Tuesday, December 19, 2006

adiabatic invariants

Research, research, research. Trying to find the right definition of action.

So, what's an adiabatic invariant?
Its something that stays constant as the environment changes.
For my undergrad thesis, I tried to figure out what happens to simple systems as they are transported from here to there. So this research provided an answer to the question: which part of Boaz is unchanged by his weekly transitions from A to B, from mom to dad. Adiabatic invariants. Just find them and I will know what to count on.

Ok, still working on these metaphors. Does it say something about isolation and self-containment that I focus on the single particle case? Does it take away the fun to give away the punchline... that I'm trying to find my own adiabatic invariants? When does one transcend the personal and create something of interest to others? Perhaps this dynamic is healthy? Continually reinterpretting in terms of trenchant metaphors?

Let me give a concrete example of an adiabatic invariant: the classic example is a pendulum which has a string the length of which is slowly shortening. As the environment changes so does the motion. The frequency of the pendulum increases as the length decreases (f~sqrt(g/l)). But the energy of the pendulum divided by the frequency stays constant. This is an adiabatic invariant.

Are there fluid dynamicists out there that think of themselves as scale models of the world and wonder what their true Reynolds number is? Are there quantum field theorists who renormalize away their existential concerns, find that the infinite difference between self and God can be swept under the rug if one cares only about practical solutions? What about economists who see all interactions as a game and try to make the market of human kindness as free as possible?

But I will be an adiabatic invariant. I will search for that which remains the same in me and allows for equilibrium. I can only be so many things at the same time. A 2n dimensional integrable Hamiltonian system has only n adiabatic invariants.
Right. right.

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