Just a few quick notes on historical trails I'm finding as I study group theory and Emmy Noether's theorem. The people involved are a bit of a group themselves. The first person I found is Lagrange. He introduced the Langrangian, one of the key concepts of analytical mechanics. It's used today, well, everywhere, from plain old mechanics to quantum mechanics, to quantum field theory. He also laid some of the foundations of group theory working on permutation groups and their use in solving polynomials.
That brings us to Évariste Galois. Galois read Lagrange's papers at the age of 15. He later went on to develop Galois theory. Galois theory relates permutation groups of the roots of polynomials to their solvability.
Sophus Lie developed Lie groups, provide a framework that is similar to Galois theory for studying the symmetries of differential equations.
And that brings us to Emmy Noether. Her paper on "Invariant Variation Problems" applied Lie's work to variational calculus revealing conservation laws applied to Lagrangians.