Can the system $\partial_x f(x,y) = \dot{y}$, $\partial_y f(x,y) = \dot{x}$ be related to some Hamiltonian system?

Question

If one has found some function $f(x,y): \partial_x f = \dot{y}, \partial_y f = \dot{x}$, is there a simple transformation or change of variables that results in Hamilton's equations $\partial_p H = \dot{q}, \partial_q H = - \dot{p}$? Is the first system in any way Hamiltonian or, more generally, symplectic?

Answer 1

As Jose Garcia said in the comments, the answer is no. The easiest way to see this is that a symplectic transformation has to be area preserving. Note that with your vector field $X = \partial_y f \frac{\partial}{\partial x} + \partial_x f \frac{\partial}{\partial y}$, $\dot f = L_X f = 2 f_x f_y$.

Consider then the following function: $f(x,y) = -xy$. Then, your vector field is $X = - x \partial_x - y \partial_y$. This gives you a radial compression of the plane, so the time $t$-flow of this vector field sends the unit disk onto the disk of radius $e^{-t}$. This is clearly not at all area preserving.