Action-angle variables on Dirac manifolds (2012)

 

18/Apr/2012:

Arxiv: http://arxiv.org/abs/1204.3865

I finally submitted this paper to the journal Geometry & Topology.

Corrections to the first version:

* One of the word “contravariant” should be changed to “covariant”

* The co-affine structures (that are induced from integrable systems on presymplectic manifolds) have been studied in the literature under the name”affine differential geometry”:

http://en.wikipedia.org/wiki/Affine_differential_geometry

I’ll have to add some appropriate references about affine differential geometry. Maybe this book:

Nomizu, K.; Sasaki, T. (1994), Affine Differential Geometry: Geometry of Affine Immersions, Cambridge University Press

For integrable systems on 3-manifolds this paper may be relevant:

Davis, D. (2006), Generic Affine Differential Geometry of Curves in Rn, Proc. Royal Soc. Edinburgh, 136A, 1195−1205.

17/Apr/2012:

The first version of this paper is now available: PDF File

Action-Angle variables on Dirac manifolds

Abstract: The main purpose of this paper is to show the existence of action-angle variables for integrable Hamiltonian systems on Dirac manifolds under some natural regularity and compactness conditions, using the torus action approach.  We show that the Liouville torus actions of general integrable dynamical systems hava the structure-preserving property with respect to any underlying geometric structure of the system, and deduce the existence of action-angle variables from this property. We also discover co-affine structures on manifolds as a by-product of our study of action-angle variables. 22 pages.

——–

Updated 16/April/2012

12:50PM 19 pages, the main theorem about the existence of A-A variables on Dirac manifolds  is now proved. Will add some final remarks, change the introduction, and that’s it.

Refs: Eisntein (1917), Bohr, Sommefeld, Epstein … (history about quantization of action variables)

1:00AM 1 day behind schedule already. 18 pages now,  still a few more to go. I’m not very satisfied with one of the main theorems in the paper, namely the one about the Hamiltonianity of the Liouville torus action. In that theorem I had to add a regularity condition, and I tried during 2 days to remove that condition or to produce a counterexample without success. That’s an excuse why i’m behind schedule, hehehe.

As always, the paper is more complicated than I first thought.

I’m not sure if I want to submit it to a journal right away after finishing the first version. Maybe I’ll put the first version on the web for a while before submitting it. If in the meantime I can improve it then I will.

The theorem of the paper about the fact that the Liouville torus action preserves any tensor field which is preserved by the system seems to be new and quite interesting by itself.

Updated 13/April/2012

13 pages so far. Proved another auxiliary theorem which says that the Liouville torus action preserves the Dirac structure if the vector fields in a integrable systems preserve the Dirac structure. This auxiliary result looks interesting by itself.

Updated 12/April/2012

Have written 10 pages so far. Finished the section about Dirac manifolds and (co-)Lagrangian submanifolds. Wish I could do faster.

still don’t know what are the precise conditions that I want to impose on an integrable system on a dirac manifold/

Updated 11/April/2012:

Where can I send this paper ?

Some choices:

* J. Symplectic Geometry (A)

* Archive for Rational Mechanics and Analysis (A*)

* Annales de Fourier ? (A)

* Crelle’s ? (A*)

* Commun. Math. Phys. ? (A*)

* J Math Soc Japan ? (A)

* Transactions AMS ? (A*) (if > 15 pages) (editor for diff geom & global analysis: Ch. Woodward). OK, write 18 pages, send to Chris, wait 2+ years for the paper to appear. It will be counted as a publication in 2014 or 2015, hehehe.

Have written about half of the paper so far. I wrote a section about generalities on Dirac structures and Hamiltonian systems on them.

It seems that some notions that I need don’t exist explicitly in the literature, so I have to introduce them, e.g. the notion of a co-Lagrangian submanifold. I also wrote down a normal form theorem for the tubular neighborhood of a co-Lagrangian submanifold, similar to Weinstein’s theorem about Lagrangian submanifolds in symplectic manifolds.

This paper will probably be 15-20 pages long. It’s not a difficult paper. Things are more or less straightforward. Just need time to write them up carefully.

Soma additional refs:

Bursztyn, Kosmann, Gelfand-Dorfman (Dirac structures)

Liouville, Mineur (classical papers)

09/April/2012 I’m now writing a small paper on the action-angle variables on Dirac manifolds.

There have been verious generalizations of the action-angle variables theorem to the Poisson case and to the contact case. I had not been very much interested in them. But now I need a version of this action-angle varibales  theorem for Dirac manifolds, in order to do the Hamiltonianization of non-Hamiltonian integrable systems. This result will be needed, in particular, for my paper on the geometry of integrable dynamical systems on 3-manifolds, so I will postpone my paper on 3-manifolds until I finish this small paper.

This week I have no teaching, so hopefully will be able to write up this paper rather quickly. First version expected on 15/April/2012.

The main point, as was already emphasized in my previous papers on this subject (Arnold-Liouville with singularities; torus actions and automorphism groups of integrable systems, etc.), is the existence of a Hamiltonian torus action. It is this Hamiltonian action which gives rise to action-angle variables. The notion of  Hamiltonian torus action can be easily adapted to the case of Dirac manifolds.

Some references:

* Classical: Liouville, Mineur, Arnold, Jost, Weinstein (quantization 1917)

* Non-commutative: Nekhoroshev, Fomenko – Mischenko, Fasso

* Almost-symplectic: Fasso – Sansonetto

* Pre-symplectic: Karshon (related) ?

* Poisson: Laurent-G, Miranda, Vanhaecke, Fernandes, Caseiro (?),

* Contact: Jovanovic, Khesin – Tabachnikov, E. Lerman ?

* Torus action characterization: my papers

* Dirac structures: Courant, Dufour-Wade, our Poisson book (Dufour-Z)

* With singularities ? (I don’t know yet if I want to discuss singularities in this paper)

* Additional refs: …

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