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Monday, August 31, 2009

moving and the symplectic camel

Yeah, yeah, more about this tedious moving process, whereby I reduce my digital, physical and mental footprint, so that I can cross the sea and find peace in a new land.
A recent paper someone forwarded to me:
"Symplectic capacities and the geometry of uncertainty: The irruption of symplectic topology in classical and quantum mechanics"
One of the main points is the story of the symplectic camel, that sad creature who could not pass through the eye of the needle. It is in fact a relatively recent rigorous result (Gromov's non-squeezing theorem) about symplectic geometry.
Basically it says that whatever your initial projection onto the different phase planes, a canonical transformation will never decrease, only increase them.
So, yes, I suppose this means that I should get rid of stuff. The connection is obvious, right?

On a different, not really related subject, I just finished reading Alan Watts' "The Book on the Taboo Against Knowing Who You Are".
Some parts really resonated with me, though at times it put me in that kind of trance where he could say just about anything, and I'd nod slowly with a blank stare. No, actually, it did that remarkably little. The image of self as the universe playing hide and seek with itself is an interesting one.

Tuesday, August 25, 2009

frequency amplitude dependence

So, consider a dynamical system that has an elliptical fixed point. The eigenvalues of the map linearized about the fixed point gives the frequency associated with rotation about that fixed point. Move away from the fixed point and the frequency changes. This is one definition of a non-linear system. A pendulum, for example, has this property; and this changing frequency explains why it doesn't make a perfect clock.

This simple property is what I am trying to get to the bottom of. It becomes less simple when one wants to think of this continuum of frequencies as the spectrum of some operator. So the operator has become rather more complicated from the simple matrix expanded about the fixed point. Suddenly it has taken on an infinity of eigenvalues! Once we say this, however, we are left with the question of what the eigenvectors mean. Are they functions that are basically delta functions except where the orbit is? This is a rather messy singular quantity. How about the eigenvectors of the adjoint operator? These are the quantities we've been looking at, but they seem to be rather strange.
This is the line thinking I've been following lately. I'm hoping I can find a simple reference that explains all this. But I can't seem to.