15. Chaos I (The Chaotic Pendulum)

The script starts with a welcome back to the office hour, where the presenter introduces the topic of chaotic dynamics, specifically focusing on the chaotic pendulum. The discussion begins with the logistic map and the emergence of chaos in non-linear systems, highlighting the underlying structure within chaotic behavior. The main problem presented involves solving the motion of a driven single pendulum with friction, emphasizing the importance of not using small angle approximations and allowing for large oscillations. Additionally, a more complex problem involving a double pendulum is introduced, which adds an extra degree of freedom and the potential for chaotic motion without the need for external driving forces or friction.

This paragraph delves into the physics of the chaotic pendulum using Newton's law of rotation to derive the equations of motion. The forces acting on the pendulum, including gravity, friction, and the external driving force, are described. The equation of motion is presented in two forms, with the second form being a standardized version obtained by dividing through by the moment of inertia. Key parameters such as the natural frequency of the pendulum, the frictional constant, and the external torque are defined. The paragraph concludes with a recommendation to solve the derived differential equation using numerical methods, specifically the fourth-order Runge-Kutta method for ordinary differential equations.

The presenter suggests beginning the analysis with the simplest case of a pendulum undergoing simple harmonic motion without friction or driving force. This approach allows for testing the numerical solution against known analytic solutions and understanding the fundamental behavior of the pendulum. The importance of starting with small angular deviations from the rest position is emphasized, and the expectation that the period of oscillation should be consistent with the natural frequency of the pendulum is discussed. The paragraph also guides on how to test the numerical algorithm for accuracy and reliability by comparing it with the expected simple harmonic motion.

The script continues by discussing the impact of amplitude on the period of a pendulum's oscillation. It explains that as the amplitude increases, the period also increases due to the non-linear relationship between the restoring force and the sine of the angle. The presenter encourages the development of an algorithm to compute the period of the pendulum's motion and to compare it with the simple harmonic motion. The concept of the separatrix is introduced, which is the critical condition where the pendulum stands still at the top before potentially rotating. The paragraph concludes with a suggestion to explore the auditory representation of periodic motion, converting the pendulum's oscillations into sound to provide a unique perspective on the system's behavior.

This paragraph introduces an interactive and sensory method of understanding the physics of motion by converting it into sound. It describes a CD and digital book resource that includes applets for listening to the sounds of different types of motion, such as harmonic and non-harmonic oscillations, as well as chaotic motion. The presenter provides a detailed walkthrough of using the applet to convert data into sound, allowing students to 'listen' to their simulations and gain a deeper understanding of the underlying frequencies and harmonics present in the motion.

The script moves on to discuss the concept of phase space, an abstract space that combines position and velocity (or momentum) to visualize the motion of a system. The presenter explains how phase space plots can simplify the analysis of complex motions by reducing them to geometric figures. The paragraph provides examples of phase space plots for simple harmonic motion, non-harmonic motion, and repulsive potentials, illustrating how these plots can reveal the nature of the motion, whether it is periodic, chaotic, or unstable.

This paragraph explores the use of phase space plots to understand the dynamics of a pendulum, particularly the concept of the separatrix, which separates oscillatory (vibrational) motion from rotational motion. The script describes how small oscillations result in closed loops in phase space, indicative of periodic vibrations, while larger oscillations that go over the top result in 'running' solutions. The phase space plot is used to illustrate the transition between these two types of motion and the unstable equilibrium at the separatrix.

The presenter discusses the effects of friction and external torque on the oscillations of a pendulum. Initially, the large oscillations result in significant energy loss due to friction. However, as the amplitude decreases, the energy loss due to friction can match the energy input from the external torque, leading to stable oscillations with constant amplitudes. The phase space representation of this process is described, showing spiraling orbits that represent energy loss and the eventual stabilization into a limit cycle where energy input and loss are balanced.

The final paragraph of the script encourages students to explore the full complexity of the pendulum problem, including friction, driving force, and large oscillations. The goal is to observe complex behavior, including the transition to chaos. The presenter advises on how to explore the system by fixing certain parameters and varying initial conditions to observe the system's flow through phase space. The paragraph concludes by emphasizing that chaos, while complex, is not random but has an underlying structure that can be understood through phase space analysis and other analytical tools.

In conclusion, the script summarizes the key points discussed and encourages students to take a break, go into the lab, and run simulations to understand the dynamics of the pendulum system. The presenter advises starting with simple cases and gradually moving to more complex scenarios to gain insights into the behavior of the system, including the emergence of chaos. The script ends with a prompt for students to return for further discussion on their findings and observations.