1. The following is Newton's second law for the motion of a non linear pendulum (see here for an explanation) : 6\" __3 sinQ. I Here, I is the length of the pendulum, g is the gravity acceleration (in the Earth) and 9 is the angle between the pendulum and the vertical axis. In this exercise we will plot the phase portrait of the solutions using the software pplane. An ODE of order two (such as the one studied in this example) can be converted into a system of two ODE of order 1, using a simple substitution: denote 6 = a: (the position), and :r' = y (the velocity). Then our second order ODE is equivalent to the system at" = y y' = sin(.7:) l (a) Use the software pplane (please download it from here) to plot the phase portrait of the solution, that is the family of curves with tangent vector eld (y, sin(:c)). Plot the portrait obtained with 5,1 = 1. Set the range of :c and y from 10 to 10 (or a larger range). 2 (b) Explain the meaning of the closed curves in the phase portrait, in terms of the motion of the pendulum. (c) Explain the meaning of the non-closed curves in the phase portrait, in terms of the motion of the pendulum. (d) Where are the xed points (equilibrium points), and which ones are stable? (e) Plot the phase portrait for different values of 9. What happens if g is very large (that is, you live in a very massive planet)