I am reading Zworski's book on semiclassical analysis and I can't make sense of the following opening paragraph the of chapter 11: Quantum And Classical Dynamics.
We assume for this section that $\{q_t\}_{0 \leq t \leq T}$ denotes a family of smooth symbols that vanish outside some fixed, bounded open set $U_0$. When $q = q_t$ is independent of $t$, the flow $\varphi_t = \mathrm{exp}(tH_q)$ induces a smooth family $\{\kappa_t\}_{0\leq t\leq T}$ of symplectomorphisms of $\mathbb{R}^{2n}$, which equal the identity outside $U_0$ and satisfy $\partial_t\kappa_t = H_q = (\kappa_t)_*H_q$, where the last equality follows from Jacobi's Theorem.
The given Jacobi's Theorem reads as:
If $\kappa$ is a symplectomorphism, then $H_f = \kappa_*(H_{\kappa^*f})$.
where $\kappa^*(f)$ is the pullback of $f$ by $\kappa$ and $\kappa_*f$ is the pushforward of $f$ by $\kappa$.
What I don't understand is the equality $\partial_t\kappa_t = H_q = (\kappa_t)_*H_q$, as the Jacobi's Theorem establishes a connection between the Hamiltonian of a function $f$ and the pushforward version of the Hamiltonian of a pullback version of $f$. But the said equality states that the Hamiltonian of a smooth symbol is the pulledback version of a Hamiltonian of the same smooth symbol. Moreover, in the proof of Jacobi's Theorem, the $\kappa^*f$ plays a crucial role due to how contractions and differentials are related.
In fact, $\kappa^*(H_f)\lrcorner \sigma = \kappa^*(H_f)\lrcorner \kappa^*\sigma = \kappa^*(H_f\lrcorner \sigma) = \kappa^*(df) = -d(\kappa^* f) = H_{\kappa^*f}\lrcorner \sigma$
where $\lrcorner$ is the contraction operator and $\sigma$ the symplectic product of $\mathbb{R}^{2n}$.
Therefore, should $H_q$ be in fact $H_{\kappa^*q}$ in the equality stated by Zworski?