Ciro Santilli
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Noether's theorem | 🗖 nosplit | ↑ parent "Lagrangian mechanics" | 199

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For every continuous symmetry in the system (Lie group), there is a corresponding conservation law.
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Furthermore, given the symmetry, we can calculate the derived conservation law, and vice versa.
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As mentioned at http://buzzard.ups.edu/courses/2017spring/projects/schumann-lie-group-ups-434-2017.pdf, what the symmetry (Lie group) acts on (obviously?!) are the Lagrangian generalized coordinates. And from that, we immediately guess that manifolds are going to be important, because the generalized variables of the Lagrangian can trivially be non-Euclidian, e.g. the pendulum lives on an infinite cylinder.
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Video 91. "The most beautiful idea in physics - Noether's Theorem" by Looking Glass Universe (2015-09-23) Source. One sentence stands out: the generated quantities are called the generators of the transforms.
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Video 92. "The Biggest Ideas in the Universe | 15. Gauge Theory" published by Sean Carroll on 2020-06-30. Source. This attempts a one hour hand wave explanation of it. It is a noble attempt and gives some key ideas, but it falls a bit short of Ciro's desires (as would anything that fit into one hour?)
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