University · Ages 18+First-year university · Physics olympiadsAdvanced

 

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Setups other learners found interesting in Double Pendulum. Try one, then tweak it.

Double Pendulum

A second pendulum hangs from the end of the first. The system has just two degrees of freedom, yet its motion is famously chaotic: run it twice from almost-identical starts and the arms soon diverge. This simulation integrates the exact Lagrangian equations for the two angles rather than using a constraint solver, so the chaos is physical and not a numerical artefact.

Key formulas

  • L=T−V\mathcal{L} = T - V

    Lagrangian

    L\mathcal{L}
    LagrangianJ
    TT
    Kinetic energyJ
    VV
    Potential energyJ
    Open the formula card
  • ddt∂L∂q˙i−∂L∂qi=0\dfrac{d}{dt}\dfrac{\partial \mathcal{L}}{\partial \dot{q}_i} - \dfrac{\partial \mathcal{L}}{\partial q_i} = 0

    Euler–Lagrange equation

    L\mathcal{L}
    LagrangianJ
    qiq_i
    Generalised coordinate (an angle, a position…)
    q˙i\dot{q}_i
    Its time derivative
    Open the formula card

What you can change

  • Length 1 and 2
  • Mass 1 and 2
  • Gravity
  • Damping
  • Initial angles

Key concepts

  • Coupled oscillators
  • Degrees of freedom and generalized coordinates
  • Deterministic chaos and sensitivity to initial conditions
  • Energy conservation in the absence of damping