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A Hamiltonian function is a smooth function whose differential determines a <Hamiltonian vector field>.

Hamiltonian function

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{title2=$\operatorname{Ham}(M,\omega)$}

A Hamiltonian isotopy is the flow of a time-dependent <Hamiltonian vector field>. Its time-one map is a Hamiltonian diffeomorphism.

= Hamiltonian diffeomorphism
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Hamiltonian isotopy

Again take a symplectic two-sphere, but let $L$ be a small latitude bounding a cap of area strictly below half the total area. A rotation carrying that cap to a disjoint cap carries $L$ to a disjoint Lagrangian circle. Rotations of the symplectic sphere are flows of <Hamiltonian vector fields>; for rotation about an axis, a height function is a <Hamiltonian function>. Thus this rotation is a <Hamiltonian isotopy>, and $L$ is Hamiltonian displaceable.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 146 / 2 / iv / Solution

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{title2=$X_H$}
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For a smooth function $H_t:M\to\mathbb R$, the Hamiltonian vector field is defined by $\iota_{X_{H_t}}\omega=-dH_t$, with the sign depending on convention.


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 Incoming links: Hamiltonian vector field