Mathematica and KAM tori reforming

Some days ago I received an email from Wolfram Research asking to me to produce a demonstration for their demonstrations project based on my last proved theorem about KAM tori reforming (see here). WolframDemosBeing aware of the power of Mathematica I have found the invitation quite stimulating.  The idea behind these demonstrations is to use Mathematica’s command Manipulate that permits to have interactive presentations. A typical application is exactly in the area of differential equations where you can have some varying parameters. But the possibilities are huge for this method and, indeed, you can find almost 5000 demonstrations at that site. Indeed, in this way you are able to explore the behavior of mathematical models interactively and this appears as a really helpful tool. If you mean to send a demo of yours, be advised that it will undergo peer-review. So, such a publication has exactly the same value of other academic titles.

In a few days I prepared the demo and I have sent it to Wolfram. It was accepted for publication last friday. You can find it here. You can check by yourself the truthfulness of my theorem. The advantage to work in classical mechanics is that you can have immediately an idea of what is going on by numerics. Manipulate of Mathematica is a powerful tool in your hands to accomplish such an aim.

Finally, you can download the source code and modify it by yourself changing ranges, equations and so on. I tried the original Duffing oscillator without dissipation and, granted the validity of KAM theorem, one can verify an identical behavior with tori reforming for a very large perturbation. A shocking evidence without experiments, isn’t it?

Update: My demonstration has been updated (see here). The code has been improved, and so the presentation, due to a PhD student, Simon Tyler, that did this work. Thank you very much, Simon!

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7 Responses to Mathematica and KAM tori reforming

  1. carlbrannen says:

    If I could afford Mathematica, I’d either contribute one demonstrating my solution for the exact Newtonian equations of motion for Schwarzschild test bodies, (for which I got an “honorable mention” at the annual gravity essay contest), or possibly contribute a Schwarzschild / Gullstrand-Painleve multiple test body simulation based on the equations, though Java does a pretty good job.

    Instead of Mathematica, I use the free tool, MAXIMA, which does symbolic manipulation but I’ve not tried to graph with it.

    By the way, the philosophy behind that gravitation paper is something you would appreciate: “an exact non linear solution is much more useful than any number of linear approximations”.

    Meanwhile, PRD kicked out my unitary matrix parameterization paper with Marni Sheppeard, while IJMPD is still grinding on my gravity paper.

  2. mfrasca says:

    Hi Carl,

    Do not worry about rejected papers, there are a lot these days out there.

    Best compliments for your publication. It is always a great success for an outsider to get through scientific publishing.

    Marco

  3. carlbrannen says:

    Marco,

    I’d don’t have a publication yet, it’s still in peer review. They publish maybe 2/3 of the honorable mentions in the December IJMPD. My guess is that they’ll make me rip out the part where I call GR science fiction and the part where I propose a new theory of gravitons, and maybe the parts where I reference the Cambridge geometry group’s gravity theory. However, I did get endorsement to load the paper on arXiv and so I will do this this week, after I’ve expanded the bibliography.

    The whole thing reminds me of how much I enjoy doing physics, how boring it is to write papers, how nervous giving presentations makes me, and how horribly painful it is to get things published. I doubt I’d have written up either paper but I was hoping to win cash with the gravity contest and my post-doc Marni needs to get some papers out. (Uh, our relationship is that she’s my post-doc and I’m her grad student, possibly depending on who you ask.)

  4. Simon says:

    Hi Marco,

    Nice demonstration…
    the only problem is that when you switch from “time series” to “phase space” it has to recalculate the NDSolve (even though nothing has changed in the system). This is a general problem when using Manipulate — it’s hard to control the evaluation or expressions.
    I’ve hacked out a quick fix for this and emailed it to you.

    Simon

  5. mfrasca says:

    Hi Simon,

    Thank you very much for your effort. I have received your email with the notebook. Please, give me a few time to check it.

    Marco

  6. […] This paper was the subject for a demonstration with Mathematica (see here) for the Wolfram Demonstration Project. You can find a post of mine about this here. […]

  7. […] This paper was the subject for a demonstration with Mathematica (see here) for the Wolfram Demonstration Project. You can find a post of mine about this here. […]

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