Local Hamiltonian evolution preserves the symplectic form

In local coordinates write $\omega =\frac{1}{2}{\omega }_{ab}d{x}^a\wedge d{x}^b$, $X=X^c{\mathrm{\partial }}_c$, and ${\mathrm{\Phi }}_t={\mathrm{\Phi }}_{t,s}$. Differentiating $({\mathrm{\Phi }}_t^{\ast }\omega {)}_{ij}={\omega }_{ab}({\mathrm{\Phi }}_t){\mathrm{\partial }}_i{\mathrm{\Phi }}_t^a{\mathrm{\partial }}_j{\mathrm{\Phi }}_t^b$ and using ${\mathrm{\partial }}_t{\mathrm{\Phi }}_t^a=X^a({\mathrm{\Phi }}_t)$ and ${\mathrm{\partial }}_t{\mathrm{\partial }}_i{\mathrm{\Phi }}_t^a=({\mathrm{\partial }}_c{X}^a)({\mathrm{\Phi }}_t){\mathrm{\partial }}_i{\mathrm{\Phi }}_t^c$ gives ${\mathrm{\partial }}_t({\mathrm{\Phi }}_t^{\ast }\omega {)}_{ij}=(X^c{\mathrm{\partial }}_c{\omega }_{ab}+{\omega }_{cb}{\mathrm{\partial }}_a{X}^c+{\omega }_{ac}{\mathrm{\partial }}_b{X}^c)({\mathrm{\Phi }}_t){\mathrm{\partial }}_i{\mathrm{\Phi }}_t^a{\mathrm{\partial }}_j{\mathrm{\Phi }}_t^b\mathrm{.}$ The bracket is the coordinate expression for the Lie derivative ${\mathcal{L}}_X\omega$.

In the same coordinates,

and

Adding them cancels $-X^k{\mathrm{\partial }}_j{\omega }_{ki}$ with $X^k{\mathrm{\partial }}_j{\omega }_{ki}$ and cancels $X^k{\mathrm{\partial }}_i{\omega }_{kj}$ with $X^k{\mathrm{\partial }}_i{\omega }_{jk}$ because ${\omega }_{kj}=-{\omega }_{jk}$. The remaining terms are $X^k{\mathrm{\partial }}_k{\omega }_{ij}+({\mathrm{\partial }}_i{X}^k){\omega }_{kj}+({\mathrm{\partial }}_j{X}^k){\omega }_{ik}$, exactly the coordinate expression in step 1.1. Hence ${\mathcal{L}}_X\omega =d({\iota }_X\omega )+{\iota }_{X}d\omega$. Since ${\iota }_X\omega =-d{H}_t$ and $d\omega =0$ by [F1], both terms vanish and ${\mathcal{L}}_X\omega =0$.

Step 1.1 now gives ${\mathrm{\partial }}_t{\mathrm{\Phi }}_{t,s}^{\ast }\omega =0$ on the local evolution domain. At $t=s$, ${\mathrm{\Phi }}_{s,s}$ is the identity, so the pullback is $\omega$ throughout that domain.