{"id": "card_joint_arnold_geodesics", "kind": "note", "title": "Euler flow is geodesic flow on the volume-preserving diffeomorphisms", "body": "Arnold (1966): the Euler equations of an ideal fluid are the geodesic equations of the group of volume-preserving diffeomorphisms with the kinetic-energy metric - the fluid as a point moving on an infinite-dimensional manifold, the same geometric analysis Perelman's proof lives in. It has not produced regularity for Navier-Stokes; the join is real and the question stays open.", "source": {"label": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "url": "", "domain": "mathematics", "authority_tier": "engine_derived"}, "shelf": "codex", "box": "joint", "bands": ["joint", "two trees", "millennium", "floor", "arnold", "geodesic flow", "diffeomorphisms"], "subject": "Euler flow is geodesic flow on the volume-preserving diffeomorphisms", "connections": [{"to_card_id": "card_floor_millennium", "relationship": "part_of", "evidence": "a joint that is proven or observed - a part of the floor"}, {"to_card_id": "card_question_navier_stokes", "relationship": "connects_at", "evidence": "Euler flow is geodesic flow on the volume-preserving diffeomorphisms. arnold 1966. Cited, not sealed: no arithmetic to recompute"}, {"to_card_id": "card_question_poincare", "relationship": "connects_at", "evidence": "Euler flow is geodesic flow on the volume-preserving diffeomorphisms. arnold 1966. Cited, not sealed: no arithmetic to recompute"}, {"to_card_id": "card_src_mill_arnold_1966", "relationship": "cites", "evidence": "the record the joint stands on"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "codex.joint", "facets": {"subject": ["arnold", "diffeomorphisms", "floor", "geodesic flow", "millennium", "two trees"]}, "presentation": {"glyph": "•", "kind_label": "joint", "by": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "authority": "engine_derived", "posted": "", "standing": ""}, "neighbors": [{"id": "card_floor_millennium", "title": "The Millennium floor - seven open questions, and where they connect", "relationship": "part of", "why": "a joint that is proven or observed - a part of the floor", "href": "/card/card_floor_millennium", "resolved": true}, {"id": "card_question_navier_stokes", "title": "Navier-Stokes existence and smoothness", "relationship": "connects at", "why": "Euler flow is geodesic flow on the volume-preserving diffeomorphisms. arnold 1966. Cited, not sealed: no arithmetic to recompute", "href": "/card/card_question_navier_stokes", "resolved": true}, {"id": "card_question_poincare", "title": "The Poincare conjecture", "relationship": "connects at", "why": "Euler flow is geodesic flow on the volume-preserving diffeomorphisms. arnold 1966. Cited, not sealed: no arithmetic to recompute", "href": "/card/card_question_poincare", "resolved": true}, {"id": "card_src_mill_arnold_1966", "title": "V. I. Arnold 1966 — Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits", "relationship": "cites", "why": "the record the joint stands on", "href": "/card/card_src_mill_arnold_1966", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_joint_arnold_geodesics", "next_step": {"do": "follow a connection: what this card rests on, and what rests on it", "door": "FIND", "tool": "card_connections", "params": {"id": "card_joint_arnold_geodesics"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}