 Compactly supported Hamiltonian translation

Given a bounded set in the symplectic plane and a translation vector $v=(a,b)$, the linear Hamiltonian $H(x,y)=bx-ay$ generates translation by $v$. Multiplying it by a cutoff that equals one on the entire swept region produces a compactly supported Hamiltonian diffeomorphism that agrees with the translation on the set.