I'm reading article about Hamiltonian's system. They defined Hamiltonian's equations as a matrix $$ \mathbf{\dot{\eta}} = \mathbf{J} \frac{\partial H(\mathbf{\eta},t)}{\partial \mathbf{\eta}}, $$ where $\mathbf{J}$ is standard symplectic matrix $$ \mathbf{J} = \begin{bmatrix} 0 & \mathrm{1}\\ \mathrm{-1} & 0 \end{bmatrix}. $$ Later, they said that state-transition matrix $\Phi$ for this equation is $$ \mathbf{\dot{\Phi}} = \mathbf{J} \frac{\partial ^2H}{\partial \mathbf{\eta^2}} \mathbf{\Phi}$$ I read something about state-transition matrix but I have no idea how the second equation came from the first one. Can somebody explain it?
Thanks for help!