2. (5) Marion & Thornton, problem 4-2. Draw the phase space diagram (xdot vs. x) right below the potential energy plot, as is done in Figure 4-5 in the book. This way one can see the correspondence between the two diagrams.
 
 

3. (5) Marion & Thornton, problem 4-6. Use conservation of energy to calculate theta-dot as a function of theta.
 
 

4. (10) Marion & Thornton, problem 4-10. To do this problem you will need to integrate the nonlinear differential equation using a computer. Download my Mathematica file to get started:  Mathematic File for Nonlinear Oscillator Simulation  It contains one simulation showing chaotic behavior and another one showing periodic behavior, both for a drive frequency of 0.5 in dimensionless units.  The only difference between the two simulations is the value of the drive amplitude.

After you finish the specific simulations asked for in problem 4-10, I encourage you to play around with the three adjustable parameters: drive amplitude, drive frequency, and damping. The driven damped pendulum is a very rich system with some surprising behavior.

Note the following points:

1) The differential equation and initial conditions are enclosed together inside a curly bracket.

2) The differential equation and initial condition equations contain double equal signs.

3) The funny set of symbols in the Evaluate function /.% is telling Mathematica to evaluate the function that was calculated in the previous step. (The % sign always refers to the previous step.)

4) If you ask Mathematica to go to very long times, it may give you an error message. I circumvented this by increasing the maximum number of steps allowed, with the MaxSteps option.  The number of steps needed varies depending on the parameters in the equation.

5) To make the phase space plots, I had to define both the solution y[t] and its first derivative y'[t].  To do this I gave the solution to the differential equation a name ("solution"), then used the Evaluate function.  I don't understand why I needed the [[1]] at the end of that line.  Apparently Mathematica thinks of the solution as a vector or matrix, and we are asking for the first component of that vector (or first column of the matrix).  If one of you understands this well, you can explain it to me.