{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:bd322781-a622-431f-a000-1be56b3c37e1", "labels": { "zh-CN": "哈密顿向量场", "en": "Hamiltonian vector field" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "bae0a54eb75eff85a638bc1c8ef1b07ceff46776626ff538594870ae1440bcb3", "occurrences": [ { "concept_id": "urn:uuid:bd322781-a622-431f-a000-1be56b3c37e1", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-16/05-classical-mechanics", "anchor": "first-0acbbd0ad314962e6deb-zh", "chapter": "16", "section": "16.5", "offset": 27413, "range": [ 35193, 35444 ], "source_revision": "aac6045c8f792acfb5c9989193835b417610f4c945fe5f70074b9b0c3fb0e939", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "哈密顿向量场(Hamiltonian vector field)", "context": { "range": [ 35057, 35607 ], "quote": "对所有向量场 $Y$。若存在 $M$ 上的函数 $H$，使得 $X = { \\overline { { \\mathrm { d } H } } }$，或等价地 ${ \\overline { { X } } } = \\mathrm { d } H$，则称向量场 $X$ 为 哈密顿向量场(Hamiltonian vector field)。函数 $H$ 称为生成该向量场的哈密顿算子。若函数 $f$ 是哈密顿向量场 $X = { \\overline { { \\mathrm { d } H } } } { \\mathrm { ~ i f ~ } } X f = 0$ 所生成相位流的第一积分，则称其为该相位流的第一积分。哈密顿算子 $H$ 是相位流的第一积分，因为", "quote_sha256": "e9eac314d30ea1274352b85d11010c3459c2512b84c6522d8c8c10e95401fb30", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-16/05-classical-mechanics", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-19fe1b0636b54b08a5537aed" }, { "concept_id": "urn:uuid:bd322781-a622-431f-a000-1be56b3c37e1", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-16/05-classical-mechanics", "anchor": "first-0acbbd0ad314962e6deb-en", "chapter": "16", "section": "16.5", "offset": 36458, "range": [ 43767, 44010 ], "source_revision": "6ba31cc0cea910b2bb1780c9e846e4e265229202d450afd615e6b7053ad533bb", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "Hamiltonian vector field", "context": { "range": [ 43705, 44484 ], "quote": "for all vector fields $Y$. A vector field $X$ is said to be a Hamiltonian vector field if there exists a function $H$ on $M$ such that $X = { \\overline { { \\mathrm { d } H } } }$ , or equivalently ${ \\overline { { X } } } = \\mathrm { d } H$ . The function $H$ is called the Hamiltonian generating this vector field. A function $f$ is said to be a first integral of the phase flow generated by the Hamiltonian vector field $X = { \\overline { { \\mathrm { d } H } } } { \\mathrm { ~ i f ~ } } X f = 0$ The Hamiltonian $H$ is a first integral of the phase flow, for", "quote_sha256": "68a6a89697c1dcb20ea53ca15903f022cb59e8ec4d2572f6111547574931c0cc", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-16/05-classical-mechanics", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-cc33e7eb0e18a186d0312a5a" }, { "concept_id": "urn:uuid:bd322781-a622-431f-a000-1be56b3c37e1", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-15/02-hamiltonian-vector-fields-and-canonical-transformations", "anchor": "first-32812f1047753eb9505d-zh", "chapter": "15", "section": "15.2", "offset": 343, "range": [ 7380, 7986 ], "source_revision": "7dad044ddf4bfa479d6eefef8474f3f2aefd1287253063ffa62891cf24ccb600", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "definition", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "books/woit-qtgr/2025-10-20/zh/chapter-15/02-hamiltonian-vector-fields-and-canonical-transformations", "lang": "zh-CN", "section": "15.2", "signal": "named-definition", "label": "哈密顿向量场", "english": "", "offset": 273 } }, "label": "> [!definition] 定义（哈密顿向量场）", "context": { "range": [ 7380, 7986 ], "quote": "\n\n> [!definition] 定义（哈密顿向量场）\n> 若 $M=\\mathbf R^2$ 上的向量场具有下列形式\n>\n> $$\n> X_f=\\frac{\\partial f}{\\partial p}\\frac{\\partial}{\\partial q}-\\frac{\\partial f}{\\partial q}\\frac{\\partial}{\\partial p}=-\\{f,\\cdot\\},\n> $$\n>\n> 其中 $f$ 是 $M$ 上的函数，则称它为哈密顿向量场，记作 $X_f$。在更高维的情形下，$M=\\mathbf R^{2d}$ 上的哈密顿向量场具有如下形式：\n>\n> \n>\n> $$\n> X_f=\\sum_{j=1}^{d}\\left(\\frac{\\partial f}{\\partial p_j}\\frac{\\partial}{\\partial q_j}-\\frac{\\partial f}{\\partial q_j}\\frac{\\partial}{\\partial p_j}\\right)=-\\{f,\\cdot\\}.\\tag{15.2}\n> $$\n>\n> 其中 $f$ 是 $M$ 上的函数。最简单的非零哈密顿向量场来自线性函数 $f$。设 $c_q,c_p$ 为常数，并令\n", "quote_sha256": "10e5120675a7298af8d7cb787a3c0b3d2e2f30c99281e017f8bee1c5ac70eae7", "environment": "definition", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-15/02-hamiltonian-vector-fields-and-canonical-transformations", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-3888fad53ad945ab6b8a3025" }, { "concept_id": "urn:uuid:bd322781-a622-431f-a000-1be56b3c37e1", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-15/02-hamiltonian-vector-fields-and-canonical-transformations", "anchor": "first-3fcaaf2a4cf66d90d2dd-en", "chapter": "15", "section": "15.2", "offset": 493, "range": [ 6687, 7501 ], "source_revision": "94d4f8c60bb4d1d5fdcbd75675ef49a2eda8acdc416fa20232ad9c7e67e06d03", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "definition", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "books/woit-qtgr/2025-10-20/en/chapter-15/02-hamiltonian-vector-fields-and-canonical-transformations", "lang": "en", "section": "15.2", "signal": "named-definition", "label": "Hamiltonian vector field", "english": "Hamiltonian vector field", "offset": 423 } }, "label": "> [!definition] Definition (Hamiltonian vector field)", "context": { "range": [ 6687, 7501 ], "quote": "\n\n> [!definition] Definition (Hamiltonian vector field)\n> A vector field on $M=\\mathbf R^2$ of the form\n>\n> $$\n> X_f=\\frac{\\partial f}{\\partial p}\\frac{\\partial}{\\partial q}-\\frac{\\partial f}{\\partial q}\\frac{\\partial}{\\partial p}=-\\{f,\\cdot\\}\n> $$\n>\n> for a function $f$ on $M$ is called a Hamiltonian vector field and is denoted $X_f$. In higher dimensions, Hamiltonian vector fields on $M=\\mathbf R^{2d}$ have the form\n>\n> \n>\n> $$\n> X_f=\\sum_{j=1}^{d}\\left(\\frac{\\partial f}{\\partial p_j}\\frac{\\partial}{\\partial q_j}-\\frac{\\partial f}{\\partial q_j}\\frac{\\partial}{\\partial p_j}\\right)=-\\{f,\\cdot\\}.\\tag{15.2}\n> $$\n>\n> Here $f$ is a function on $M$.The simplest nonzero Hamiltonian vector fields arise from linear functions $f$. 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