Existence of a Hamiltonian system

Question

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!

Answer 1

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.