Gradient system but not Hamiltonian

Question

We know that Hamiltonian is gradient system if and only if $H$ is harmonic. So we can easily find an example that is Hamiltonian but not gradient. But this proposition does not say every gradient system is Hamiltonian. Is there any examples?

Answer 1

It's pretty easy to cook up such example. Take $V(x, y) = -\frac{1}{2}(x^2+y^2)$ and consider the system $\dot{x} = \frac{\partial V}{\partial x} = -x, \; \dot{y} = \frac{\partial V}{\partial y} = -y$. This system is gradient, but it can't be a Hamiltonian or even a conservative system: having asymptotically Lyapunov stable equilibrium (namely, the one at the origin) forbids having non-trivial first integral.