Ah, the joys of math. So now I can ramble away with equations! (
Thanks to)
So, we can represent the symplectic part of the dynamics of an accelerator by a one turn map:
![](http://www.forkosh.dreamhost.com/mimetex.cgi?%5Cmathcal%7BM%7D%20=%20e%5E%7B:H:%7D)
where
![](http://www.forkosh.dreamhost.com/mimetex.cgi?:H:)
is the operator saying to take the Poisson bracket with its argument.
![](http://www.forkosh.dreamhost.com/mimetex.cgi?H)
is the effective Hamiltonian and is a function of
![](http://www.forkosh.dreamhost.com/mimetex.cgi?x)
and
![](http://www.forkosh.dreamhost.com/mimetex.cgi?p)
. In the linear case,
![](http://www.forkosh.dreamhost.com/mimetex.cgi?%5Cmathcal%7BM%7D)
is just a matrix
![](http://www.forkosh.dreamhost.com/mimetex.cgi?%5Cmathbb%7BM%7D)
and if the Hamiltonian is
![](http://www.forkosh.dreamhost.com/mimetex.cgi?H%20=z_i%20z_j%20S_%7Bij%7D)
with
![](http://www.forkosh.dreamhost.com/mimetex.cgi?%5Cvec%20z%20=%20%28%5Cvec%20x,%20%5Cvec%20p%29)
, then
![](http://www.forkosh.dreamhost.com/mimetex.cgi?%20%5Cmathbb%7BM%7D%20=%20e%5E%7BJS%7D)
where
![](http://www.forkosh.dreamhost.com/mimetex.cgi?J%20=%20%5Cleft%28%5Cmatrix%7B0&1%5Ccr%20-1&0%7D%5Cright%29)
is the symplectic inner product matrix and its multidimensional
extensions. Actually its kind of ugly. Ah well, I may or may not write more equations.
3 comments:
@#$$%@!??? - aka: I don't get it.
I never was very good at Math. I worked really hard on it in high school. School, private lessons, self-taught...
Now when I think of math I think of money. Ha - ha.
Yeah, a bunch of mumbo jumbo.
Funny, cause I'm the exact opposite. For all the math I've looked at, I seem to have a major block when it comes to money. Just doesn't compute.
I only like symplectic equations because it helps my little enzymes to conserve energy in a 160 nanosecond simulation. But sometimes, for reasons that I'm trying to track down now, the darn symplectic equation fails me!!!!! Whyyyyyy!
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