As mentioned at 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-Euclidean geometry, e.g. the pendulum lives on an infinite cylinder.
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Noether's theorem is a fundamental result in theoretical physics and mathematics that establishes a profound relationship between symmetries and conservation laws. Named after the German mathematician Emmy Noether, the theorem essentially states that for every continuous symmetry of a physical system, there corresponds a conserved quantity. In more precise terms: 1. **Continuous Symmetries**: These are transformations of a physical system that can be performed smoothly and without abrupt changes.