The phase space model is a powerful mathematical tool in physics that visualizes every possible state of a dynamic system by plotting its position against its momentum. When applied to a pendulum, phase space transforms its physical back-and-forth swing into geometric shapes like circles, ellipses, and wavy lines. This allows physicists to track the total energy and long-term behavior of the pendulum at a glance. 1. Construct the Phase Coordinates
To build a phase space for a pendulum, we plot two independent variables on a two-dimensional coordinate system: The horizontal axis ( -axis) represents the angular displacement (
), which is the angle of the pendulum relative to its lowest vertical equilibrium point. The vertical axis ( -axis) represents the angular momentum ( pθp sub theta ) or angular velocity ( ), which describes how fast the pendulum is moving.
Every unique snapshot of the pendulum’s motion corresponds to exactly one point in this phase space. 2. Map Trajectories by Energy Levels
As time passes, the shifting state of the pendulum traces a continuous path called a trajectory. The shape of this path depends entirely on the system’s total mechanical energy ( ), which is the sum of its kinetic energy ( ) and potential energy (
E=12mL2ω2+mgL(1−cosθ)cap E equals one-half m cap L squared omega squared plus m g cap L open paren 1 minus cosine theta close paren is the mass, is the length of the string, and
is the acceleration due to gravity. Because total energy is conserved in an ideal system, trajectories can never cross each other. 3. Analyze Key Motion Regimes
A complete phase space diagram of an ideal, frictionless pendulum reveals three distinct behavioral zones:
Closed Ellipses (Oscillation / Libration): At low energy levels, the pendulum gently swings back and forth around the bottom equilibrium point (
). In phase space, this forms a series of closed concentric loops. For tiny angles, these loops are nearly perfect ellipses.
Open Wavy Lines (Rotation / Whirling): At very high energy levels, the pendulum has enough momentum to loop completely over the top bar ( 180∘180 raised to the composed with power
radians) and spin continuously in one direction. Since the velocity never drops to zero, the trajectory forms a continuous, undulating wave stretching infinitely to the left or right.
The Separatrix (The Borderline): This critical, cat-eye-shaped boundary separates the closed swinging loops from the open spinning waves. It represents the exact energy threshold where the pendulum has just enough speed to reach the absolute top point (
) and balance there perfectly upside down in an unstable equilibrium. Visualizing the Ideal Phase Space
The plot below demonstrates how these trajectories look mathematically. The center nested loops represent low-energy oscillations, the top and bottom wavy lines represent continuous spinning, and the bold outer loop outlining the “eye” shape is the separatrix. 4. Observe the Effects of Friction (Damped Systems)
When you add real-world factors like air resistance or friction, energy is no longer conserved. Every time the pendulum swings, it loses mechanical energy to its surroundings.
In the phase space model, this causes the trajectories to warp. Instead of closing into perfect loops, the path spirals inward over time. The system eventually loses all momentum and lands exactly at the origin
, which represents the pendulum hanging completely still at the bottom. This resting destination point is known as a fixed-point attractor. Summary of Core Behaviors Pendulum Motion State Physical State Description Geometric Appearance in Phase Space Stable Equilibrium Hanging straight down and completely motionless A single stagnant point at the origin Low-Energy Oscillation Standard back-and-forth ticking or swinging Closed, concentric elliptical loops around the origin Threshold Energy Swings up to stand completely upside down The Separatrix, a distinct cat-eye border loop High-Energy Whirling Spun with enough speed to flip around continuously Open, horizontal wavy lines stretching infinitely Damped Motion Swing slowing down due to friction or air drag A continuous inward spiral leading back to ✅ Summary of Phase Space Utility
The phase space model fundamentally converts the complex time-based changes of physical pendulum motion into a geometric map, allowing researchers to evaluate the total stability and energy limitations of dynamic systems without solving rigorous differential equations over time.
To explore this concept further, you can interact with a simulation on the PhET Interactive Simulations platform or study the foundational mathematical derivations on the Wolfram MathWorld Pendulum Page.