Please provide a clear detailed explanation (with examples) of the meaning and calculation of phase space and phase diagram.
Also solve the following problem:
A mass m moves in one dimension and is subjected to a constant force +F1 when
x<0 and to a constant force -F1 when x>0.
Find the following:
a) construct a phase diagram
b) calculate the period of the motion in terms of
m, F1, and Amplitude (disregard damping).
I'll explain the phase diagram by looking at the problem using energy considerations.
First, let's find the potential energy that will yield the correct force. The force is minus the derivative of the potential energy, so you find that:
V(x) = F1 Abs(x) (1)
If the amplitude is A then if the particle is at x = A or x = -A its velocity will be zero. This means that the total energy, E, must be:
E = V(A) = F1 A (2)
At any position in between -A and A its kinetic energy is:
E_kin = E - V(x) = F1[A - Abs(x)] (3)
The phase diagram is the plot of the momentum of the particle as a function of position (or vice versa). You can write the kinetic energy as:
E_k = 1/2 m v^2 = p^2/(2m) (4)
Using (4) and (3) you get:
p = plus or minus sqrt[2m F1 [A - Abs(x)]] (5)
Now both plus and minus signs will have to be included in the graph. If you follow the motion of the particle, you see that it goes from -A to A and then back to -A. So, it reaches all the points in between with a positive velocity and with a negative velocity. At A and -A the momentum is zero so at these points the two graphs meet.
If you follow the motion of the particle on the phase diagram graph, starting at -A then you move to the point +A along the "upper part" (the plus sign in Eq. (5)) because the momentum is positive. When you reach +A the momentum is zero again and then you return to -A via the lower part of the graph ...
A detailed solution is given. We also discuss the concept of phase-space in statistical physics.