Double pendulum

The double pendulum is a popular model for the demonstration of chaotic processes. On the arm of a pendulum another pendulum is hanged.

This simple design creates an unpredictable movement pattern that responds exponentially to disturbances. This behavior arises from the non-linear dynamics, which has elliptic and hyperbolic fixed points.

Clearly, in certain states ( phase space positions ) a small change crucial for the immediate future development.

Derivation of the equations of motion

Using the Lagrangian formalism

With

And

Here, the kinetic energy of the pendulum masses. describes the potential energy of the pendulum mass in the constant gravitational field. The point on the angles represents the time derivative represents the state variables and variables with subscript 1 represent the inner pendulum of the double pendulum and those with index 2 respectively represent the outer pendulum.

From the Lagrange equations are obtained for the angle and the equations of motion to

And

And the lengths of said ( massless ) connecting rods, and the pendulum mass and the acceleration of gravity is. In the equations of motion occur angle functions of the state variables. It is a non-linear system. In the special case of small deflections, the equations of motion can, however, simplify using the small angle approximation. Then, for example, other special cases can be as or consider having an approximate harmonic solution.

Solution of the equations of motion

The equations of motion for the generalized coordinates, and represent a nonlinear system of two differential equations, which is not analytically solvable. It can be solved in four known initial values ​​with numerical methods.

Using trigonometry, the angle and are transferred to the Cartesian coordinates of the mass points.