Chad Orzel has a very sensible piece at Forbes, headlined What Math Do You Need For Physics? It Depends, which addresses the question of what math a physicist like him (experimental AMO physics) really needs. I’m glad to see that he emphasizes the same basic courses my department offers aimed at non-majors:

Multivariable calculus

Differential equations

Linear algebra

together with statistics (which here at Columbia is handled by a separate department). He disses complex analysis, for reasons that I can understand. That’s a beautiful subject, and the results you can get out of analytic functions and contour integration are often unexpected and seemingly magic, but they’re not of as general use as the other subjects.

One subject he doesn’t mention that I would argue for is Fourier Analysis, which is the class I’ll be teaching next semester. That’s an incredibly useful as well as profound subject which every physicist should know, but it is true some of its basics often gets taught in other courses (for example in ode courses as a method for solving differential equations).

Orzel starts off with an amusing discussion of a physics version of “Humiliation”, admitting that he’s never used or really worked through a proof of Noether’s theorem, widely considered “the most profound idea in science”. I’ve argued here for the Hamiltonian over Lagrangian method, in which case a different set of ideas about symmetry is fundamental, with Noether’s theorem playing no role. In the Hamiltonian case symmetries are generated by functions on phase space, and finding the function that generates any symmetry is a matter of Poisson brackets (as an experimentalist, maybe Orzel has never calculated a Poisson bracket either though…).

One says that a function F on phase space generates an infinitesimal transformation if such an infinitesimal transformation changes the function G by {G,F} (the Poisson bracket of the functions G and F). A basic example is the Hamiltonian function, with {G,H} the infinitesimal change of G by time translation, or in other words, Hamilton’s equation

dG/dt={G,H}

When {H,F}=0, we say that the infinitesimal transformation generated by F is a symmetry, since H is left unchanged by such transformations. Using the antisymmetry of the Poisson bracket, this can also be read as {F,H}=0, with Hamilton’s equation then the conservation law that F doesn’t change with time.

All this seems to me much more straightforward than the Lagrangian Noether’s theorem approach to symmetry transformations and conservation laws. My own analog of Orzel’s admission would be admitting (which I won’t do) how long it took me in life to understand this fundamental point (I blame my teachers).

For lots and lots more about this, see chapters 14 and 15 here.