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) ?
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 ?
- 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*)