Looking around I see one version of Noether's Theorem that creates conserved quantities from symmetries that preserve the Lagrangian (e.g. http://math.ucr.edu/home/baez/noether.html), and another theorem also called Noether's Theorem that finds conserved quantities by seeing if they Poisson commute with the Hamiltonian (e.g. page 29 of http://www.math.ucla.edu/~tao/preprints/chapter.pdf).
Are these two results actually the same results disguised?
For example, Terry Tao in http://www.math.ucla.edu/~tao/preprints/chapter.pdf on page 83 derives the total charge $\int |u|^2 \, dx$ as an invariant of the free Schroedinger equation, He says this is a consequence of the map $u \mapsto e^{i\phi} u$. I tried writing the Schroedinger equation in Lagrangian form, by deciding that the real part of $u$ would be "position", and the imaginary part of $u$ would be "momentum." but then the map $u \mapsto e^{i\phi} u$ ended up being a map that didn't map position to a new position, but rather mixed position and velocity in a rather unpleasant manner. Would there be some other smart choice of deciding which variables are "position" and "momentum" that would cause the total charge to arise as the conserved quantity given by http://math.ucr.edu/home/baez/noether.html?
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