Harmonic Oscillator
The ubiquitous model of physics: a mass on a spring. Whether describing molecules, pendulums, or quantum fields, the mathematics of restoring forces are universal. Here, we pair the physical space (a driven, damped spring) with its phase space portrait ($x$ vs $v$), mapping the trajectory from transient spirals into steady-state limit cycles.
The Equation of Motion
The damped, driven harmonic oscillator is governed by a second-order linear ordinary differential equation:
Where $m$ is mass, $c$ is the viscous damping coefficient, $k$ is the spring constant, $F_0$ is the amplitude of the external driving force, and $\omega$ is the angular frequency of the drive.
Phase Space
The state of the system at any instant is completely described by two variables: position $x$ and velocity $v$. Plotting these against each other produces a phase space portrait.
Without damping or driving forces, the system conserves energy and traces closed ellipses. When damping is introduced ($c > 0$), energy dissipates, and the trajectory spirals inward to a fixed point at the origin (an attractor).
When a periodic driving force is added, the system eventually settles into a steady-state oscillation, represented in phase space as a stable limit cycle. The transient response—the initial spiral before it locks onto the drive—depends on the initial conditions.
Resonance
Every spring-mass system has a natural frequency, given by $\omega_0 = \sqrt{k/m}$. When the external driving frequency $\omega$ matches this natural frequency, the system experiences resonance. The driving force continually does positive work on the system, leading to maximum amplitude.