I. THE PENDULUM. y 1 (t) = θ(t) (2) y 2 (t) = dθ(t) dt

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1 1 I. THE PENDULUM In this section, we will explore using Mathematica as a tool, several aspects of the pendulum that you might have done as anexperiment inthe lab. We begin by looking at the forces that govern the motion of the Pendulum and we have the following second order differential equation (Newton s second law): d 2 θ(t) 2 = ω 2 0 sin(θ(t)) αdθ(t) +Acos(ωt) (1) where ω 0 is the natural frequency of the pendulum, the α term arises due to friction and the A term is the driving force with the driving frequency ω. Using the following definitions: y 1 (t) = θ(t) (2) y 2 (t) = dθ(t) (3) show that we can re-write Eq. 1 into the following set of coupled first order differential equations: dy 1 (t) dy 2 (t) = y 2 (t) = ω 2 0sin(y 1 (t)) αy 2 (t)+acos(ωt) (4) We will now solve the system of coupled first order differential equation using Mathematica. Open a new notebook. This you can do by clicking on the Mathematica icon. Using the NDSolve function (you can look at the documentation center for examples), solve the Eqns. 4 for the following cases using ω 0 = 1, T = 2π/ω 0 Replace sin(y 1 (t)) y 1 (t) and choose A = 0, α = 0 - Ideal pendulum for different initial values of y 1 (0) and y 2 (0). Make a plot (using the function Plot) of the phase

2 2 space i.e. a plot of y 2 (t) as a function of y 1 (t). Study the motion from t = 0 to t = 4T. What do you get? Increase the value of y 1 (t) from say 0.1 to 2.0. Let us go back to using the sin(y 1 (t) term as in Eq. 4 and let A = 0, α = 0. Run for the same set of y 1 (t) as the previous question and make plots (using the function Plot) of the phase space i.e. a plot of y 2 (t) as a function of y 1 (t) as well as y 1 (t) and y 2 (t) versus t in each case. Study the motion from t = 0 to t = 4T. What do you get? How is this different from the previous case? From here on, we will use the Eq. 4 for all our analysis. Choose for a fixed value of y 1 (t) increasing values of y 2 (t) and re-do the phase space plots as well as plots of y 1 (t) and y 2 (t) as a function of t. Comment on what you observe. Keeping A = 0, set α = 0.01,0.1,0.5,2.0,5.0 and once again plot the phase space as well as y 1 (t) and y 2 (t) as functions of t. Study the motion from t = 0 to t = 4T. What

3 3 do you get and how would you interpret your result? Let us add a driving force: A = 0.52, α = 0.5. Explore the phase space and the solutions y 1 (t) and y 2 (t) for different driving frequencies ω = ω 0,2ω 0,2/3,. Study the motion from t = 0 to t = 4T. Explain your results: We will now explore Chaos in the driven and damped pendulum. Note that in these examples, you will have to solve for at least 100T in order to see effects. Solve the Eqns. 4 for the following conditions and make a plot of the phase space. 1. Set A = 2.0, α = 0.5, ω = 2/3. Explore the phase space by first beginning with y 1 (0) = y 2 (0) = 1. Then change y 2 (0) = 0.998,0.995,0.997 and observe the changes in the phase space. Explain your results: 2. What happens to your results in 1, if you ran for less time, say from t = 0 to t = 4T? 3. Set α = 0.2, ω = 2/3. Change A in the following range:

4 4 {0.1,0.16,0.2,0.4,0.5,0.6}. For the following initial conditions, y 1 (0) = ,y 2 (0) = , plot the phase space in each case and explain what you see. How does the phase space compare with the one in the previous case? Finally we will do some Poincare section: Here we will sample the phase space at fixed intervals (for example), say when the driving force goes through a zero. If we now plot y 1 (t) as a function of y 2 (t) we get a Poincare section. These sections capture important characteristics of the dynamics. We will explore some of these here. As a first step, open the sample mathematica notebook poincare section example.nb, go through the notebook and run it to evaluate the poincare section of the duffing s equation. Answer the question below before you proceed to the pendulum. Can you correlate the phase space plot and the poincare section? What would you expect for a poincare section of the motion is periodic? Your job is to adapt the notebook for the Pendulum. Find the poincare section for the following cases and in each case describe what you understand: For the simple periodic case without damping and external force.

5 5 For the case when the pendulum is damped. Set α = 0.5. For the chaotic cases that you have already studied: a. Chaotic b. Limit cycles What do you understand from the Poincare sections?