If I have some autonomus hamiltonian, then it is a constant of the motion, and all the trajectories will lie on constant set levels.
For example, for the double well potential problem
$$H(x,p) = p^2-\frac{1}{2}x^2+\frac{1}{4}x^4$$
the equations of motion will be given by hamilton's equations
$$\frac{\partial H(x,p)}{\partial p} = \dot{x} = 2p$$ $$-\frac{\partial H(x,p)}{\partial x} = \dot{p} = x-x^3$$
and the solutions to these ODE's will be trajectories that will lie on the curves
How will the equations of motion change when the hamiltonian turns non-autonomous?
$$H(x,p,t)=p^2-\frac{1}{2}x^2+\frac{1}{4}x^4+\epsilon t$$
will the trajectories lie on constant surface levels?