Friday, June 17, 2011

Integrability and Chaos in Hamiltonian Systems

   This last semester, I took a course intitled Nonlinear Dynamics and Chaos (in  French). It was quite a novelty, as undergraduates in physics mostly deal with linear problems (they are much easier to solve analytically. Most nonlinear systems do not possess such solutions and they do, the analytic expression is so complex that it offers no qualitative understanding of the subject.

    A dynamicist must make use of a completely different set of tools, mostly numeric, to try to grasp the inner workings of a particular system. Phase space, which shows the relationship between a variable and its nth order derivatives (where n is the dynamical order of the system, if I can use this vocabulary), is an important example of these tools.

   In the special context of Hamiltonian mechanics, where momenta and position are related through what is called a symplectic symmetry, the topology of the phase space trajectories are significantly restricted; they are N-tori. That is, as long as the system is integrable (has a closed form solution).

   As soon as a perturbation is added that renders the system non-integrable, then the phase space qualitatively evolves as described by what is called the KAM Theorem.

This last theorem was the main topic of a mini research project that was conducted by Raphaël Dubé-Demers and myself throughout the second half of the semester. Following these last few words are the text produced during this research and the Beamer presentation used to introduce the subject to our peers.

Happy reading!

[Presentation]  (in French)
[Text]  (in French)

To anyone that will ask nicely, I could translate it in English. Asked very nicely!

UPDATE: I made a flipbook out of it. It looks pretty cool, but that's pretty much it!
UPDATE 2: Here's the link to the complete thing.




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