The 3D Double Spherical Pendulum: Modeling, Analysis, and Simulations

Authors

Alexander Gofen, Taylor CenterFollow

Publication Date

9-22-2025

Keywords

3D double pendulum, ODE solver, Taylor Series, Euler-Lagrange Equations

Disciplines

Educational Technology | Mathematics | Physical Sciences and Mathematics | Science and Mathematics Education

Abstract

This paper presents an inquiry-based research project that develops and analyzes the system of ordinary differential equations governing the motion of the 3D double pendulum, blending computational experiments with applied and theoretical mathematics. We formulate the Lagrangian for the three-dimensional double spherical pendulum and use Maple to derive the four coupled ordinary differential equations (ODEs) in angular variables; for completeness, we also present an equivalent Cartesian formulation. We then compare and visualize the models using the Taylor Center high-order Taylor-series solver, which delivers high-accuracy trajectories and real-time animations in 2D and anaglyph 3D (red/blue). We place the system in context by comparing the spherical double pendulum with the planar double pendulum and the spherical single pendulum, highlighting structural similarities, differences, and special cases of uniform motion. The Taylor Center environment functions as a virtual laboratory, enabling students to vary parameters and initial conditions and to observe qualitative changes in the dynamics. Intended for undergraduates with solid backgrounds in ODEs and numerical analysis, the module shows how symbolic derivation and modern solvers make complex three-dimensional dynamics accessible and engaging for students.

Recommended Citation

Gofen, Alexander (2025) "The 3D Double Spherical Pendulum: Modeling, Analysis, and Simulations," CODEE Journal: Vol. 19, Article 4.
Available at: https://scholarship.claremont.edu/codee/vol19/iss1/4

Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License