Consider a system of two identical pendulums swinging in parallel planes and connected at the top by a flexible string. (A

recently published Demonstration considered the related version of two pendulums with their bobs connected by a massless spring.) We describe here a more classic form of the problem, going back to the time of Huygens.

Each pendulum consists of a mass

suspended from a fixed support by a massless string of length

, with its coordinate described by the angle

from the vertical. Under the action of gravity, the equation of motion is given by

. Restricting ourselves to small-amplitude oscillations, we can approximate

, leading to the elementary solution

, where

, the natural frequency of the pendulum, independent of

. For the coupled pendulum system, to the same level of approximation, the Lagrangian can be written

where

is a measure of the coupling between the two pendulums mediated by the connecting string. The equations of motion are readily solved to give

.

The system has two normal modes of vibration. In the in-phase mode, starting with

, the two pendulums swing in unison at a frequency of

. In the out-of-phase mode, starting with

, they swing in opposite directions at a frequency of

.

A more dramatic phenomenon is resonance. If one pendulum is started at

and another at

, then energy will periodically flow back and forth between the two pendulums.