2. Consider the pendulum equation Lx" + g sin r = 0. It is a common practice to write the above equation as x"+ksin = 0, where k? = g/L. Now draw the phase plane portrait of this equation. Here are some directions. (a) Write as a first order system assuming y=x'. (b) Find the equilibria points and the Hamiltonian. (c) Show that the Hamiltonian is nonnegative. It may be helpful to draw the graph of --F(2). (d) Discuss the set Ac. You should consider 4 cases based on c. (e) Draw the phase portrait. (f) You should see the connection between the graph of - F(x) and the phase portrait