1. Chaotic motion in the double pendulum Dan Hampton and Wolfgang Christian Abstract The double pendulum provides an ideal system in which to study chaotic motion. The system is relatively simple, but an infinitesimally small change in the initial conditions of the pendulum produces a drastically different trajectory for energies that exhibit chaotic motion. I use a phase-space plot, a Poincaré section, and a 3D trace to graphically represent the pendulum’s motion. The simulation allows for the initial energy and the masses on each pendulum to vary. The Feldberg eighth order numerical method serves as the algorithm for computing the motion. The program shows the energies at which the motion becomes chaotic while sometimes still exhibiting quasiperiodic trajectories, and it quantitatively demonstrates the unpredictability of this simple system. Dynamics The equations of motion can be derived from the Lagrangian: Normal Modes of Oscillation Discussion These calculations yield: Poincaré section for Energy total = 15 Joules. Most of the allowed states on this map give chaotic trajectories. Some periodic states act as attractors to the chaotic motion as shown by the pink points. Poincaré section for Energy Total = 8 Joules. Here there is some chaos present, but most states still are periodic. Poincaré sections provide complete information about the system at a particular codition. This is due to conservation of energy, which can give the value of ω 2, and the fact that θ 2 = 0. The Poincaré sections show how chaos changes with energy in this system. Chaotic motion dominates at energies above 25 J with this pendulum, but at higher energies than 125 J periodic motion returns. Chaos is highly dependent on initial conditions. It is not random motion, but it is unpredictable. Chaos and Poincaré Sections The configuration of the double pendulum can be described by two variables: θ 1 and θ 2. The motion of the entire system can be described by the rates of change in θ 1 and θ 2. Once the angular accelerations for both pendulums are found, the double pendulum’s motion can easily be simulated computationally. This simulation used the Feldberg 8 th order numerical algorithm to model the system, where L 1 = L 2 = 1 (meter), m1 = m2 = 1 (kg), and g = 9.81 (meters/second). Apart from the two normal modes, the double pendulum exhibits quasiperiodic motion and chaotic motion. The trajectory of the pendulum can be viewed as a 3-dimensional trail traveling through spatial coordinates of θ 1, ω 1, and θ 2 with time. Here M = m1+m2. To get α 1 and α 2, solve the Euler-Lagrange equations that follow: and Computer Model where λ = L 1 /L 2, g1 = g/L 1, g2 = g/L 2, μ = m2/M Normal modes of oscillation are possible in this system at low energies (small angles of oscillation). There are two normal modes that can be described by the maximum amplitudes of θ 1 and θ 2. If θ 1max = +/-θ 2max *Sqrt(2), then both pendulums will oscillate with the same frequency. Normal Mode 1:θ 1max =.2 = Sqrt(2)* θ 2max Normal Mode 2: θ 1max =.2 = - Sqrt(2)* θ 2max Comparison of normal mode phase space and non-normal mode phase space: Normal mode 2 phase space Non-normal mode, but quasiperiodic (not chaotic) motion. This way of viewing the pendulum is interesting and can give some qualitative information about the pendulum’s dynamics, but few people can get much use from 4-dimensional trails. A way to simplify this trail is to plot the points on the plane θ 2 = 0 when second pendulum is moving to the right. These conditions are satisfied when θ 2 = 0 and ω 2 + λ* ω 1 *cos(θ 1 ) > 0. This produces a Poincaré section. The above picture corresponds to the periodic trajectory that makes the three islands of stability around the center of the 15 J Poincaré map (to the right). Poincaré section for Energy total = 5 Joules. At this low energy there are no chaotic trajectories. The pendulum will remain in quasiperiodic orbits. Notice how the points in the map are line- filling and not area-filling. Acknowledgments Francisco Esquembre for the development of EJS. References Gould, Tobochnik, Christian. An Introduction to Computer Simulation Methods San Francisco : Pearson Education, c2006. 3rd ed. Thorton, Marion. Classical Dynamics of Particles and Systems. Thomson, Books/Cole, 5 th ed. Vank´o. “Investigation of a chaotic double pendulum in the Basic Level Physics Teaching Laboratory”