Lagrangian Mechanics of a Triple Pendulum System

Educational Live Script applying Lagrangian mechanics to a system of physical pendulums to simulate quasi-periodic and chaotic motion.

A double pendulum (one simple pendulum suspended from another) is well-known to exhibit quasiperiodic small-angle motions and chaotic large-angle motions. This educational Live script explores Lagrangian mechanics with a triple physical pendulum that exhibits chaotic motion. Variable parameters include a length scale, mass, and a moment of inertia per unit mass for each pendulum. Two visualization options are provided - equilateral triangles and, for fun, a hoop, cross-shape, and arrow, the components of symbols for Mars and Venus.

Symbolic methods are used to construct the Lagrangian and Euler equations of motion and to reduce the equations to a linear system for small-angle oscillations about equilibrium. The normal modes are found and illustrated. Numerical methods are employed to simulate and visualize large-angle motions and a suite of metrics is applied to classify these motions as periodic, quasi-periodic, or chaotic.

This Live Script may interest students and instructors of physics and requires familiarity with intermediate-level mechanics. The symbolic Lagrangian methods may be applied to similar systems. 'Try this' suggestions, coding 'Challenges' , hyperlinks, and references are included for exploration. Additional educational Live Scripts by the author are available here..