The double pendulum is a classical system known for its chaotic dynamics. Two mathematical tools allow us to model and simulate its motion:
Lagrangian mechanics was developed in the 18th century by Joseph-Louis Lagrange. Rather than focusing on forces (as in Newton's laws), it concentrates on energy. The central quantity is the Lagrangian, \( L \), defined
\[ L = T - V, \]
where \( T \) is the kinetic energy and \( V \) is the potential energy of the system.
The path taken by the system between two points in time is such that the action \( S \) is stationary (often a minimum in classical systems): \[ S = \int_{t_1}^{t_2} L\, dt. \] Requiring that small variations in the path do not change \( S \) leads to the Euler-Lagrange equations: \[ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0, \] where \( q_i \) are the generalized coordinates (for a double pendulum, these are the angles \( \theta_1 \) and \( \theta_2 \)), and \( \dot{q}_i \) their time derivatives.
This approach captures the dynamics of the system and leads to the derivation of the coupled differential equations that describe the motion of each pendulum.
More information: Lagrangian mechanics on Wikipedia.
The differential equations obtained via the Lagrangian method for a double pendulum are nonlinear and generally lack closed-form solutions. To simulate the system, we use numerical integration. One of the most popular methods is the fourth-order Runge-Kutta (RK4) integration.
Given a state vector \( \mathbf{y}(t) \) and its time derivative \( f(\mathbf{y}(t)) \), RK4 approximates the state after a small time step \( \Delta t \) by computing intermediate slopes: \[ \begin{aligned} k_1 &= f(\mathbf{y}(t)), \\ k_2 &= f\left(\mathbf{y}(t) + \frac{\Delta t}{2} \, k_1\right), \\ k_3 &= f\left(\mathbf{y}(t) + \frac{\Delta t}{2} \, k_2\right), \\ k_4 &= f\left(\mathbf{y}(t) + \Delta t \, k_3\right). \end{aligned} \]
The new state is then given by: \[ \mathbf{y}(t + \Delta t) = \mathbf{y}(t) + \frac{\Delta t}{6} \left( k_1 + 2k_2 + 2k_3 + k_4 \right). \]
RK4 is widely used because it strikes an good balance between computational efficiency and accuracy, particularly in systems with sensitive dynamics like the double pendulum.
More information: Runge-Kutta methods on Wikipedia.
In the simulation, the state of the double pendulum is represented by: \[ \mathbf{y} = \begin{bmatrix} \theta_1 \\ \theta_2 \\ \omega_1 \\ \omega_2 \end{bmatrix}, \] where \( \theta_1 \) and \( \theta_2 \) are the angles and \( \omega_1, \omega_2 \) their angular velocities. The time evolution of this state is computed using the RK4 method.
Damping (to simulate energy loss due to friction or air resistance) is included. This is implemented by multiplying the angular velocities by a damping factor after each integration step.
The double pendulum simulation.