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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.

Translation by a differential one-form $\beta$ is $(x,\xi )\mapsto (x,\xi +{\beta }_x)$. It changes the canonical one-form on a cotangent bundle by ${\pi }^{\ast }\beta$. A closed differential form gives a symplectomorphism; an exact differential form gives a Hamiltonian diffeomorphism when compact support is not required.

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.

A Hamiltonian isotopy is a smooth path of symplectomorphisms ${\varphi }_t$, with ${\varphi }_0=\mathrm{id}$, generated by a smooth time-dependent Hamiltonian function. In the convention ${\iota }_{X_{H_t}}\omega =-d{H}_t$, it satisfiesThe flow must exist throughout the chosen time interval. This is automatic on a closed symplectic manifold; on an open one, uniformly compactly supported Hamiltonians also suffice. The Hamiltonian flow preserves the symplectic form by Cartan's magic formula.

Its time-one map is a Hamiltonian diffeomorphism, an element of the group of Hamiltonian diffeomorphisms. A path and its endpoint are different data: a full rotation of the symplectic plane gives a nonconstant Hamiltonian path whose endpoint is the identity. Distinct paths can therefore represent the same Hamiltonian diffeomorphism.

Write the exact differential form as $\alpha =du$ for a smooth real-valued function $u$ on $N$. On $T^{\ast }N$ take the Hamiltonian function $H=u\circ \pi$. With $\omega ={\sum }_{j}d{q}_j\wedge d{p}_j$ and ${\iota }_{X_H}\omega =-dH$, the Hamiltonian vector field isIts complete flow is explicit:The base point stays fixed, so this flow exists for all real $t$, even if $u$ is unbounded on a noncompact base. Its time-one map is exactly $f$, proving that the cotangent fiber translation is a Hamiltonian diffeomorphism in the unrestricted complete-flow group.

This construction also justifies the hint for this family, without treating exactness as a general substitute for a Hamiltonian isotopy. For any Hamiltonian isotopy ${\psi }_t$ on this cotangent bundle, Cartan's magic formula, $d\theta =-\omega$ and ${\iota }_{X_{H_t}}\omega =-d{H}_t$ giveIntegrating proves that ${\psi }_1^{\ast }\theta -\theta$ is exact. For the present translation this difference is ${\pi }^{\ast }\alpha$, which is exact if and only if $\alpha$ is exact: one direction is pullback of $du$, and the other is pullback by the zero section. Necessity therefore follows from the differential identity, while sufficiency follows from the complete flow just constructed.

Compact support is a separate issue. If $du$ is nonzero at some $q$, the map moves every point of the entire unbounded fiber over $q$, so it is not a member of the compactly supported Hamiltonian diffeomorphism group. For example $N=\mathbb{R}$, $u(q)=q$ gives $(q,p)\mapsto (q,p+1)$. Thus the claim about this global translation uses the full Hamiltonian group, not ${\mathrm{Ham}}_c$; its flow can be cut off to agree on a prescribed compact swept set, but such a cutoff does not give the same map on the entire cotangent bundle.

Write the exact differential form as $\beta =dh$ and take the cotangent fiber translation family ${\rho }_t(x,\xi )=(x,\xi +td{h}_x)$. Its generating vector field is vertical and in local coordinates isWith $\omega ={\sum }_{j}d{x}_j\wedge d{\xi }_j$ and the Hamiltonian vector field convention from Question 3,Thus ${\rho }_t$ is a Hamiltonian isotopy and $f={\rho }_1$ is a Hamiltonian diffeomorphism. The local flow is explicitly defined for every real $t$, regardless of compactness. With the alternative convention ${\iota }_{X_H}\omega =-dH$, the same family is generated by $H=h\circ \pi$.

The definition in the question does not require compact support for the Hamiltonian function; the displayed Hamiltonian diffeomorphism uses that definition.