If I have been given a graph of potential energy, how do I sketch typical trajectories in the phase plane?

Question

The particle is moving in the x-axis under the action of a conservative force with a potential
energy V (x).

Answer 1

One method starts with a choice for the total energy , E.   

Once E is selected, find a value of x for which V(x)  = E.     (call that value x₀ )

Then since p =  ± √2m  sqrt( E - V(x)),    a first point on the phase plot (p,x)  is  (0, x₀)

Since the argument of the square root cannot be allowed to be negative, the next points the next x value must be

either  to the left or right of x₀ and there will be two corresponding p values (± something)

This process can be continued to get additional points.  

 

Selecting different values of E will generate a family of trajectories.

 

Of course there might be more than one value of x for which V(x) = E.    Generally the trajectories started from different 

such points will join up.   A good example of this is the simple harmonic oscillator.