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  "body": "\n## Definition\n\nLet $(M, \\omega)$ be a [[symplectic manifold|CAN490EPRT]].\nFor a smooth function $f \\colon M \\to \\R$, let $X_f$ be its [[Hamiltonian vector field|C5EH3B4BEP]], so $\\iota_{X_f} \\omega = df$.\nThe *Poisson bracket* of smooth functions $f, g \\colon M \\to \\R$ is the smooth function\n$$\\{f, g\\} = \\omega(X_f, X_g).$$\n\n## Remarks\n\nEquivalently $\\{f, g\\} = df(X_g) = X_g(f) = -X_f(g)$.\nThe bracket is bilinear and antisymmetric, satisfies the Leibniz rule $\\{f, gh\\} = \\{f, g\\} h + g \\{f, h\\}$, and satisfies the Jacobi identity because $d\\omega = 0$.\nWith these conventions $X_{\\{f, g\\}} = -[X_f, X_g]$.\n\nExample: on $\\R^{2n}$ with $\\omega = \\sum_i dq_i \\wedge dp_i$,\n$$\\{f, g\\} = \\sum_{i=1}^n \\left( \\frac{\\partial f}{\\partial q_i} \\frac{\\partial g}{\\partial p_i} - \\frac{\\partial f}{\\partial p_i} \\frac{\\partial g}{\\partial q_i} \\right).$$\nFor instance $X_{p_j} = \\partial / \\partial q_j$, so $\\{q_i, p_j\\} = X_{p_j}(q_i) = \\delta_{ij}$, while $\\{q_i, q_j\\} = \\{p_i, p_j\\} = 0$.\n\nNon-example: on $\\R^4$ the closed $2$-form $dq_1 \\wedge dp_1$ is degenerate, so it does not define a bracket in this way: the function $q_2$ has no Hamiltonian vector field for it.\n\nAlong an integral curve $\\gamma$ of $X_H$, a function $g$ satisfies $\\frac{d}{dt} g(\\gamma(t)) = \\{g, H\\}(\\gamma(t))$.\nSome authors use the opposite sign convention for $\\{f, g\\}$.\n",
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