Existence of a Hamiltonian system It is well known that a two dimensional system to first order is locally Hamiltonian from Darboux' theorem. For example, 
\begin{equation}
\dot x = f(x,y), \qquad \dot y = g (x,y) 
\end{equation}
Admits the following Poisson structure, 
\begin{equation}
\{x,x\}=\{y,y\} =0 , \qquad \{x, y\}= -\{y,x\} = F(x,y) 
\end{equation}
Where $F\neq 0$ and Hamilton's equations being, 
\begin{equation}
\dot x = F(x,y) \frac {\partial H}{\partial y}, \qquad \dot y =-F(x,y) \frac{\partial H}{\partial x}
\end{equation}
If now we have an $n$-dimensional system $\dot x_i=f(x_1,\dots , x_n)$ where $i=1,\dots , n$, can we in general give conditions for the admission of a Hamiltonian system? 
If I had a system and wanted to solve it's dynamics, is there a way I could test to see if it is Hamiltonian? By this I mean, let us assume I have a collection of variables and can monitor their time evolution in a computational experiment. Is there a way I can use the very powerful theory of Hamiltonian mechanics to someway solve my own system? i.e how can I take it beyond the use of $q$s and $p$s to solve my own problems! 
 A: There is the well-known condition for a system 
$$\dot{x}_i = f_i(x_1,\ldots, x_{2n})\:,\quad i =1,\ldots, 2n \tag{1}$$
to admit a local Hamiltonian re-formulation:
$$\dot{x}_i = \frac{\partial H}{\partial x_i}(x_1,\ldots, x_{2n})\:,\quad \dot{x}_k = -\frac{\partial H}{\partial x_k}(x_1,\ldots, x_{2n}) \quad i =1,\ldots, n\quad k=n+1,\ldots, 2n\tag{2}$$
Assuming the $f_i$ of class $C^1$ on the open set $\Omega \subset \mathbb R^{2n}$,  define the new functions
$$ F_j := \sum_{k=1}^{2n}S_{jk}f_k\quad j=1,\ldots, 2n $$
where 
$$S=\left[\begin{matrix}0 & -I\\ I & 0\end{matrix}\right]$$
and $I$ and $0$ are viewed as $n \times n$ submatrices (the identity and the zero matrix respectively).
Then (1) admits a local Hamiltonian reformulation of the form (2) if and only if, everywhere on $\Omega$,
$$\frac{\partial F_i}{\partial x_k} = \frac{\partial F_k}{\partial x_i}\quad i,k = 1,\ldots , 2n\:.$$
Obviously this is a very particular case where we also suppose that the coordinates $x_1,\ldots, x_{2n}$ are canonical. It is however possible that an Hamiltonian re-formulation of the initial system arises after having also changed the initial coordinates.
