The Hamiltonian may depend on time as well as the phase coordinates but for Newtonian systems in which the force is conservative and time independent the the Hamiltonian is the total energy and is conserved.
An autonomous Hamiltonian system has a Hamiltonian function that is independent of time,
This function takes a constant value along any phase curve. We can show this by differentiating
with respect to
If the curve is a phase curve then Hamilton's equations are satisfied:
and
hence
Conversely if the system is conservative then the Hamiltonian is autonomous, since
Hence
and the Hamiltonian is autonomous.
Sinceis conserved,
a constant, defines a curve in phase space. It is enough, when drawing the phase diagram for a Hamiltonian system, to draw a selection of curves
for various constants
The difference
for any two phase curves will be constant throughout the phase plane.
Once the phase curve is drawn we can find the speed of the system on a phase curve by finding
If necessary a velocity field can be drawn using the velocity vector field.