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_s7WnCyMqmW3PPHwptcLacp0e8yGEiViP2eT3R6TS2x-u9dy2HzLtj52l1SUU80FnJAw3bUbR1OnMt6HYo54yrqZOC1MrSGYMFlY_rFi-1sIoH2iga2TAoq5FFM6mmTR6EfPZUjU2yxvaa5WqvzYjXSxw=s0-d)
where
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vsqTSnayd2OTbk_LscbTGzFowwgAbgHLYCPdoSDSlaMdC1i6FrmmmZVzFgSQzckcDdqATrFwcJyPko6kYxQYe3XsHzAsvJMp8jiXe1LHHm34T7Xw=s0-d)
is the operator saying to take the Poisson bracket with its argument.
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uozm4o2-YDS0zB61mh3aHVZlMeo_DfncDzBJOMl9n9zRN8I045v6OwKZjTfcXNX7gp8fF_cMyrGp27MSSvOaLjUtDRu-7_XCHa4Oio45ancrk=s0-d)
is the effective Hamiltonian and is a function of
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tRRTcJszvkQGSZj8fa-OAczrYvz7sf3s4KYzRW8PCUxhEgRmG_c92BJgt42Yn6JMIWD514PcnU7x0GTJyD1_QudnZKBDjUMQvUWgltGSL6Og=s0-d)
and
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tULQ7OzQOa354X2mXGhoQiXoBoHR4CqenDf-REhnTV-z5JP4rJeTyuKNJRpxZGs9etJ8Vlm6GVVhWFE6LPB04-S07S4STZmw1LtGt9V4BNaG0=s0-d)
. In the linear case,
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uCQGLYjujfgcfOdnoj841TzkiJlY7gSxujkMTA4A50dQQzMAEiAeV_L3Fq83BzdIhCWCg_K1G-M8v-6B8vQ44HB-VYZRyUi2o7WPBllwBtAswZfVOn5Sq2rVVSbOIb5GSZ=s0-d)
is just a matrix
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v2l0x9wOp6HEMqSxJeMxYsOpIeRpZeAWG0rdwQkkx4RvbZQwzrV5q9KQNT7tL6c5Srvnthtx6mbOhjzeoiN48NVJV-zgbxxhoXhDtcLi5lwWk4NcQg3ujAR7qSSbHr-4g=s0-d)
and if the Hamiltonian is
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uKoocm9YEVXTh0Q69GdZbDYOs-zwijZAi1MrQpy2MfwJcgUVoi4AdeCQaJYNrjgcNPJivdGisRi7M778r_rAwhLsq5WU5pfcL2uM9zbpjK-FzT_dfGdp7SE1r5JU7w_wMFwutm=s0-d)
with
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tCB1u1sJpnpybpizWoH_9K7E78OcN9RFhUo9xB7RmG0T-bt1RmN6RuBV8QVVW3x6rppy404Bk-GcKBTbKv3OIDukPckAksdMdxAwRa18qK0rWYkQZoBi6j0LSDW_PgNjF1_zfgnRzKWLFo3-jVkqw=s0-d)
, then
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vlvQvDAOWDmFpeJgzEY5wbDQLOUH9AxJH5bgbmrQCinrsKvPC13VdIAS09cVI_i6cf8ynpwET7eKL9RcSJLUt22lsb6qdBRXerJ24VNM7VK5nUMqsQDfM6VBrvk1qqCvRd4iI_C9nzEH7n4WjmCfOw=s0-d)
where
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tNgZ4BWf3VPr1lRCRD_8Pgg8c6znYZqKq-4PtUYDMfhZsYFthB9qQ4tig7p8U1R8GZW7MQwZhtt1QOgQ9tsBbwsOmghKTyqHt3abjtMMTOx0J0gW9X2YEFYSB-RJfTEdTps3AYEio9qySHuCXbyChM4hxSO-xA6vZtvCID4JL_bw8=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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