Homework 07: due Thursday, December 7, 2017 (HW07)

1. Consider the double pendulum. For the naming conventions and governing equations look at this web site. It's easiest to use the Hamiltonian form (equations 24-27), where the independent variables are (t1,t2,p1,p2), i.e., the angles and the generalized momenta. Write a fortran code that integrates the equations from some time t=0 to t=TF. The other parameters are M1, M2, L1, L2, and g=9.81. Write the code so you can use either Euler forward or 4th order Runge-Kutta to integrate the equations. Make sure you do this in a group. It's very easy to introduce typos since the equations are a bit complicated.
   
2. Pick some sensible parameters and initial conditions t1 and t2, (p1=0, p2=0, that is, you don't push the pendulum initially). With all SI units, sensible masses would be of the order of 1 (kg) and lengths also of order 1 (meter). We don't know any analytic solutions, but we know that energy (E=T+V) should be conserved. How well E is conserved will tell you how good the solution is. Run the code with Euler and RK4 integration and see how well energy is conserved. Vary the time step and find out how small it must be for E to be conserved within 30%, for each of the methods. Is Euler useful?
   
3. Run the double pendulum case with these parameters: L1=0.1, L2=0.03, M1=1.0, M2=0.5, and initial conditions: p1=0, p2=0, t1=0.1*pi, t2=0.03*pi, from t=0 to final time TF=20. Output the time series of t1 and compute the spectra with the code you wrote for HW05. As in HW05 make plots for the time series, the power as a function of frequency, and the power as a function of period. I can make out at least 4 power peaks. What is their physical meaning? Is the system in a chaotic stage?
   
4. Now change parameters to: L2=0.05, M2=5.2, and initial conditions to t1=0.7*pi, t2=0.21*pi. Do the same analysis as above. Is the system still periodic? Start with a step size of h=0.001. Then increase/decrease the step size. Do you always get the same result?