Let $(M,\omega)$ be a symplectic manifold. Let $(\varphi_t)_{t\in[0,1]}$ be a smooth one-parameter family of diffeomorphisms $\varphi_t:M\to M$ such that $\varphi_0=\operatorname{id}_M$. Define the time-dependent vector field $X_t\in\mathfrak X(M)$ by