I am currently doing the following exercise in the book Modern Geometry - Methods and Applications Part I by Dubrovin, Fomenko, and Novikov.
Exercise 33.4.1: Consider the differential equation \begin{align}\label{1}\tag{1} \dot{u}=\frac{\partial}{\partial x} \frac{\delta S}{\delta u}, \end{align} on the space of functions $u(x)$ periodic with period $T$, and satisfying $$ \int_{x_0}^{x_0+T} u(x) d x=0, $$ where $S=S[u]$ is a functional of the form $$ S[u]=\int_{x_0}^{x_0+T} L\left(u, u^{\prime}, \ldots\right) d x . $$ Transform the differential equation into standard Hamiltonian form. (Hint. Work with the Fourier coefficients $u_n$ of the function $u$, determined by $u=$ $\sum_{n=-\infty}^{\infty} u_n e^{(2 \pi i n x) / T}$.)
First, I think \eqref{1} is $$ u'(x)=\frac{d}{dx} \left(\frac{\delta S}{\delta u}(u(x),u'(x),u''(x),\ldots)\right). $$ Then, the equation is equivalent to \begin{align}\tag{2}\label{2} u(x)=\frac{\delta S}{\delta u}(u(x),u'(x),u''(x),\ldots)+C \end{align} for some constant $C$.
Question 1: I wonder if my understanding is correct. The exercise write $\partial / \partial x$ instead of $d/dx$.
Pretend that what I understand is correct. Then, I have the following attempts:
The space we consider is identified as $\ell^{2}$ space with elements $(u_{n})_{n\in\mathbb{Z}}$ (Note that $u_{0} = 0$ and $u_{-n} = \overline{u_{n}}$.) via the Fourier expansion.
Then, I plug the Fourier expansion $u=\sum u_ne^{(2\pi inx)/T}$ into the differential equation \eqref{2} and compare the coefficient of $e^{(2\pi inx)/T}$. The LHS of \eqref{2} is easy. However, I have no idea how to deal with the RHS of \eqref{2} (instead of brute Force expansion and calculation).
Question 2: How to deal with the RHS of \eqref{2} after plug in $u=\sum u_ne^{(2\pi inx)/T}$.
Alternatively, I think we may directly transform the ODE \eqref{2} into a system of $1$-st order Hamiltonian equations. That is, I need to find the Hamiltonian $H(p_1,\ldots,q_1,\ldots)$ such that \begin{align} q_{i}'(x) = \dfrac{\partial H}{\partial p_i}(p(x),q(x))\quad p_{i}'(x) = -\dfrac{\partial H}{\partial q_i}(p(x),q(x)) \end{align} I attempt to consider $H(p_i,q_i) = \sum p_i q_{i} + \{\text{blabla}\}$ and set $q_{i} = u^{(i)}(x)$. However, I cannot find the correct $H$ and $p_{i}s$.
Question 3: How to find $H$ and $p_{i}$ so that the ODE \eqref{2} transform into the Hamilton type equation.