= Double pendulum
{wiki}
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> $m>0$ and two massless rigid rods of equal <length> $\ell>0$, with joints that allow free rotation in one vertical <plane>. The upper pivot is fixed; <gravity> is uniform, with <gravitational acceleration> $g>0$. Neglect joint <friction> and <drag force>. Both <angles> $\theta_1,\theta_2$ are measured from the downward vertical, positive toward the right, in <radians>. In particular, $\theta_2$ is an absolute <angle>, not the bend between the rods: that bend is $\theta_2-\theta_1$.
\Image[/double-pendulum-point-mass-geometry.png]
{title=Equal point masses and massless rods: both angles are measured from the downward vertical}
{description=Original diagram of the ideal <double pendulum>. Dashed vertical lines are the separate references for $\theta_1$ and $\theta_2$; the lower <position vector> is the sum of the two rod <vectors>. The drawn <angles> illustrate geometry, rather than the <small-angle approximation>.}
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.
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