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:
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tbPobklyTwxyk2iafT-NFIiBlUytLSronUsHOIkSOxaqJsVuiW3IjTTbZ79tq5PLOv42VfD3OnycJOXYpYmDzM3QhQZfCF2Hyc4P_NeOEEjKQIzD9JcTtR7Ecjj0FJcCpldRQhW-8sxotoSB6S-qa-fQ=s0-d)
where
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v6vgt5WazIEoHNv6y12epp_kGfYgMo8rWELGEgYM9UwYKXBLNUkihDBRhx4UxAKr3AVkywh2WkORWX8XNeEYMWTbL_M5-5_yTDkFE-9PJf4blqlw=s0-d)
is the operator saying to take the Poisson bracket with its argument.
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uLslC3SZtY_EaTzRYWNYwhu-JrKx8aGqzNCRCGc4Ns1ZwqpHm9k5Q5NeTgvlSLCTx-25dN5_V0P2b3g5FZR6CQkZWmmAcY_Sjex8nYJyUS9Dg=s0-d)
is the effective Hamiltonian and is a function of
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v0gy-QF5lDfVpfW6HT-qhx9CYTn260i-t6D7WbIZScHqqVV0YITeonP-C2i-h85lzXiBm61KYQozdrnVlKXG5UlE_qWZErWpVTTHxrYPOhkg=s0-d)
and
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vp-oKwdq08gqqGO_jkjJS6aOIKNuXPgRpQVO0tAI7fJRcJ61RlUYZaDQ_GTLlqARSo5-kRZhbKMpbEpEvrTWhYQh9LS7dGIFOZv0UAldUdHLA=s0-d)
. In the linear case,
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sd7cyaQpVZ7UMhkl7M4kYIe_Tel6XAKJ8CPHpZ7OGTmrXrmymSqSbEQGZEtaBUsSqFpFbywM-QlL2Pl9mUeFWkraHChaA2fITZdxe1LIozwmgPbLAj0bn-jq0zqAOFGzU0=s0-d)
is just a matrix
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_utXF-YxnXUX907MmcitnHmGAlDftfWFBWIXYKoGv8cl5hugdj07czs0Hry_hclDSCZVga91zXq7RZif6sxPIBNZa_TZ7eBIEaMv9dR0WWTNEZ54EFEHLD2bVNpuF_6Fvg=s0-d)
and if the Hamiltonian is
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vXvSmm9ClkrNhR94KB8W_jXZB4awcgfeuhpjjudFbByY1SYE9IhWBwlNCCOM2CKhrqO7sIZvV-tKEoMls1PCOtLwhN0DSu2FoHebTb39BeBGDunU8oGxdUlHdiE6nJ410uXI7s=s0-d)
with
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t7TXxe69GHNUsXwjRMiz1YGpe6fXA-mpYPsu4xHpd1bKBX5TbRXaxhKqBPI-J6fda_BQ7_iC5bOxCbAHvv1BLkj2DBf490M535Vr1vRRBwL1SufVmaWdHjb2bW-2mT3-_q3oO2FVb-bUQwoUNjLRk=s0-d)
, then
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tQtqI0JFeDoUN8noeLPhE9U01GR3PEBGk2krZ3lhUOMhJOXELopYzofAaJ1eMw9eNio3yN_7cI4sw-ep5rNrDlfkjjUxmdB37u057Bdq7UB_2h8_J3o-dpOGu5gP2Ilr40pnFSLt-UzNO75JG_gIfb=s0-d)
where
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vBUyFlpdlEAjrB_m1FD7qd1wbdoON_2mHP4fbgfmRuytLwBpM39NZD0iS4QJlLvGoFJ87NUTTMsa3W3A8xa1kSI68S2bINkQElvrAz55lwM6C090jRr6ggdBvVIWVsWovCXuFBCOiN3Fj-O018lBlRhWkqpi8h617Bdtodk0sbEY0=s0-d)
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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