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        "quote": "若 $T ^ { * } \\bar { M } \\to \\mathbb { R }$ 是另一个同维数的扩展相位，具有正则坐标 $\\bar { q } ^ { i } , \\bar { p } _ { i }$ 和哈密顿算子 $\\bar { H } ( \\bar { \\mathbf { q } } , \\bar { \\mathbf { p } } , t )$，则微分同胚 $\\phi : T ^ { * } M \\times \\mathbb { R } \\to T ^ { * } { \\bar { M } } \\to R$ 称为 <u class=\"concept-term\" data-concept-identity=\"urn:uuid:368c43a6-d587-4e9d-96c7-99da4d648aaa\" data-concept-route=\"/concept-identities#concept-368c43a6-d587-4e9d-96c7-99da4d648aaa\" id=\"first-050ffff04cc9cd79f7f8-zh\">接触变换(contact transformation)</u> $i \\mathbf { \\dot { \\varphi } } \\phi ^ { * } \\mathbf { d } \\bar { \\lambda } = \\mathbf { d } \\lambda$。由于 $\\phi ^ { * } \\circ \\mathsf { d } = \\mathsf { d } \\circ \\phi$，在任意一点的邻域内存在 $T ^ { * } M \\times \\mathbb { R }$ 上的函数 $F$，使得 $\\phi ^ { * } { \\bar { \\lambda } } = \\lambda - \\mathrm { d } F$。若我们将函数 $F$ 写成依赖于变量 $q ^ { i }$ 和 $\\bar { q } ^ { i }$ 的形式，这在局部通常是可能的，",
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        "quote": "If $T ^ { * } \\bar { M } \\to \\mathbb { R }$ is another extended phase of the same dimension with canonical coordinates $\\bar { q } ^ { i } , \\bar { p } _ { i }$ and Hamiltonian $\\bar { H } ( \\bar { \\mathbf { q } } , \\bar { \\mathbf { p } } , t )$ , then a diffeomorphism $\\phi : T ^ { * } M \\times \\mathbb { R } \\to T ^ { * } { \\bar { M } } \\to R$ is called a <u class=\"concept-term\" data-concept-identity=\"urn:uuid:368c43a6-d587-4e9d-96c7-99da4d648aaa\" data-concept-route=\"/concept-identities#concept-368c43a6-d587-4e9d-96c7-99da4d648aaa\" id=\"first-050ffff04cc9cd79f7f8-en\">contact transformation</u> $i \\mathbf { \\dot { \\varphi } } \\phi ^ { * } \\mathbf { d } \\bar { \\lambda } = \\mathbf { d } \\lambda$ . Since $\\phi ^ { * } \\circ \\mathsf { d } = \\mathsf { d } \\circ \\phi$ there exists a function $F$ on $T ^ { * } M \\times \\mathbb { R }$ in the neighbourhood ofany point such that $\\phi ^ { * } { \\bar { \\lambda } } = \\lambda - \\mathrm { d } F$ . If we write the function $F$ as depending on the variables $q ^ { i }$ and $\\bar { q } ^ { i }$ , which is generally possible locally,",
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