Small Oscillations of the $n$-Pendulum and the “Hanging Rope” Limit $n \rightarrow \infty$

Single	Triple	10-pendulum

This is my term paper for my undergraduate classical mechanics course at the University of Rochester.

Topics: Derivation of the equations of motion for a system of arbitrary $n$-pendulums each hanging below the previous. Equations of motion for small oscillations about the equilibrium position, from a Newtonian Mechanics and Lagrangian Dynamics perspective. The behavior of a hanging rope of constant mass density is explored by both taking the limit $n \rightarrow \infty$ in the $n$-pendulum solution, and by formulating Lagrangian Dynamics for a continuous system. Additionally, a numerical solution to the nonlinear equations of motion for the n-pendulum is provided in the Mathematica notebooks below, in which one can observe the phenomenon of chaotic motion.

Numerical solution for full n-pendulum: npendulum.nb

Small-oscillations (solved analytically): smalloscillationsnpendulum.nb