## symplectic mappings

Let the symplectic form $\omega$ be defined on the finite-dimensional vector space $\mathcal{D}$ (see Sec. 1.4 in [Tao06]).
**Definition** (Symplectic mapping, see [Hai10]) A differentiable map $u:\mathcal{D}\mapsto\mathcal{D}$ is called symplectic if
$$ \omega((\grad u) \xi, (\grad u) \eta) = \omega(\xi,\eta),$$
for all $\xi,\eta\in\mathcal{D}$.
Let $\phi(t,u_0)$ is the solution of the Hamiltonian flow $\p_t\phi(t)=\grad_\omega H(\phi(t))$ with initial condition $\phi(0,u_0)=u_0$. Then we have
**Theorem 1** (Poincar$\acute{\text{e}}$ 1899) If $H$ is a twice continuously differentiable function, then for each fixed $t$, the flow $\phi_t(u_0)=\phi(t,u_0)$ is a symplectic mapping.
**//Proof//** Since $\omega$ is anti-symmetric, then direct calculation shows that
$$\f{d}{dt} \omega( \grad\phi_t \xi, \grad\phi_t \eta ) = 0,$$
by observing that $\phi_t$ satisfies the Hamiltonian equation. Therefore
$$\omega( \grad\phi_t \xi, \grad\phi_t \eta )= \omega( \grad\phi_0 \xi, \grad\phi_0 \eta ) = \omega(\xi,\eta).$$
The inverse of **Theorem 1** is also true, see **Theorem 3** in [Hai10] where the author showed it in the special case when $\omega$ is the standard symplectic form in classical mechanics. For general symplectic form, we only need to notice that
$$\omega(u,v)= u^t\Omega v,\hspace{6pt}\text{with }\Omega=(\omega(e_i,e_j))_{ij},$$
where $\Omega$ is anti-symmetric ($\Omega = J$ when we talk about the standard symplectic form in classical mechanics). Note that the following Integrability Lemma plays an essential role in proving **Theorem 3** in [Hai10].
**Lemma 1** (Integrability Lemma, see [Hai10]) Let $f:\mathcal{D}\mapsto\mathcal{D}$ be continuously differentiable, and assume that $\grad f(y)$ is symmetric for all $y\in\mathcal{D}$. Then for every $y_0\in\mathcal{D}$ there exists a neighbourhood and a function $H(y)$ such that
$$f(y) = \grad_\omega H(y)$$
on this neighbourhood.
**Remark 1** In our case, the function $H$ can be globally defined.
=== References:
[Tao06] Terence Tao, Nonlinear dispersive equations: local and global analysis, CBMS regional conference series in mathematics, July 2006.
[Hai10] Ernst Hairer, [[http://www.unige.ch/~hairer/poly_geoint/week1.pdf|Lecture 1: Hamiltonian systems]]