symmetry in classical field theory
from Lagrangian density we can algorithmically get equations of motion, but the Lagrangian density is a more compact way of representing the equations of motion
definition of symmetry in context: keeps Lagrangian unchanged up to a total derivative
Noether's theorem
youtu.be/cj-QpsZsDDY?list=PLDfPUNusx1EpRs-wku83aqYSKfR5fFmfS&t=3062 Lagrangian and conservation example under translations
youtu.be/cj-QpsZsDDY?list=PLDfPUNusx1EpRs-wku83aqYSKfR5fFmfS&t=3394 same but for Poincaré transformations But now things are harder, because it is harder to describe general infinitesimal Poincare transforms than it was to describe the translations. Using constraints/definition of Lorentz transforms, also constricts the allowed infinitesimal symmetries to 6 independent parameters
youtu.be/cj-QpsZsDDY?list=PLDfPUNusx1EpRs-wku83aqYSKfR5fFmfS&t=4525 brings out Poisson brackets, and concludes that each conserved current maps to a generator of the Lie algebra
This allows you to build the symmetry back from the conserved charges, just as you can determine conserved charges starting from the symmetry.

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