A double pendulum is a system consisting of two pendulums attached end to end. It is an example of a complex mechanical system that exhibits chaotic behavior. The first pendulum is fixed at one end and swings freely, while the second pendulum is attached to the end of the first pendulum and also swings freely. The double pendulum is notable for its rich dynamics; its motion depends on several factors, including the initial angles and velocities of each pendulum.
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A double pendulum consists of two pendula coupled because the second pivot moves with the first mass. It has two degrees of freedom: even when the upper pivot is fixed, moving the upper arm carries the lower pivot along. This makes the double pendulum a compact example of how mechanical constraints couple motion, and how Lagrangian mechanics converts geometry into dynamics.
Use two equal point masses and two massless rigid rods of equal length , with joints that allow free rotation in one vertical plane. The upper pivot is fixed; gravity is uniform, with gravitational acceleration . Neglect joint friction and drag force. Both angles are measured from the downward vertical, positive toward the right, in radians. In particular, is an absolute angle, not the bend between the rods: that bend is .
Equal point masses and massless rods: both angles are measured from the downward vertical
. Original diagram of the ideal double pendulum. Dashed vertical lines are the separate references for and ; the lower position vector is the sum of the two rod vectors. The drawn angles illustrate geometry, rather than the small-angle approximation. Details.The double pendulum Lagrangian below gives the exact dynamics of this ideal model. Near the downward stable equilibrium, normal modes of a double pendulum replace the coupled motion with two independent harmonic oscillators. Their fixed amplitude ratios explain which releases make the rods swing together and which make them swing in opposite directions.
