Let $(M,\omega)$ be a symplectic manifold, and let $\varphi:[0,1]\times M\to M$ be a smooth symplectic isotopy with $\varphi_0=\operatorname{id}_M$. Let $X_t\in\mathfrak X(M)$ be its time-dependent generating vector field, defined by
the isotopy $(\varphi_t)_{t\in[0,1]}$ is Hamiltonian if and only if there exists a smooth map $H:[0,1]\times M\to\mathbb R$, with $H_t:M\to\mathbb R$ given by $H_t(x)=H(t,x)$, such that