Displacement energy (symplectic geometry, $e(A)={\inf }_{\varphi (A)\cap A=\varnothing }{d}_H(\mathrm{id},\varphi )$)

The displacement energy of a subset of a symplectic manifold is the infimum of the energy norms of compactly supported Hamiltonian diffeomorphisms moving it off itself. The infimum is $+\infty$ if no such map exists. One must use the same chosen norm convention in both this definition and any energy lower bound. A compact set can have zero displacement energy even when every nonidentity map has positive norm.

Use the standard symplectic form ${\sum }_{j}d{q}_j\wedge d{p}_j$ and convention ${\iota }_{X_H}\omega =-dH$. The Hamiltonian $H=\delta q_n$ translates $p_n$ by $\delta$. A fixed compactly supported cutoff equal to one near the set swept for $|\delta |\le 1$ leaves that translation unchanged on the original compact set. Its supremum norm is at most $C|\delta |$, so the translation displaces the set at arbitrarily small cost. This proves zero Hamiltonian displacement energy for either Hofer normalization.