{"id": "card_floor_noethers_theorem", "kind": "note", "title": "Noether's theorem — symmetry is the source of conservation", "body": "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).", "source": {"label": "Narrow Highway — Noether's theorem", "url": "", "authority_tier": "engine_derived", "ref": "stick_noether_s_theorem_every_symmetry_a_conservation_law"}, "shelf": "codex", "box": "noether", "bands": ["floor", "noether", "symmetry", "conservation", "invariant", "quantum", "one map"], "subject": "Noether's theorem — symmetry is the source of conservation", "connections": [{"to_card_id": "card_noether_symmetry_conservation", "relationship": "has_part", "evidence": "a pillar of Noether's theorem (Every continuous symmetry a conservation law)"}, {"to_card_id": "card_noether_the_invariant", "relationship": "has_part", "evidence": "a pillar of Noether's theorem (The invariant)"}, {"to_card_id": "card_noether_time_gives_energy", "relationship": "has_part", "evidence": "a pillar of Noether's theorem (Time-translation symmetry gives energy)"}, {"to_card_id": "card_noether_gauge_gives_charge", "relationship": "has_part", "evidence": "a pillar of Noether's theorem (Gauge symmetry gives charge)"}, {"to_card_id": "card_k_floor_of_discovery", "relationship": "part_of", "evidence": "the symmetry-conservation seam on the one map"}, {"to_card_id": "card_maxwell_gauge_charge", "relationship": "connects_at", "evidence": "the U(1) gauge symmetry -> conserved charge, via Noether's theorem"}, {"to_card_id": "card_thermo_first_law", "relationship": "connects_at", "evidence": "the first law is Noether's conserved energy, accounting heat and work"}, {"to_card_id": "card_rel_poincare", "relationship": "connects_at", "evidence": "the Poincare symmetry group -> conserved energy, momentum, angular momentum, via Noether"}, {"to_card_id": "card_chem_conservation", "relationship": "connects_at", "evidence": "conservation of atoms and mass - the balanced equation is a Noether conservation law"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "codex.noether", "facets": {"subject": ["conservation", "floor", "invariant", "one map", "quantum", "symmetry"]}, "presentation": {"glyph": "•", "kind_label": "noether", "by": "Narrow Highway — Noether's theorem", "authority": "engine_derived", "posted": "", "standing": ""}, "neighbors": [{"id": "card_noether_symmetry_conservation", "title": "Every continuous symmetry a conservation law", "relationship": "has part", "why": "a pillar of Noether's theorem (Every continuous symmetry a conservation law)", "href": "/card/card_noether_symmetry_conservation", "resolved": true}, {"id": "card_noether_the_invariant", "title": "The invariant — the same end carried through every path", "relationship": "has part", "why": "a pillar of Noether's theorem (The invariant)", "href": "/card/card_noether_the_invariant", "resolved": true}, {"id": "card_noether_time_gives_energy", "title": "Time-translation symmetry gives energy — the Hamiltonian", "relationship": "has part", "why": "a pillar of Noether's theorem (Time-translation symmetry gives energy)", "href": "/card/card_noether_time_gives_energy", "resolved": true}, {"id": "card_noether_gauge_gives_charge", "title": "Gauge symmetry gives charge — the Standard Model's currents", "relationship": "has part", "why": "a pillar of Noether's theorem (Gauge symmetry gives charge)", "href": "/card/card_noether_gauge_gives_charge", "resolved": true}, {"id": "card_k_floor_of_discovery", "title": "The Floor of Discovery — one floor, and by its design the fear of God", "relationship": "part of", "why": "the symmetry-conservation seam on the one map", "href": "/card/card_k_floor_of_discovery", "resolved": true}, {"id": "card_maxwell_gauge_charge", "title": "U(1) gauge symmetry → conservation of charge", "relationship": "connects at", "why": "the U(1) gauge symmetry -> conserved charge, via Noether's theorem", "href": "/card/card_maxwell_gauge_charge", "resolved": true}, {"id": "card_thermo_first_law", "title": "The first law — energy is conserved", "relationship": "connects at", "why": "the first law is Noether's conserved energy, accounting heat and work", "href": "/card/card_thermo_first_law", "resolved": true}, {"id": "card_rel_poincare", "title": "Poincaré symmetry — the group that feeds Noether", "relationship": "connects at", "why": "the Poincare symmetry group -> conserved energy, momentum, angular momentum, via Noether", "href": "/card/card_rel_poincare", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_floor_noethers_theorem", "next_step": {"do": "follow a connection: what this card rests on, and what rests on it", "door": "FIND", "tool": "card_connections", "params": {"id": "card_floor_noethers_theorem"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}