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Pendulum Lab theory

The equations and assumptions below are rendered directly from the theory document pinned with the simulation source. There is no separate web copy to drift out of date.

Physics Theory — Project 2: Simple and Double Pendulums

Objective

This project studies rotational oscillating systems using angular coordinates instead of direct Cartesian coordinates. It focuses on three areas:

  1. rotational dynamics;
  2. numerical integration of ordinary differential equations;
  3. the transition between regular and chaotic behavior.

Conventions

  • Length unit: meter (m)
  • Mass unit: kilogram (kg)
  • Time unit: second (s)
  • Angle unit in the interface: degree (deg)
  • Angle unit in the integrator: radian (rad)
  • Gravity: m/s²
  • Angular velocity: rad/s

The angle θ=0\theta = 0 represents the rod pointing downward in stable equilibrium. Positive values rotate counterclockwise when viewed in the conventional Cartesian plane.

Simple pendulum

Model

The simple pendulum is modeled as a point mass attached to a rigid, massless rod of length LL.

Its exact equation of motion is:

θ¨=−gLsin⁡(θ)\ddot{\theta} = -\frac{g}{L}\sin(\theta)

The project also uses a linear damping term proportional to angular velocity:

θ¨=−gLsin⁡(θ)−cθ˙\ddot{\theta} = -\frac{g}{L}\sin(\theta) - c\dot{\theta}

where:

  • gg is gravity;
  • LL is the rod length;
  • cc is the angular damping coefficient.

Small-angle approximation

When ∣θ∣≪1|\theta| \ll 1 rad, sin⁡(θ)≈θ\sin(\theta) \approx \theta, and the equation becomes linear:

θ¨+gLθ=0\ddot{\theta} + \frac{g}{L}\theta = 0

In this regime, the theoretical period is:

T≈2πLgT \approx 2\pi\sqrt{\frac{L}{g}}

This expression is used as a numerical validation reference in the tests.

Mechanical energy

For mass mm, the simple pendulum’s total energy can be written as:

E=12mL2ω2+mgL(1−cos⁡(θ))E = \frac{1}{2}mL^2\omega^2 + mgL\left(1 - \cos(\theta)\right)

where ω=θ˙\omega = \dot{\theta}.

Without damping, the energy should remain approximately constant. With damping, it should decrease over time.

Double pendulum

Model

The double pendulum has two point masses, m1m_1 and m2m_2, coupled by two rigid rods of lengths L1L_1 and L2L_2. Its degrees of freedom are θ1\theta_1 and θ2\theta_2.

The exact equations are nonlinear and coupled, and they depend simultaneously on the angles and angular velocities. One standard form for the angular accelerations is:

θ¨1=−g(2m1+m2)sin⁡θ1−m2gsin⁡(θ1−2θ2)−2sin⁡(θ1−θ2)m2(ω22L2+ω12L1cos⁡(θ1−θ2))L1(2m1+m2−m2cos⁡(2θ1−2θ2))\ddot{\theta}_1 = \frac{-g(2m_1+m_2)\sin\theta_1 - m_2g\sin(\theta_1 - 2\theta_2) - 2\sin(\theta_1-\theta_2)m_2\left(\omega_2^2L_2 + \omega_1^2L_1\cos(\theta_1-\theta_2)\right)}{L_1\left(2m_1+m_2-m_2\cos(2\theta_1-2\theta_2)\right)} θ¨2=2sin⁡(θ1−θ2)(ω12L1(m1+m2)+g(m1+m2)cos⁡θ1+ω22L2m2cos⁡(θ1−θ2))L2(2m1+m2−m2cos⁡(2θ1−2θ2))\ddot{\theta}_2 = \frac{2\sin(\theta_1-\theta_2)\left(\omega_1^2L_1(m_1+m_2) + g(m_1+m_2)\cos\theta_1 + \omega_2^2L_2m_2\cos(\theta_1-\theta_2)\right)}{L_2\left(2m_1+m_2-m_2\cos(2\theta_1-2\theta_2)\right)}

The project also adds linear damping terms:

θ¨1←θ¨1−c1ω1\ddot{\theta}_1 \leftarrow \ddot{\theta}_1 - c_1\omega_1 θ¨2←θ¨2−c2ω2\ddot{\theta}_2 \leftarrow \ddot{\theta}_2 - c_2\omega_2

Mechanical energy

The mass positions are:

x1=L1sin⁡θ1,y1=−L1cos⁡θ1x_1 = L_1\sin\theta_1, \qquad y_1 = -L_1\cos\theta_1 x2=x1+L2sin⁡θ2,y2=y1−L2cos⁡θ2x_2 = x_1 + L_2\sin\theta_2, \qquad y_2 = y_1 - L_2\cos\theta_2

The velocities are obtained by differentiating these expressions. The total energy is:

E=12m1(vx12+vy12)+12m2(vx22+vy22)+UE = \frac{1}{2}m_1(v_{x1}^2 + v_{y1}^2) + \frac{1}{2}m_2(v_{x2}^2 + v_{y2}^2) + U

with gravitational potential measured relative to the lowest-energy configuration:

U=m1gL1(1−cos⁡θ1)+m2g(L1(1−cos⁡θ1)+L2(1−cos⁡θ2))U = m_1gL_1(1-\cos\theta_1) + m_2g\left(L_1(1-\cos\theta_1) + L_2(1-\cos\theta_2)\right)

Numerical integration

The project implements two integrators:

Symplectic Euler

  • Updates the angular velocities first;
  • Then updates the angles using the corrected velocities.

It is simple and inexpensive, and it often preserves the system’s qualitative structure better than standard explicit Euler.

Fourth-order Runge-Kutta (RK4)

It uses four intermediate evaluations per time step and produces much lower local error than first-order methods.

For the simple pendulum, RK4 should closely approximate the theoretical small-angle period. For the double pendulum, it helps reduce local error in a system that is highly sensitive to initial conditions.

Chaos and sensitivity to initial conditions

The double pendulum is a classic example of a deterministic system that can exhibit chaotic behavior. This means that:

  • the equations are fully deterministic;
  • small differences in the initial state can grow rapidly over time;
  • long-term predictions become highly sensitive to numerical error.

Therefore, double-pendulum tests should prioritize:

  • absence of NaN and Inf;
  • numerical stability for reasonable time steps;
  • basic invariants when the system has no damping;
  • simple special cases, such as exact rest at θ1=θ2=0\theta_1 = \theta_2 = 0.

Practical validation targets

Simple pendulum

  • small-angle period;
  • approximate energy conservation without damping;
  • energy decay with damping;
  • stability with different integration methods.

Double pendulum

  • stable rest in the downward vertical configuration;
  • finite evolution without numerical blow-up;
  • coherent trails for both bobs;
  • qualitatively distinct behavior between regular and chaotic regimes.

Model limitations

This project deliberately simplifies several aspects:

  • rods are rigid and massless;
  • there is no pivot friction beyond the model’s imposed linear damping;
  • masses and rods do not collide;
  • there is no distributed air resistance along the rods;
  • the problem is two-dimensional.

These simplifications are appropriate for studying numerical integration, energy, periodicity, and chaos without introducing unnecessary complexity too early.