== Section 1.4 in [Tao06]: === Exercise 1.32 Since $\p_t u = \grad_\omega H$, we have $$$$\begin{split} \f{d\omega(\p_x u,\p_y u)}{dt} &= \omega(\p_x\grad_\omega H, \p_y u) +\omega(\p_x u, \p_y\grad_\omega H)~\\ &=\p_x \omega(\grad_\omega H,\p_y u) + \p_y\omega(\p_x u, \grad_\omega H)~\\ &=\p_x(dH\cdot \p_y u) - \p_y(dH\cdot\p_x u)~\\ &=0. \end{split}$$$$ where the second equality results from that $\omega$ is anti-symmetric, and the equality is by definition. === Exercise 1.35 By definition, we first have $$$$\begin{split} \{ H,E\}&=\omega(\grad_\omega H, \grad_\omega E)=(dH\cdot\grad_\omega E)~\\ &=-\omega(\grad_\omega E, \grad_\omega H) = -(dE\cdot\grad_\omega H). \end{split}$$$$ We then calculate $$$$\begin{split} \{ H_1, \{H_2, H_3\}\} & = -d(dH_2\cdot\grad_\omega H_3)\cdot\grad_\omega H_1 ~\\ &=-[d^2H_2\cdot\grad_\omega H_3 + dH_2\cdot d\grad_\omega H_3]\cdot \grad_\omega H_1, \end{split}$$$$ Similarly, $$\begin{split} \{ H_3, \{H_1, H_2\}\}=-\{ H_3, \{H_2, H_1\}\}=[d^2H_2\cdot\grad_\omega H_1 + dH_2\cdot d\grad_\omega H_1]\cdot \grad_\omega H_3. \end{split}$$ Adding the above two identities gives $$$$\begin{split} \{ H_1, \{H_2, H_3\}\} + \{ H_3, \{H_1, H_2\}\} &= - [d^2H_2\cdot\grad_\omega H_3]\cdot \grad_\omega H_1 + [d^2H_2\cdot\grad_\omega H_1]\cdot \grad_\omega H_3 ~\\ &=-dH_2\cdot \omega(\grad_\omega (\grad_\omega H_3), \grad_\omega H_1) + dH_2\cdot \omega( \grad_\omega (\grad_\omega H_1), \grad_\omega H_3 ) ~\\ &=dH_2\cdot \grad_\omega\{H_1,H_3\}~\\ &=-\{ H_2, \{H_3, H_1\}\}, \end{split}$$$$ which verifies the Jacobi identity. === References: [Tao06] Terence Tao, Nonlinear dispersive equations: local and global analysis, CBMS regional conference series in mathematics, July 2006.