🌀 N‑Mass Points Pendulum Simulator

A scientific and visual exploration of multi‑pendulum dynamics, inspired by research on chaotic motion of coupled pendulums. This simulator models pendulums with 3 or more masses, solves their motion using rigorous physics, and provides a real‑time visualizer with velocity‑based trails.

pendu.mp4

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🔬 Research Basis

This project is grounded in scientific research and builds on key references:

• D. Assêncio, “Double pendulum: Hamiltonian formulation” (Link)
  Derivation of double pendulum equations from Hamiltonian/Lagrangian mechanics.

• J. Jiménez‑López & V.J. García‑Garrido, “Chaos and Regularity in the Double Pendulum with Lagrangian Descriptors” (arxiv)
  Quantification of chaos using Lagrangian descriptors.

• B. Yesilyurt, “Equations of Motion Formulation of a Pendulum Containing N-point Masses” (arxiv)
  General formulation for n‑mass pendulums, providing the framework used for simulations with 3 or more masses.

Key points from these works implemented:

• Equations of motion derived from Lagrangian / Hamiltonian mechanics.
• Double pendulum (n = 2): classical coupled nonlinear ODEs.
• Multi-mass pendulums (n ≥ 3): generalized via mass matrix and coupling terms (see PendulumSolver.cs).
• Runge‑Kutta 4th order (RK4) integration with configurable sub-steps for stability and accuracy.
• Sensitivity to initial conditions and emergence of chaotic dynamics naturally appear in simulations.

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🧮 Mathematics Behind It

• Double pendulum (n = 2):

• General n‑mass pendulum (n ≥ 3):

• Integration: RK4 with subSteps per frame:

• Chaos and energy: Following Jiménez‑López & García‑Garrido, chaos fraction depends on energy, mass ratios, and length ratios.

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🎮 Features

• ✅ Supports 3 or more pendulum masses
• ✅ Real-time visualization with velocity-based trail colors
• ✅ Custom shader for smooth, colorful trails
• ✅ Adjustable rod width, mass radius, trail length, and velocity gradient
• ✅ Sensitive dynamics, allowing chaos and regularity studies

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⚙️ Usage

1. Open the project in Unity.
2. Add PendulumSolver and PendulumFullRenderer to an empty GameObject.
3. Configure mass points (Mass, AttachedRodLength, InitialAngleDegrees, AngularVelocity).
4. Adjust subSteps in PendulumSolver for numerical stability.
5. Customize visual parameters in PendulumFullRenderer (trail length, gradient, rod width, mass radius).
6. Press Play to simulate and visualize pendulum motion.

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🎨 Visualization

• Trails colored by instantaneous angular velocity.
• Rods connecting masses with configurable colors and widths.
• Mass points rendered as spheres, with color indicating speed.
• Z-offsets per mass ensure distinct trails for multiple pendulums.

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📚 References

• Assêncio, D., Double pendulum: Hamiltonian formulation, https://dassencio.org/33
• Jiménez‑López, J. & García‑Garrido, V.J., Chaos and Regularity in the Double Pendulum with Lagrangian Descriptors, arXiv:2403.07000
• Yesilyurt, B., Equations of Motion Formulation of a Pendulum Containing N-point Masses, arXiv:1910.12610