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  "body": "\n## Definition\n\nLet $Q$ be a [[smooth manifold|F45VQ47GPV]] with cotangent bundle $\\pi \\colon T^*Q \\to Q$.\nA point of $T^*Q$ is a pair $(q, p)$ with $q \\in Q$ and $p \\in T_q^* Q$.\nThe *tautological one-form* is the [[$1$-form|0E3G0GRPZ6]] $\\theta$ on $T^*Q$ given by\n$$\\theta_{(q, p)}(v) = p\\big(d\\pi_{(q, p)}(v)\\big) \\quad \\text{for } v \\in T_{(q,p)}(T^*Q),$$\nwhere $d\\pi$ is the [[differential|6P9PXQDSEN]] of $\\pi$.\n\n## Remarks\n\nIn local coordinates $q^1, \\ldots, q^n$ on $Q$ with the induced coordinates $p_1, \\ldots, p_n$ on the fibres, $\\theta = \\sum_i p_i \\, dq^i$.\nThe form $\\omega = -d\\theta = \\sum_i dq^i \\wedge dp_i$, with $d$ the [[exterior derivative|004FRCXH6M]], is the canonical [[symplectic form|CAN490EPRT]] of $T^*Q$; some authors use the opposite sign.\n\nCharacterization: for every $1$-form $\\alpha$ on $Q$, viewed as a section $\\alpha \\colon Q \\to T^*Q$, one has $\\alpha^* \\theta = \\alpha$, and $\\theta$ is the only $1$-form on $T^*Q$ with this property.\n\nExample: for $Q = \\R$ with coordinate $q$ and fibre coordinate $p$, $\\theta = p\\, dq$; for $\\alpha = c\\, dq$ the section is $q \\mapsto (q, c)$ and $\\alpha^*(p\\, dq) = c\\, dq = \\alpha$.\n\nNon-example: the form $q\\, dp$ on $T^*\\R$ is not the tautological form.\nFor the section $\\alpha = dq$, that is $q \\mapsto (q, 1)$, the pullback of $q\\, dp$ is $q \\cdot d(1) = 0 \\ne dq$.\n",
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