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  "body": "\n## Summary\n\nOn a [[symplectic manifold|CAN490EPRT]], every smooth function $H$ determines a [[vector field|0299VCW1TH]] $X_H$, its Hamiltonian vector field.\nThe symplectic form turns the differential $dH$ into a tangent vector at each point.\nThe flow of $X_H$ is the motion of a [[Hamiltonian system|0EDKGA2FAK]] with energy $H$, so this notion carries classical mechanics onto curved phase spaces.\n\n## Explanations\n\n### L1 human\n\nThink of $H$ as the energy of a mechanical system, written as a function of position and momentum.\nThe gradient of $H$ points to where the energy grows fastest.\nThe Hamiltonian vector field is the gradient turned by a quarter turn, so it points along the level sets of $H$ instead of across them.\nFor $H(q, p) = \\frac{1}{2}(q^2 + p^2)$ on the plane, the field is $X_H = p\\,\\partial_q - q\\,\\partial_p$.\nIts flow turns the plane around the origin and keeps the energy constant, which is the motion of a harmonic oscillator.\n",
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