Double pendulum

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In mathematics, in the area of dynamical systems, a double pendulum is a pendulum with another pendulum attached to its end, and is a simple physical system that exhibits rich dynamic behavior. The motion of a double pendulum is governed by a set of coupled ordinary differential equations. For certain energies its motion is chaotic.

Contents

Analysis

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.

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.

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 θ1 and θ2. 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, then the centre of mass of this pendulum is at:

and the centre of mass of the second pendulum is at

This is enough information to write out the Lagrangian.

Lagrangian

The Lagrangian is

The first term is the linear kinetic energy of the center of mass of the bodies and the second term is the rotational kinetic energy around the center of mass of each rod. The last term is the potential energy of the bodies in a uniform gravitational field. The dot-notation indicates the time derivative of the variable in question.

Substituting the coordinates above and rearranging the equation gives

There is only one conserved quantity (the energy), and no conserved momenta. The two momenta may be written as

and

These expressions may be inverted to get

and

The remaining equations of motion are written as