Several variants of the double pendulum may be considered; the two limbs may be of equal or unequal lengths and masses, they may be simple pendulums or compound pendulums (also called complex pendulums) and the motion may be in three dimensions or restricted to the vertical plane. In the following analysis, the limbs are taken to be identical compound pendulums of length \ell and mass m, and the motion is restricted to two dimensions.
An example of a double pendulum.

In a compound pendulum, the mass is distributed along its length. If the mass is evenly distributed, then the centre of mass of each limb is at its midpoint, and the limb has a moment of inertia of \textstyle I=\frac{1}{12} m \ell^2 about that point. The moment of inertia of a rod rotating around an axis attached to one of its ends equals \textstyle I=\frac{1}{3} m \ell^2.

It is convenient to use the angle between each limb and the vertical as the generalized coordinates defining the configuration of the system. These angles are denoted ?1 and ?2. The position of the centre of mass of each rod may be written in terms of these two coordinates. If the origin of the Cartesian coordinate system is taken to be at the point of suspension of the first pendulum.
Data from