Non-Symplectic Flows: Excluded Dynamics in Symplectic Mechanics

[mma]

Symplectic mechanics deals with symplectic flows. I wonder, how general this description of the possible (or imaginable) dynamical phenomena is. If a given flow on a symplectic manifold turns to be non-symplectic, then perhaps we can find another symplectic form on the manifold that is invariant to the flow, that is, this flow can remain inside the relam of symplectic mechanics.
Are there flows on even dimensional orientable manifolds that don't have any invariant symplectic form? If yes, then what look they like? In other words what kind of dynamical phenomena are excluded from symplectic mchanics?

 

[zhentil]

I guess the clearest delineation would be behavior of fixed points. A clear example (though perhaps not what you're looking for) is that if we look at a smooth Lefschetz map on S^2, there will be infinitely many periodic points. This is not the case if you weaken the condition to homeomorphism, or even a C^1 diffeomorphism (there is one with exactly two periodic points).

I guess this question falls under the umbrella of Hamiltonian diffeomorphisms. One looks for invariants for any Hamiltonian diffeomorphism, so that one can get away from a fixed symplectic form.

 

[mma]

This sounds interesting, but I'm afraid that i don't really understand it. What is Lepschetz map? Is it that there is in the http://en.wikipedia.org/wiki/Lefschetz_manifold" artcle in Wikipedia?Then where is the flow we talk about?

 

[zhentil]

A lefschetz map is a smooth map from a manifold to itself with only non-degenerate fixed points (i.e. the Jacobian does not have 1 as an eigenvalue). Any Lefschetz map has only isolated fixed points. A symplectomorphism would be such a map, but there are others.

The interplay between diffeomorphisms and flows of vector fields on a compact manifold is very straightforward. If we have a vector field, it generates a one-parameter group of diffeomorphisms. Going the other way, if we can represent a given diffeomorphism as the time-one (say) realization of the flow of a vector field, you'd say that the diffeomorphism is isotopic to the identity.

 

[mma]

It begins to dawn on me... You mean something like the http://en.wikipedia.org/wiki/Symplectomorphism#Arnold_Conjecture". A non-symplectic flow could be a flow that violates this. I guess, it wouldn't be too easy to give a specific example for this.

 

[zhentil]

Precisely.

 

[simeonsen_bg]

Wouldn’t it be easier to take a flow on S^2 that is not area-preserving? So is for example a gradient flow with 2 fixed points - source and sink. Then it cannot be a symplectic one, as any symplectic flow is volume-preserving.

(http://homepages.cwi.nl/~jason/Classes/numwisk/ch16.pdf)