Spring Pendulum
A mass on a spring that can both swing and bounce. The elastic pendulum couples two oscillation modes — angular and radial — and energy flows chaotically between them. Tiny changes in initial conditions lead to wildly different trajectories.
m·r̈ = m·r·θ̇² − k(r − L₀) + m·g·cosθ | m·r·θ̈ = −2m·ṙ·θ̇ − m·g·sinθ
The elastic pendulum
The elastic pendulum (or spring pendulum) is one of the simplest physical systems that exhibits genuine chaos. A mass m hangs from a spring of natural length L₀ and spring constant k, free to both swing like a pendulum and bounce up and down. The system has two degrees of freedom: the angle θ from vertical and the length r of the spring.
Equations of motion
In polar coordinates with the pivot at the origin and θ measured from the downward vertical: the radial equation is m·r̈ = m·r·θ̇² − k(r − L₀) + m·g·cosθ, and the angular equation is m·r·θ̈ = −2m·ṙ·θ̇ − m·g·sinθ. The coupling comes from the r·θ̇² term (centrifugal effect) and the 2·ṙ·θ̇ term (Coriolis effect).
Energy transfer and chaos
When the radial oscillation frequency ωr = √(k/m) is close to twice the pendular frequency ωθ = √(g/L), a parametric resonance occurs. Energy flows back and forth between the two modes in an unpredictable way. Small differences in initial conditions lead to exponentially diverging trajectories — deterministic chaos.
Phase space
The phase space plot shows θ vs θ̇ (angular position vs angular velocity). For regular motion, the trajectory traces closed loops. For chaotic motion, it fills a region densely without repeating — the hallmark of a strange attractor in the energy surface.
Conservation of energy
The total energy E = ½m(ṙ² + r²θ̇²) + ½k(r − L₀)² − mgr·cosθ is conserved (no friction). The energy bar shows how kinetic, gravitational potential, and elastic potential energy trade with each other in real time.