Last updated: 01/Apr/2011

This is a research project on which a PhD student of mine is working with me. Please don’t steal the ideas and results that I discuss here.

The problem is to study the topology and geometry of proper non-Hamiltonian integrable dynamical systems on manifold. A non-Hamiltonian integrable system consists of:

* p commuting vector fields X1, …, Xp (p \geq 1)

* q common first integrals F1,…,Fq (q \geq 0)

such that p+q = m is the dimension of the manifold

(in the Hamiltonian case on a symplectic manifold, one has p=q=n where n is half the dimension of the manifold)

Natural examples of non-Hamiltonian systems include systems with non-holonomic constraints, or non-conservative systems (energy-losing), etc.

For Hamiltonian systems, we already know a lot about their topological / geometric invariants.

We want to obtain similar results for non-Hamiltonian systems. In particular, we ant to extend the known structural theorems (for example, of nondegenerate singularities) to the non-Hamiltonian case.

## The (1,1) case:

The geometry of non-Hamiltonian system is very rich, already in the 2-dimensional case of type (1,1) (i.e. 1 vector field and 1 function):

In the Hamiltonian case, there are only two generic (non-degenerate) singularities: elliptic (x^2 + y^2) and hyperbolic (x^2 – y^2)

In the non-Hamiltonian case,the classification up to essential isomorphisms of generic singularities is much more complicated:

* Isolated singularities: they are similar to the Hamiltonian case, but there is an infite family of them now, instead of just two (elliptic & hyperbolic):

1) elliptic: F =x^2 + y^2, X = X_F (like the hamiltonian case)

2) hyperbolic: also like the Hamiltnian case, but the “Hamiltonian” function can be F = x^k y^h (instead of just F=xy)

and the vector field is X = hx d/dx – k y d/dy

For local problems, k and h are coprime. But when we consider the global picture (perturbations of systems on a global surface) then even the case when k and h are not coprime can be rigid !

* Curves of non-isolated singularities:

3)The F= constant curve has multipilicty:

X = d/dx regular, F = y^k + constant

In local problems, then only the case k=2 is rigid, the case k>=3 can be perturbed into lower multiplicities. But in global problems, even the case k > 2 can be rigid.

4) X = +- x d/dx, F = y (X changing sign)

5) X = +-(x^2 – y) d/dx, F=y (turning of the curve X=0)

6) X = +- x d/dx, F = y^k + constant (X changing sign & F folding simultaneously)

**Definition: **Two singularities (X,F) and (X’,F’) are called essentially equivalent, if, after a diffeomorphism, we can write:

X = gX’ where g > 0 is a smooth function, and F = h(F’) where h is a invertible function

**Theorem:** i) The above singularities are rigid, i.e. any small perturbation will yield the same type of singularity up to essential isomorphisms.

ii) Any (1,1) system can be perturbed (by an arbitrarily small perturbation) into a (1,1) system which admits only above 6 types of singularities.

Remark: The above 6 types of singularities play the role analogous to Morse singularities for functions. Singularities of type 3) are actually Bott singularities. Bott singularities (of fist integrals) are a common feature of integrable system, due to commuting vector fields.

The set of singular points in a generic (1,1) system can be divided into 3 subsets:

a) isolated singular points, where both X and F are singular (type 1 and type 2)

b) smooth (closed) curves where dF=0. This curve may have a finite number of points where X=0 (points of type 6), the remaining points are of type 3.

c) smooth (closed curve) where X = 0. This curve have have nondegenerate (quadratic) tangencies with a non-singular level of F (points of type 5), may intersect transversally curves of dF=0 (points of type 6); the other points are of type 4.

Remark: It may happen that folds of F cannot be avoided (by changing the first integral but keeping the vector field unchanged), it the surface is non-orientable. If the surface is orientable, then one can change the first integral to kill the folds (thus killing singularities of type 3 and type 6)

Question: how to define invariants, e.g. rank and corank, of the above 6 types of singularities ?

Maybe it’s more appropriate to talk about bi-corank ?

bicorank (1,1): types 1,2, 6

bicorank (0,1): type 3

bicorank (1,0): type 4, 5

Note: type 5 is generic but “degenerate” ?! (degenerate among points of bicorank (1,0) )

## The (2,0) case:

Two commuting vector fields on a compact surface.

* If there is no singular point, then the surface is a torus, with constant vector fields on it. Up to essential isomorphisms, this case is rigid.

* corank-1 case: X1 non-singular, X2 singular: then X2 remains singular on the orbit by X1 through the singular point, and we have a singular curve.

nondegenerate corank-1 case essentially isomorphic to: X1 = d/dx; X2= y d/dy

* corank-2 case: both X1 and X2 are singular.

Definition of essential isomorphisms: (X1,X2) is essentially isomorphic to (X1′,X2′) if one can turn (X1,X2) into (X1′,X2′) by a combination of the following operations:

– Diffeos

– change of linear base: X1′ = aX1 + bX2, X2′ = cX1 + dX2

nondegenerate corank-2 case ? Use results on intrinsic torus actions -> can arrange so that X1 = x d/dx and X2 = y d/dy (in the analytic case).

Question: is it possible to have an integrable (2,0) system on a hyperbolic surface (genus greater than 1) ?

(to be continued)

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