Geometry of nonsingular integrable systems on 3-manifolds (2012)

 

28/Apr/2012

This 3D project is not prioritary, so I’ll postpone it until I’m out of other ideas :-)

Need to work on something more exciting, or I’ll feel bored and tired.

Right now maybe I’ll write up some stuff about stability of commuting singular foliations (extenstion of Reeb-Thurston-…-Crainic-Fernandes-…-Scardua-Seade-… to commuting foliations) ?

26/Apr/2012:

I had to postpone this paper to finish the other two papers first (the one one smooth linearization of completely integrable vector fields, and the one on action-angle variables). The results of both of them will be used in this one, namely:

* Singularities in of systems of type (1,1) will become singularities of systems of type (2,1) with one regular vector field after a reduction

* Action-angle variables on presymplectic manifold –> problem of Hamiltonianization by a compatible presymplectic or Dirac structure.

* Quasi-convex co-affine structures and compatible copntact structures ?!

New deadline for this paper: 08/Mai ?

This semester I worked as a mercenary, teaching a class that normally I didn’t have to teach. Hopefully they will pay me correctly for that. But the teaching finishes this week, so no more teaching in May, and I’ll have more time to reorganize a bit. (Too many things are in disorder now), and to finish some old stuff like this one quickly.

08/Aprl/2012: This paper in progress is part of my program of a systematic study of the geometry and topology of integrable non-Hamiltonian system.

Start date: 08/Apr/2012; expected finish date: 20/Apr/2012 (I will not be a long paper, so I’ll give it just 12 days).

This paper is about nonsingular systems (i.e. vector fields which are non-zero everywhere) on 3-manifolds which are integrable in the Hamiltonian sense.

Things to be discussed:

* Picewise T1-action and reduction to the 2D case

* Generic singularities (not of the vector field, but of the whole system which is a triple)

* Hamiltonian (isoenergy) and non-Hamiltonian examples (e.g. steady inviscid hydrodynamic flows)

* Relation with  nonsingular Morse-Smale systems

* A bit of topology: graph manifolds, knots and links of special orbits.

* Bifurcation theory of integrable systems (in 1-parameter families) ?

* Local, semi-local and global invariants

* Sytems will be studied up to: orbital equivalence, geometric (fibration) equivalence.

* Compatible contact structures ? Twisted/nontwisted ?

* etc.

References:

- Bolsinov, Fomenko, Matveev, Sharko, …

- Etnyre, Ghryst, …

- Morgan, Franks, …

- Wada, Yano, …

- Campos, Cordero, Martinez, Vindel, Nunes, Casasayas, Kidambi, Newton, …

- Eliashberg, Hofer, … ?

There is a 20-year old unpublished preprint of mine on compatible contact structures, which I will include in this paper.

What journal to send the paper to ?

Some possible choices (depending on how the finished paper will look like):

* Discrete and Continuous Dynamical Systems – Series A. (A)

* Nonlinearity (A*)

* Journal of Nonlinear Science. (A)

* Annali di Mat Pura & Applicata (A)

* Annali di Pisa

* J Dynamics Diff Eqns (A)

* ZAMP (A)

* Archive for Rational Mechanics and Analysis (A*)

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