Illustrations of
New Remarkable Features Found in the Double Pendulums
By Alexander Gofen and James Sochacky
(An excerpt from the extensive research to be published soon)
Here you can watch a few movies illustrating new remarkable features newly found in the planar and spherical double pendulums. This movie presentation is intended for those who did not download and install the Taylor Center software for Windows running these and many other video simulations in a more efficient way.
An extensive collection of dynamic visualizations of the pendulum motion is provided here as the data files for the Taylor Center software (rather than movies in the mp4 format).
Double pendulums are known for their notorious chaotic motion whose ODEs are given here. In this excerpt we are to demonstrate several cases of the regular motion of the double pendulum with quite unexpected shapes.
In the planar double pendulum, the mass m1 is restricted to move along a circumference of the radius L1 typically in a chaotic manner, while the mass m2 usually moves along a chaotic trajectory and in a chaotic manner. However, we discovered such initial values and parameters of the double pendulum, that both masses make some kinds of remarkable uniform motion.
In the simulations below the mass m1 is nearly zero.
Trajectories of the two masses |
Moving rods |
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Here the mass m2 moves along a regular triangle with constant high velocity. At that, when m2 approaches a vertex, the mass m1 dramatically accelerates, or decelerates so much that it moves backward. This behavior alternates from vertex to vertex. The initial values may be set for making any regular polygon. |
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Here the masses make the Ψ-shape. At that, m2 vertically oscillates like the Maxwell pendulum, while the mass m1 dramatically accelerates approaching the lowest position on the circumference. |
The spherical double pendulum
In the spherical double pendulum, the mass m1 is restricted to move along a surface of a sphere of the radius L1, though in a chaotic manner, while the mass m2 usually moves along a chaotic spatial trajectory and in a chaotic manner. We discovered such initial values and parameters of the spherical double pendulum, that the trajectories of both masses make some kinds of remarkable uniform motion.
The simulations below are in 3D stereo requiring the Red/Blue glasses.
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Trajectories of the two masses |
Moving rods |
This is the 3D version of what was the Ψ-shape in 2D. Consequently, instead of the Ψ-shape, the masses outline a "glass" in 3D. As before, m2 vertically oscillates like the Maxwell pendulum. At that, the linear velocity of m1 (having near zero mass), and the angular velocity of the rods dramatically accelerate to infinity approaching the lowest position on the sphere. |
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This is one of the cases discovered by Mersden and Scheurle in 1993. Both masses move along circumferences. At that, the rod L2 may be either stretched-in or stretched-out. |