Virtual Labs

Double Pendulum

Theory

The double pendulum is a classic example in physics that involves the study of a system consisting of two interconnected pendulums. It is a simple yet chaotic system that demonstrates sensitive dependence on initial conditions and exhibits complex, non-linear behavior.

A double pendulum consists of two pendulums connected to each other. The first pendulum (called the "upper" or "inner" pendulum) is attached to a fixed point, and the second pendulum (called the "lower" or "outer" pendulum) is connected to the end of the first pendulum. The motion of the double pendulum is more complex and exhibits chaotic behavior.

The double pendulum system behaves like a linear system for small angles, generating a Lissajous curve trajectory. The pendulum is non-linear for large angles, and the phase graph becomes more complex.

The behavior of the double pendulum is governed by a set of coupled, non-linear differential equations known as the Lagrangian equations. These equations describe the conservation of energy and angular momentum in the system. Solving these equations analytically is extremely challenging due to their non-linear nature. Therefore, numerical methods or computer simulations are typically used to study the behavior of the double pendulum.

We regard the pendulum rods as being massless and rigid. We regard the pendulum masses as being point masses.

The equation of motion of a double pendulum can be obtained using the Lagrangian method.

Fig. 1.

Double pendulum

Consider a double bob pendulum with masses m₁ and m₂ attached by rigid massless wires of lengths l₁ and l₂. Further, let the angles the two wires make with the vertical be denoted as 𝜭₁ and 𝜭₂, as illustrated above. Finally, let gravity be given by g. Then the positions of the bobs are given by