Is the motion truly chaotic?

The ideal conservative spring pendulum is not chaotic in the strict sense, but trajectories can be extremely sensitive to initial conditions and look irregular. Adding light damping stabilizes long runs; very low damping can show slow energy drift in any discrete integrator.

Spring Pendulum

This simulation integrates a planar spring pendulum: a point mass connected to a fixed anchor by a Hooke spring of rest length L₀. Gravity acts along the vertical; in Cartesian coordinates with y downward, the spring force is proportional to extension and directed along the line to the anchor. The model mixes pendular swing and radial oscillation — energy can move between gravitational potential, elastic potential, and kinetic energy in complex ways.

Who it's for: Intro mechanics and coupled oscillations; motivation for normal modes and Poincaré sections in more advanced courses.

Key terms

Hooke’s law
elastic pendulum
coupled degrees of freedom
Lissajous-like paths
sensitive dependence on initial conditions

Live graphs

How it works

A point mass hangs from a fixed anchor on a massless Hooke spring in a vertical plane. Motion mixes pendulum-like swinging with radial bounce along the spring — energy sloshes between stretch potential, gravitational potential, and kinetic energy. With little damping, trajectories can look regular or quite intricate depending on initial conditions.

Key equations

mẍ = −k(r − L₀)x/r − cẋ, mÿ = −k(r − L₀)y/r + mg − cẏ, r = √(x² + y²)

U = ½k(r − L₀)² − mgy (y downward from anchor); chaotic appearance is sensitive to ICs, not numerical noise.

Frequently asked questions

Is the motion truly chaotic?: The ideal conservative spring pendulum is not chaotic in the strict sense, but trajectories can be extremely sensitive to initial conditions and look irregular. Adding light damping stabilizes long runs; very low damping can show slow energy drift in any discrete integrator.