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        "quote": "$n$称为该约束系统的自由度数目。坐标$q ^ { i } \\ ( i = 1 , \\ldots , n )$通常称为该约束系统的<u class=\"concept-term\" data-concept-identity=\"urn:uuid:58e807c5-1ecd-41cb-b5cb-d975822e7d10\" data-concept-route=\"/concept-identities#concept-58e807c5-1ecd-41cb-b5cb-d975822e7d10\" id=\"first-d75b07f69fb525ba0ca3-zh\">广义坐标(generalized coordinate)</u>。它们甚至可用于无约束系统，此时$n = 3 N$且$V$是$ { \\mathbb { R } } ^ { 3 N }$的开子流形；在这种情况下，我们本质上是用一般坐标表示原来的牛顿动力系统。始终假定$( M , C )$是$\\mathbb { R } ^ { 3 N }$的嵌入子流形，从而切映射$C _ { * }$处处为单射。这意味着矩阵$[ \\partial \\mathbf { r } _ { a } / \\partial q ^ { i } ]$处处具有秩$n$（无临界点）。",
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        "quote": "and $n$ is called the number of degrees offreedom of the constrained system. The coordinates $q ^ { i } \\ ( i = 1 , \\ldots , n )$ are commonly called <u class=\"concept-term\" data-concept-identity=\"urn:uuid:58e807c5-1ecd-41cb-b5cb-d975822e7d10\" data-concept-route=\"/concept-identities#concept-58e807c5-1ecd-41cb-b5cb-d975822e7d10\" id=\"first-ec2c3e7e75ab34f57f10-en\">generalized coordinates</u> for the constrained system. They may be used even for an unconstrained system, in which $n = 3 N$ and $V$ is an open submanifold of $ { \\mathbb { R } } ^ { 3 N }$ ; in this case we are essentially expressing the original Newtonian dynamical system in terms of general coordinates. It will always be assumed that $( M , C )$ is an embedded submanifold of $\\mathbb { R } ^ { 3 N }$ , so that the tangent map $C _ { * }$ is injective everywhere. This implies that the matrix $[ \\partial \\mathbf { r } _ { a } / \\partial q ^ { i } ]$ has rank $n$ everywhere (no critical points).",
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