Differential Equation Systems and Nature

[fresh_42]

Abstract
“Mathematics is the native language of nature.” is a phrase that is often used when it comes to explaining why mathematics is all around in natural sciences, especially in physics. What does that mean? A closer look shows us that it primarily means that we describe nature by differential equations, a lot of differential equations. There are so many that it would take an entire encyclopedia to gather all of them in one book. The following article is intended to take the reader through this maze along with examples, many pictures, a little bit of history, and the theorem about the existence and uniqueness of solutions: the theorem of Picard-Lindelöf.
A Glimpse of Our Descriptions of the World
It all began with Isaac Newton in 1687, or if we are picky, with Gottfried Leibniz during his stay in Paris 1672-1676 [34]. The reference to Paris is more than a side note since it will be especially French mathematicians who will develop the field of analysis in the following centuries...

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[jedishrfu]

Great article @fresh_42 !

I especially like the fact that you've included the D'Alembertian operator. I remember first seeing it in college and then again in a sci-fi movie (don't remember the name) and thought wow this movie got a real physicist or mathematician assisting the production. I think they squared it too ala the Laplacian.

The Navier/Stokes term labeling was pretty cool too. A theme of the Gifted movie was a side thesis around the Navier Stokes millenium prize.

What software did you use to generate the 3D plots? Matlab / Julia / Python plots?

Jedi

 

[fresh_42]

What software did you use to generate the 3D plots? Matlab / Julia / Python plots?

Jedi

No, sorry. There is only one proper 3D picture which I took from a lecture note. The pictures I made myself used graph

and mspaint, and the linearization of the predator-prey model used the Lotka-Volterra calculator that can be found in
https://www.physicsforums.com/threa...h-physics-earth-and-other-curiosities.970262/
I tried to use a German math graphic program but I got stuck, and even WA didn't produce nice pictures, so I returned to my "keep it simple" approach. I calculated the vectors for one flow and added them into the program as arrows, made a screenshot, and copied it into mspaint. Then I zoomed out, made another screenshot, and so on.

All pictures that I did not produce myself, e.g. the graphic about Moore's law, are referenced in the chapter "Sources".

 

[jedishrfu]

Kudos to you! You did a lot of work for those charts.

Thanks for sharing.

Have you thought of writing a popular or college math book?

 

[fresh_42]

Have you thought of writing a popular or college math book?

I like the insight format. You can tell in a couple of pages the basic ideas and facts without having to write a novel. There is an (?) insight article I wrote because some kids have asked me about an overview of "differentiation". It resulted in five parts!
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/
And it is still only an overview. I like to quote the 10-point list here:
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/
where I gathered some perspectives on a derivative. And I didn't even use the word slope.

I uploaded the 500 pages of my solutions to the math challenge problems here and I think there are a lot of typos and certainly quite some mistakes in it. This would be rather boring to proofread if it were a book. And the title "Differential Equations and Nature" could easily be a book, but were to stop? I have a book (~1000 pages) from Jean Dieudonné about the history of mathematics between 1700 and 1900 (roughly). I like it very much and appreciate the work he has done to write it. I wouldn't have had the patience.

If I ever write a book it would be: "Is it hard or are we stupid?" I'm fascinated by the fact that we cannot decide NP<>P, and that there seems to be a gap between dimensions 2 (easy) and 3 (impossible), e.g. the complexity of matrix multiplication. We can determine the rank of a matrix (1-1-tensor) in linear time but fail to do the same for bilinear functions (2-1-tensors). Why? Fermat's theorem for n=3 had been solved early, arbitrary n took 350 years. Nevertheless, we still do not have a proof that NP is difficult. Same with Navier-Stokes. Is it hard or are we stupid? And if it is hard, why can't we prove it is hard?

 

[jedishrfu]

Personally, I'm fascinated by the simplicity of the Collatz Conjecture and why we can't find a proof.

I'm also amazed at how well math describes the universe we observe. You article is great in its survey of differential equations. I recall taking a couple of courses, always amazed at the solution strategies used. Some made sense with deeper understanding of Calculus but others just mystified me.

 

[jedishrfu]

You could just collect your articles together into a book of insights and self-publish. One book I liked was the Math 1001 by Elwes, its a collection of many math topics with enough written to pique your interest.

 

[fresh_42]

I'm also amazed at how well math describes the universe we observe. You article is great in its survey of differential equations. I recall taking a couple of courses, always amazed at the solution strategies used. Some made sense with deeper understanding of Calculus but others just mystified me.

I remembered that I had read an article about timber management in some Asian country, Indonesia, Vietnam, or somewhere there. Unfortunately, I don't remember the country so I couldn't find it again. Instead, I found a dissertation about regional timber management and I was totally amazed and intrigued by the sheer size of the system: 221 coupled non-linear integral and differential equations, 182 parameter functions, and 371 single parameters! Just wow!

I was also amazed by the beauty and simplicity of $F\sim \ddot x$ and what it already implicates without any other equation.

 

[jedishrfu]

Was one of these the forestry article you mentioned?

https://www.researchgate.net/public...al_Equations_in_Sustainable_Forest_Harvesting

https://www.sciencedirect.com/science/article/abs/pii/0377221786902122

 

[fresh_42]

Was one of these the forestry article you mentioned?

https://www.researchgate.net/public...al_Equations_in_Sustainable_Forest_Harvesting

https://www.sciencedirect.com/science/article/abs/pii/0377221786902122

No. Wrong continent and IIRC also wrong language, however, yes, along these lines. Seems timber management and forestation is a much better example for differential equations than Lotka-Volterra. But a horror to draw, I guess. The paper I quoted in the article was a dissertation in Switzerland based on 200 years of data! I just checked, and CC was apparently not of as much interest in 1998 as it is today. At least the word didn't appear in the thesis.

Btw., it is the second time I repaired the first post. Do you know whether it is recreated when I update the article? I corrected a typo.

 

[WHT]

"linearization of the predator-prey model used the Lotka-Volterra calculator"

In certain cases differential equations may be overrated in explaining the predator-prey dynamics. I've been conversing with an academic ecologist who was able to explain the famous fox-lemming cycles of 3.8 years by invoking a synchronization of lunar cycles with an annual cycle (spring tides within 5 days of the vernal equinox). No way to really validate this given the fragility of ecosystems nowadays.

 

[ebg]

Continue reading...

If science is the observation of nature, then math is the explaination of those observations; can we always be sure of our observations and explainations? The need to be inquisitive is the answer.

 

[Drakkith]

If science is the observation of nature, then math is the explaination of those observations; can we always be sure of our observations and explainations? The need to be inquisitive is the answer.

... and being inquisitive is literally what scientists do.

 

[PAllen]

If you are going to reference the priority dispute early in your document (no need to reference it at all), are you aware of:

https://pages.cs.wisc.edu/~sastry/hs323/calculus.pdf

 

[fresh_42]

If you are going to reference the priority dispute early in your document (no need to reference it at all), are you aware of:

https://pages.cs.wisc.edu/~sastry/hs323/calculus.pdf

Thank you, I added the reference and a reference to the reference. You were right, it wasn't specifically important, but - to be honest - I am a fan of my list of sources and every additional source that is meaningful is highly welcome. Those articles are overviews by their nature, so the list of sources is not only a good scientific habit people should get used to early, but also helpful in the rare case that someone is inspired by such an article and wants to dig deeper. I can say for myself that my juvenile curiosity in STEM fields (and the history of them) was mainly triggered by reading books and articles like the ones we have in the insight department and I was always looking for more.