Week Unknown: "Open" Problems

The four problems on which I want to work when I have time:

On the circle, S¹, with its standard Riemannian metric it inherits as a subspace of ℝ², find a second-order ODE that is

1. gradient but not Hamiltonian
2. Hamiltonian but not gradient
3. both gradient and Hamiltonian ("harmonic") (I think I have a heuristic proof there are no non-trivial ones)
4. neither gradient nor Hamiltonian