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## Main Question or Discussion Point

I'm trying to better understand how people refer to symmetry in Physics and Differential Geometry. In "Exterior Differential Systems and Euler Lagrange Partial Differential Equations," by Bryant, Griffiths and Grossman, it seems a vector field is a symmetry of a Lagrangian if the Lie derivative of the Lagrangian with respect to the vector field vanishes. I don't see how this connects to the notion of a symmetry as a conserved quantity arising from invariance of a Lagrangian under a group of transformations as these seem to be disjoint conceptually; what is the link I wonder?