# Noether's Theorem

**Type:** Theorem  
**Domain:** Mathematical Physics  
**Proved:** 1915 (published 1918), Emmy Noether  
**Codex URL:** /mathematics/mathematical-physics/noethers-theorem/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Every continuous symmetry of a physical system corresponds to a conserved quantity. Time symmetry→energy conservation, translation→momentum, rotation→angular momentum.

## The three classic examples
- Time-translation symmetry → Energy conservation (Hamiltonian)
- Spatial-translation symmetry → Momentum conservation
- Rotational symmetry → Angular momentum conservation
- U(1) gauge symmetry → Electric charge conservation

## Noether's two theorems
First Theorem: global symmetries → standard conservation laws.
Second Theorem: local (gauge) symmetries → identities among equations of motion (relevant to GR).

## Historical context
Hilbert and Klein invited Noether to Göttingen to resolve an energy-conservation puzzle in Einstein's general relativity. Faced institutional gender discrimination (barred from formal habilitation); dismissed 1933 under Nazi racial laws; emigrated to Bryn Mawr College, died 1935.

## Standard Model connection
U(1)×SU(2)×SU(3) gauge symmetries → electric charge, weak isospin, colour charge conservation.

## Sources
### Tier 1
- Noether, E. (1918). Invariante Variationsprobleme. *Nachr. Ges. Wiss. Göttingen*.
- Goldstein, H., Poole, C. and Safko, J. (2002). *Classical Mechanics*. 3rd ed.
### Tier 2
- Kosmann-Schwarzbach, Y. (2011). *The Noether Theorems*.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*