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# Hamiltonian vector field

Let $(M,\omega)$ be a symplectic manifold, and $\tilde{\omega}:TM\to T^{*}M$ be the isomorphism from the tangent bundle to the cotangent bundle

$X\mapsto\omega(\cdot,X)$ |

and let $f:M\to\mathbb{R}$ is a smooth function. Then $H_{f}=\tilde{\omega}^{{-1}}(df)$ is the Hamiltonian vector field of $f$. The vector field $H_{f}$ is symplectic, and a symplectic vector field $X$ is Hamiltonian if and only if the 1-form $\tilde{\omega}(X)=\omega(\cdot,X)$ is exact.

If $T^{*}Q$ is the cotangent bundle of a manifold $Q$, which is naturally identified with the phase space of one particle on $Q$, and $f$ is the Hamiltonian, then the flow of the Hamiltonian vector field $H_{f}$ is the time flow of the physical system.

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## Mathematics Subject Classification

53D05*no label found*

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