# 2010 MAT495H1 Group: Hamiltonian Systems

This list is a summary of topics on Hamiltonian Systems described during class on May 18. Group members are expected to contribute to developing this page. Further instructions will be provided by Prof. Colliander in the "discussion" section (near the top of the page).

# Topics

After meeting on May 19, we have divided some of the topics as follows. Group members should feel free to adjust these assignments.

Mustazee Rahman has written a Proof of Liouville`s Theorem (phase space volume is preserved under the flow of an autonomous ODE with a divergence free vector field) and a discussion of the Poincaré Recurrence theorem. From here, perhaps we could interpret some nonlinear PDE as infinite-dimensional Hamiltonian systems, and discuss the existence of invariant measures? Geordie Richards plans to prepare some digestible notes (and a talk) on this subject.

This looks great. I would also like to learn more about the Poincaré Recurrence phenomenon. In particular, I would like to learn to better quantify the recurrence times. It might also be interesting to look at this in the setting of the FPU problem. For some discussion, go here. As far as I know, the ergodicity hypothesis issue remains a central question in statistical physics and might relate to mixing questions in the PDE setting? I also wonder if one can see advantages of the Gibbs measure over the Lebesgue measure in the large but finite dimensional setting. Hamiltonian dynamics in large but finite dimension is also closely related to quantum many body questions currently being studied by Ioannis and Peter.

John Collett is planning to study Birkhoff normal form, as introduced by Alessandro Selvitella in class on May 20. Some material from this talk can be found in the lecture notes of D. Bambusi. For those of us unfamiliar with Birkhoff normal form, perhaps we would best understand these results by considering some simple examples?

This sounds excellent. In fact, Alessandro, Hiro and I are looking into aspects of the I-method and normal forms by looking at Bourgain's paper on that topic so this is indeed a research direction. As a warm up exercise, I suggest we look at the problem of N oscillators (for example a spring mass system with hook's law springs). I believe the key insight there is that the associated Hamilton's equation is a linear system spawned from purely quadratic terms in the Hamiltonian. A change of variables allows one to diagonalize this matrix so that, in the new coordinates, we observe a system of decoupled oscillators. This is essentially the process of converting to normal modes in a many oscillator system. Now, go back to the original system and tweak the restoring forces to include cubic terms (as appearing in the Taylor expansion of the gravitational $\sin \theta$) term in the pendulum restoring force. Since the resulting system is no longer linear, we can't proceed directly to the new coordinates. This path follows closely the FPU stuff. For numerical experiments, I suggest we try to simulate a system of two coupled oscillators and explore the dynamics in phase space as we tweak the restoring forces. This might be a good project to develop in collaboration with Ekaterina. Colliand 15:37, 20 May 2010 (UTC)

Ekaterina Kluyeva will prepare some code (Matlab, maybe another programming language?) to simulate the 2-d Harmonic oscillator. From here we could perturb the Hamiltonian with cubic corrections, and witness (for small perturbations) the kind of dynamical stability which is established with Birkhoff normal form.

See discussion above. Colliand 15:37, 20 May 2010 (UTC)

Ioannis Anapolitanos and Steven Corkey have both expressed an interest in studying completely integrable systems. In particular, it would be nice to begin with a proof of the Darboux-Arnold Theorem (called the Liouville-Arnold Theorem, or just the Darboux Theorem in some places?) on the existence of action-angle coordinates for a 2N-dimensional Hamiltonian system in the presence of N conserved quantities. It would also be helpful to learn more about completely integrable Hamiltonian PDE. In particular, an exposition of the inverse scattering method (perhaps just in the context of KdV) would be very nice. Lecture notes on this subject by Ablowitz and Clarkson can be found here.

I like the idea to develop a nice annotated proof of the Darboux theorem. A nice presentation appears in the textbook by Arnold (where it is indeed called Darboux's theorem). Along the way, please try to clarify the contributions of Darboux and Arnold so that we make proper attributions. Before going to infinite dimensions, I suggest we linger in finite dimensions and consider what happens when we bump the action-angle dynamics by a small Hamiltonian perturbation. This is the doorway to the Birkhoff normal form, Nekoroshev stability, and KAM theory. Towards infinite dimensions, it might be useful to look at the book by Kuksin and Poschel entitled KdV and KAM.Colliand 15:37, 20 May 2010 (UTC)