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  "body": "\n## Summary\n\nThe tautological one-form is a canonical $1$-form on the cotangent bundle of any smooth manifold.\nIn coordinates $(q, p)$ it is $\\sum_i p_i \\, dq^i$, and it needs no choice of coordinates or metric.\nIts exterior derivative gives the canonical [[symplectic form|CAN490EPRT]] of the cotangent bundle, which makes every cotangent bundle a symplectic manifold.\nIt is also called the Liouville form or the canonical one-form.\n\n## Explanations\n\n### L1 human\n\nA point of the cotangent bundle is a pair: a position $q$ in the manifold and a covector $p$ at $q$, which in mechanics is a momentum.\nThe tautological form takes a small motion of such a pair, looks only at how the position moves, and feeds that motion to $p$.\nSo it uses the point itself as the instructions for what to measure, which is why it is called tautological.\nOn the cotangent bundle of the line, with coordinates $(q, p)$, it is $p \\, dq$.\n",
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