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Noether's theorem — symmetry is the source of conservation
noether
For every continuous symmetry of the action there is a conserved quantity (Emmy Noether, 1918). The seam between the Lagrangian and the Hamiltonian: the Lagrangian's time-symmetry PRODUCES the Hamiltonian's conserved energy. Four pillars rest on the floors already seeded - the theorem (on the Lagrangian), the invariant (on the capstone, the same end through every path), time gives energy (on the Hamiltonian), and gauge gives charge (on the Standard Model). Conservation laws are found, never decreed - compelled as the shadow of a symmetry. The seals are on the stick (stick_noether_s_theorem_every_symmetry_a_conservation_law); this card places it on the map (tools/seed_noethers_theorem.py).
- has part → Every continuous symmetry a conservation law — a pillar of Noether's theorem (Every continuous symmetry a conservation law)
- has part → The invariant — the same end carried through every path — a pillar of Noether's theorem (The invariant)
- has part → Time-translation symmetry gives energy — the Hamiltonian — a pillar of Noether's theorem (Time-translation symmetry gives energy)
- has part → Gauge symmetry gives charge — the Standard Model's currents — a pillar of Noether's theorem (Gauge symmetry gives charge)
- part of → The Floor of Discovery — one floor, and by its design the fear of God — the symmetry-conservation seam on the one map
- connects at → U(1) gauge symmetry → conservation of charge — the U(1) gauge symmetry -> conserved charge, via Noether's theorem
- connects at → The first law — energy is conserved — the first law is Noether's conserved energy, accounting heat and work
- connects at → Poincaré symmetry — the group that feeds Noether — the Poincare symmetry group -> conserved energy, momentum, angular momentum, via Noether
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