I have some difficult of dealing with time dependent vector fields and I am studying hamiltonian symplectomorphisms.
I have two questions
1) If $\psi_t$ is a Hamiltonian isotopy genereted by $H_t$ and $\phi_t$ is a hamiltonian isotopy generated by $G_t$ so, $\psi_t\circ\phi_t$ is a hamiltonian isotopy generated by $H_t + G_t\circ \psi_t^{-1}$.
2) $\psi_t^{-1}$ is a hamiltonian isotopy generated by $-H_t\circ\psi_t$.
Thank you!
I am doing just 2), 1) is likely to be very similar.
2) $\psi_t^{-1} \circ \psi_t=Id$.
Differentiate wrt $t$: $$\frac{d\psi_t^{-1}}{dt}(\psi_t(x))+(d_{\psi_t(x)}(\psi_t^{-1}))(\frac{d\psi_t}{dt}(x))=0_{T_xM}. $$
In other words, up to conventions, (ie assuming that $\omega(\frac{d\psi_t}{dt}(y),\cdot)=d_{\psi_t(y)}H_t$), $$\omega((\frac{d\psi_t^{-1}}{dt})(\psi_t(x)),\cdot)=-\omega((d_{\psi_t(x)}\psi_t^{-1})(\frac{d\psi_t}{dt}(x)),\cdot)=-\omega((d_x\psi_t)^{-1}\frac{d\psi_t}{dt}(x),\cdot)=\omega(-\frac{d\psi_t}{dt}(x),(d_x\psi_t)(\cdot)),$$ because $\psi_t$ is a symplectomorphism.
This can be rewritten as $$d_{\psi_t^{-1}(\psi_t(x))}(-d_{\psi_t(x)}(-H_t \circ \psi_t)=d_x{\psi_t}=\omega(\frac{d\psi_t^{-1}}{dt})(\psi_t(x)),\cdot).$$
In other words, $d_{\psi_t^{-1}(y)}(-H_t \circ \psi_t)=\omega(\frac{d\psi_t^{-1}}{dt}(y),\cdot)$, QED.
@Mindlack thank you very much.
The first case will be something like this?
Differentiating $\psi_t\circ\phi_t$ with respect to $t$, we have \begin{equation} \frac{d}{dt}\psi_t\circ\phi_t = \frac{d}{dt}\psi(\phi_t(x) ) + (d\psi_t)_{\psi_t\circ\phi_t}\left(\frac{d}{dt}\phi_t(x)\right). \end{equation}
So, \begin{eqnarray} i_{\frac{d}{dt}(\psi_t\circ\phi_t)}\omega &=& i_{\frac{d}{dt}\psi_t(\phi_t(x) ) + (d\psi_t)_{\psi_t\circ\phi_t}\left(\frac{d}{dt}\phi_t(x)\right)}\omega \\ &=& i_{\frac{d}{dt}\psi_t(\phi_t(x) )}\omega + i_{(d\psi_t)_{\psi_t\circ\phi_t}\left(\frac{d}{dt}\phi_t(x)\right)}\omega \\ &= &dH_t + i_{\frac{d}{dt}\phi_t(x)}\omega \\ &=& dH_t + dG_t\circ\psi_t^{-1} \end{eqnarray}
And we use the fact that $\psi_t^{-1}$ is a symplectomorphism.