Simulation Results
ReadyRun a simulation to calculate angles, velocities, positions, energy, and chaos indicators.
Real-Time Animation
Graphs and Chaos Analysis
Simulation Data Table
The table uses the selected sampling interval.
| Time | θ₁ | θ₂ | ω₁ | ω₂ | x₁ | y₁ | x₂ | y₂ | Kinetic | Potential | Total |
|---|---|---|---|---|---|---|---|---|---|---|---|
| No simulation data yet. | |||||||||||
Formula Used
The simulator solves the coupled nonlinear equations for two point masses connected by rigid, massless rods. Angles are measured from the downward vertical direction.
α₁ = [−g(2m₁+m₂)sinθ₁ − m₂g sin(θ₁−2θ₂)
− 2m₂sin(θ₁−θ₂)(ω₂²L₂ + ω₁²L₁cos(θ₁−θ₂))]
÷ [L₁(2m₁+m₂−m₂cos(2θ₁−2θ₂))]
α₂ = [2sin(θ₁−θ₂)(ω₁²L₁(m₁+m₂) + g(m₁+m₂)cosθ₁
+ ω₂²L₂m₂cos(θ₁−θ₂))]
÷ [L₂(2m₁+m₂−m₂cos(2θ₁−2θ₂))]
x₁ = L₁sinθ₁ y₁ = −L₁cosθ₁
x₂ = x₁ + L₂sinθ₂ y₂ = y₁ − L₂cosθ₂
T = ½(m₁+m₂)L₁²ω₁² + ½m₂L₂²ω₂² + m₂L₁L₂ω₁ω₂cos(θ₁−θ₂)
V = −(m₁+m₂)gL₁cosθ₁ − m₂gL₂cosθ₂
E = T + V
Damping, driving torque, and spring coupling are added as optional generalized torque terms. Numerical integration advances the four-state system through time.
How to Use This Calculator
- Enter both rod lengths and bob masses.
- Select gravity or enter a custom value.
- Set initial angles and angular velocities.
- Choose damping, driving, and coupling options.
- Select a numerical solver and time step.
- Enable nearby-condition comparison for chaos analysis.
- Press Calculate and simulate.
- Use animation controls, charts, and export buttons.
Smaller time steps usually improve energy conservation. They also require more calculations and browser work.
Example Data
| Example | L₁ | L₂ | m₁ | m₂ | θ₁ | θ₂ | Expected behavior |
|---|---|---|---|---|---|---|---|
| Small-angle | 1 m | 1 m | 1 kg | 1 kg | 10° | 5° | Mostly regular oscillation |
| Chaotic | 1 m | 1 m | 1 kg | 1 kg | 120° | −10° | Strong sensitivity and complex motion |
| Damped | 1 m | 0.8 m | 1 kg | 0.7 kg | 90° | 30° | Energy gradually decreases |
Worked Example
Choose one-meter rods and one-kilogram bobs. Set the angles to 120° and −10°.
Use Earth gravity, RK4, and a 0.01-second step. Enable a 0.01° nearby-condition comparison.
Run thirty seconds and inspect trajectory separation. Compare energy drift before changing solver accuracy.
Assumptions and Limitations
- Rods are rigid and massless.
- Bobs are treated as point masses.
- The pivots remain fixed.
- Motion stays in one vertical plane.
- Air resistance uses simplified linear damping.
- Extreme settings can reduce numerical accuracy.
- The chaos label is an estimate, not a proof.
Frequently Asked Questions
Why is a double pendulum chaotic?
Its nonlinear coupled equations amplify small differences in initial conditions. Similar starts can create very different trajectories.
Which solver is best?
RK4 is a strong general choice. Adaptive RK4 improves accuracy when motion changes rapidly.
Why does total energy drift?
Numerical methods introduce approximation error. Use RK4, smaller time steps, or symplectic integration.
What does the Lyapunov estimate mean?
A larger positive estimate suggests faster divergence between nearby trajectories. Short simulations provide only rough estimates.
Can I simulate the Moon or Mars?
Yes. Select a gravity preset or enter another positive acceleration value.
What is a Poincaré section?
It samples phase-space points at repeated crossings. The pattern can reveal regular or chaotic behavior.
What does damping change?
Damping removes mechanical energy. Oscillations generally shrink until the pendulum approaches rest.
Why does Euler become unstable?
Euler has larger accumulated error. Large time steps can make energy grow unrealistically.
Can I save my setup?
Yes. Export settings as JSON or copy a shareable configuration link.