Constructing ODEs from a Hamiltonian

Question

I'm reading this book and came across this specific Hamiltonian in equation (5.2). My question is how to construct the corresponding system of ODEs explicitly. I am familiar with doing so in a simple 2D system where for instance $x'(t) = \partial H/\partial y$ and $y'(t) = -\partial H/\partial x$, but I get confused with a larger system like (5.2).

Answer 1

The general system of equations is, in vector notation ($q,p \in \mathbb{R}^N$) $$\begin{cases} q'=\nabla_p H \\ p'=-\nabla_q H \end{cases} $$ which means $$\begin{cases} q_i'=\partial H / \partial p_i \\ p_i'=-\partial H / \partial q_i \end{cases} $$ for every $i=1,..,N$.