear-cut.
Most importantly, he had co to appreciate Qiao Ze's boldness in tackling mathematical problems.
Or it might be said that his thinking was incredibly flexible; perhaps it was because he had a deeper understanding of transcendental geotry.
In any case, the approach he offered was utterly revolutionary.
Of course, the actual effectiveness would still have to be verified by a supercomputer, and his team would need to design the specific algorithm.
Although Qiao Ze didn't ntion it explicitly, Roth Dugan was well aware that Qiao Ze had already been very accommodating by explaining his thinking to such an extent. If the problem could be solved in a few simple minutes, the N-body problem would be far too simple...
More importantly, who would be listed as the first author if there was a future publication?
...
anwhile, the three people in the small courtyard had also gathered to chat casually.
"What are those two foreigners discussing with Qiao Ze? Zhou Shun, you have the best English, translate for us."
"My English is definitely good, but that's useless given what they're talking about. It seems like they're discussing so celestial computation or sothing like that? It's like Qiao Ze offered them a solution."
"Nonsense, if you put it that way, I can tell from those two foreigners' expressions that they benefited greatly. Isn't your English very fluent when you accept tasks? Why do you drop the ball at crucial monts?"
"No, it's not that, Lv Ge, but they used words I've never heard before. There's a type of English called 'academic jargon' that ordinary people don't usually encounter, you know?"
"Oh, let's not talk about this. I feel like Qiao Ze doesn't have this much patience for dostic professors. Could it be that the old man really persuaded him and he's considering going abroad? That would be a big problem. If this guy kicks up a fuss, I always feel like it won't end well."
"That won't happen. I understood the few sentences they exchanged when they first t. The old man wants Qiao Ze to go to Princeton, Qiao Ze said Xilin is good and invited him over. In the end, the two people kept going back and forth, just like in the previous video."
"Ah... I'm tired of this! In my opinion, this kind of one-sided communication should be directly canceled."
"Ahem, it was reported. Even though we don't understand, didn't they say at the eting? The achievents of theoretical mathematicians are for the benefit of all mankind..."
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