{"id": "card_floor_the_lagrangian", "kind": "note", "title": "The Lagrangian — the Hamiltonian's dual, the path chosen", "body": "L = T - V, and the action S = integral L dt. Its law is the principle of least action: of all paths, nature takes the one where the action is stationary. Four pillars rest on the floors already seeded - least action (on the capstone and the solve path), the Legendre dual (on the Hamiltonian), the path integral (on the capstone), and one scalar for all of physics (on the Standard Model). The Lagrangian is the Hamiltonian's other face: H generates time, L governs the path. The seals and the fits are on the stick (stick_the_lagrangian_the_hamiltonian_s_dual_and_the_path); this card places it on the map (tools/seed_the_lagrangian.py).", "source": {"label": "Narrow Highway — the Lagrangian", "url": "", "authority_tier": "engine_derived", "ref": "stick_the_lagrangian_the_hamiltonian_s_dual_and_the_path"}, "shelf": "codex", "box": "lagrangian", "bands": ["floor", "lagrangian", "least action", "action", "path integral", "quantum", "one map"], "subject": "The Lagrangian — the Hamiltonian's dual, the path chosen", "connections": [{"to_card_id": "card_lag_least_action", "relationship": "has_part", "evidence": "a pillar of the Lagrangian (Least action)"}, {"to_card_id": "card_lag_legendre_dual", "relationship": "has_part", "evidence": "a pillar of the Lagrangian (The Legendre dual of the Hamiltonian)"}, {"to_card_id": "card_lag_path_integral", "relationship": "has_part", "evidence": "a pillar of the Lagrangian (The path integral)"}, {"to_card_id": "card_lag_field_theory", "relationship": "has_part", "evidence": "a pillar of the Lagrangian (One scalar for all of physics)"}, {"to_card_id": "card_k_floor_of_discovery", "relationship": "part_of", "evidence": "the path side of the dynamics on the one map"}, {"to_card_id": "card_noether_symmetry_conservation", "relationship": "connects_at", "evidence": "the theorem is about symmetries of the ACTION - the Lagrangian side of the dynamics"}, {"to_card_id": "card_maxwell_covariance_relativity", "relationship": "connects_at", "evidence": "the EM field Lagrangian; manifest Lorentz covariance -> special relativity"}, {"to_card_id": "card_rel_general", "relationship": "connects_at", "evidence": "general relativity is a Lagrangian field theory: the Einstein-Hilbert action"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "codex.lagrangian", "facets": {"subject": ["action", "floor", "least action", "one map", "path integral", "quantum"]}, "presentation": {"glyph": "•", "kind_label": "lagrangian", "by": "Narrow Highway — the Lagrangian", "authority": "engine_derived", "posted": "", "standing": ""}, "neighbors": [{"id": "card_lag_least_action", "title": "Least action — of all paths, the stationary one", "relationship": "has part", "why": "a pillar of the Lagrangian (Least action)", "href": "/card/card_lag_least_action", "resolved": true}, {"id": "card_lag_legendre_dual", "title": "The Legendre dual of the Hamiltonian — H = p q' − L", "relationship": "has part", "why": "a pillar of the Lagrangian (The Legendre dual of the Hamiltonian)", "href": "/card/card_lag_legendre_dual", "resolved": true}, {"id": "card_lag_path_integral", "title": "The path integral — all paths, e^(iS/ℏ), the classical one by interference", "relationship": "has part", "why": "a pillar of the Lagrangian (The path integral)", "href": "/card/card_lag_path_integral", "resolved": true}, {"id": "card_lag_field_theory", "title": "One scalar for all of physics — the Standard Model is a Lagrangian", "relationship": "has part", "why": "a pillar of the Lagrangian (One scalar for all of physics)", "href": "/card/card_lag_field_theory", "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 path side of the dynamics on the one map", "href": "/card/card_k_floor_of_discovery", "resolved": true}, {"id": "card_noether_symmetry_conservation", "title": "Every continuous symmetry a conservation law", "relationship": "connects at", "why": "the theorem is about symmetries of the ACTION - the Lagrangian side of the dynamics", "href": "/card/card_noether_symmetry_conservation", "resolved": true}, {"id": "card_maxwell_covariance_relativity", "title": "Four equations, one field — Lorentz-covariant, the root of relativity", "relationship": "connects at", "why": "the EM field Lagrangian; manifest Lorentz covariance -> special relativity", "href": "/card/card_maxwell_covariance_relativity", "resolved": true}, {"id": "card_rel_general", "title": "General relativity — gravity is the curvature of spacetime", "relationship": "connects at", "why": "general relativity is a Lagrangian field theory: the Einstein-Hilbert action", "href": "/card/card_rel_general", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_floor_the_lagrangian", "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_the_lagrangian"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}