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Choose coordinates $(q_1,\dots ,q_n,p_1,\dots ,p_n)$ with standard symplectic form ${\omega }_0={\sum }_{j}d{q}_j\wedge d{p}_j$, and regard the given sphere $S$ as lying in $\{p_n=0\}$. With ${\iota }_{X_H}{\omega }_0=-dH$, the linear Hamiltonian function $H_{\delta }=\delta q_n$ has Hamiltonian vector field $\delta {\partial }_{p_n}$ and translates in the missing coordinate.

To make this a permissible compactly supported map, take a smooth cutoff $\chi$ equal to one on an open neighbourhood of the fixed compact swept setChoose $\chi$ with compact support and let $R={\sup }_{\mathrm{supp}\chi }|q_n|<\infty$. For $0<\delta <1$ put $G_{\delta }=\delta q_n\chi$. On the whole swept region the cutoff is locally constant, so its derivatives vanish and $X_{G_{\delta }}=\delta {\partial }_{p_n}$. Thus the flow carries every $z\in S$ to $z+t\delta e_{p_n}$ for $0\le t\le 1$.

The time-one image lies in $\{p_n=\delta \}$ and is disjoint from the original sphere. Its Hofer metric from the spatial supremum norm obeysIn the oscillation convention the upper bound is $2R\delta$. Since $\delta$ can approach zero while the cutoff and $R$ stay fixed,This proves the zero displacement energy for a compact set in a hyperplane; it does not assert that any one nonidentity displacing map has zero norm.