One of the things about accelerator physics modeling that I'd found out recently was that the mathematical maps that track particles around the accelerator use a technology related to so-called "differential algebras". In particular, the claim was that there was a relationship to a subject known as "non-standard analysis". Its a pretty cool topic. The idea is to extend the real numbers to include "infinitesimals" dx, such that dx < y for all real values y. Kind of like extending real numbers to the complex plane. Anyway, the result is that one can find derivatives and integrals using algebraic methods, rather than needing to define limits and do these annoying "delta-epsilon" proofs.
So, its still true that non-standard analysis is cool. And I'm happy to have learned a bit more about it. But the guy I'm working for has the viewpoint that pushing this perspective of "Differential Algebra" methods in accelerator physics has been a bit of a fraud. His point is that when you go ahead and implement the algorithms in the computer code- you don't actually use such structures. This would be a bit disappointing because it removes some of the sex appeal of the subject... On the other hand, I'm starting to get more used to the idea of building up nice things from elementary, not so exciting pieces. (Growing up beyond exoticism, if you will.)
I haven't yet gone through the code myself to understand the extent of the claim. My suspicion is that the truth may lie somewhere in between, with "DA methods" inspiring algorithms without being used explicitly.
Anyway, I'm sure everyone will be waiting with baited breath for an update on this topic...
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