Short answers, pulled from the story.
Emmy Noether published the paper titled Invariante Variationsprobleme in 1918. The publication appeared later that year in the journal Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen.
The theorem states that every differentiable symmetry of the action corresponds to a conserved quantity. This rule links continuous symmetries to conservation laws in physical systems through the integral over time of a Lagrangian function.
Emmy Noether began her work on this theorem in 1915 while assisting Felix Klein and David Hilbert. They were struggling with Albert Einstein's theory of general relativity when she developed most key ideas by March 1918.
Time invariance produces conservation of total energy H when N equals one and T equals one. If the Lagrangian does not depend explicitly on time, the Hamiltonian stays conserved within the system.
Dissipative systems often break the link because they cannot be modeled by a Lagrangian alone. The theorem applies strictly to systems with conservative forces but fails for those involving dissipation.