{"id": "card_lag_legendre_dual", "kind": "reference", "title": "The Legendre dual of the Hamiltonian — H = p q' − L", "body": "L and H are Legendre transforms of one another: L = T - V (the generator of the path), H = T + V (the generator of time). Free particle: L = (1/2) m v^2, p = m v, H = p v - L. Sealed on this stick and on stick_the_legendre_transform_mark_the_points_create_the_line.", "source": {"label": "Narrow Highway — the Lagrangian", "url": "", "authority_tier": "engine_derived"}, "shelf": "codex", "box": "lagrangian", "bands": ["lagrangian", "pillar", "legendre_dual", "one map"], "subject": "The Legendre dual of the Hamiltonian — H = p q' − L", "connections": [{"to_card_id": "card_floor_the_lagrangian", "relationship": "part_of", "evidence": "a pillar of the Lagrangian (The Legendre dual of the Hamiltonian)"}, {"to_card_id": "card_floor_the_hamiltonian", "relationship": "connects_at", "evidence": "the two faces of one dynamics: L is T - V, H is T + V, joined by the Legendre transform"}], "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": ["legendre_dual", "one map", "pillar"]}, "presentation": {"glyph": "•", "kind_label": "lagrangian", "by": "Narrow Highway — the Lagrangian", "authority": "engine_derived", "posted": "", "standing": ""}, "neighbors": [{"id": "card_floor_the_lagrangian", "title": "The Lagrangian — the Hamiltonian's dual, the path chosen", "relationship": "part of", "why": "a pillar of the Lagrangian (The Legendre dual of the Hamiltonian)", "href": "/card/card_floor_the_lagrangian", "resolved": true}, {"id": "card_floor_the_hamiltonian", "title": "The Hamiltonian — the law of the flow", "relationship": "connects at", "why": "the two faces of one dynamics: L is T - V, H is T + V, joined by the Legendre transform", "href": "/card/card_floor_the_hamiltonian", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_lag_legendre_dual", "next_step": {"do": "follow a connection: what this card rests on, and what rests on it", "door": "FIND", "tool": "card_connections", "params": {"id": "card_lag_legendre_dual"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}