The Space of State
Phase Space is a multi-dimensional space where every point represents a complete "State" of the system (both position \(q\) and momentum \(p\)). For a single particle in 1D, phase space is a 2D plane.
A system's evolution is a Path in phase space. Because the equations are 1st-order, phase space paths can never cross—if they did, the future would be unpredictable.
Worked Examples
Example 1: The Harmonic Oscillator
In phase space, the harmonic oscillator follows an ellipse. As it moves, energy sloshes back and forth between position (potential) and momentum (kinetic), but the "point" stays on the same energy ellipse forever.
The Bridge to Quantum Mechanics
Quantum Mechanics is the "blurring" of phase space. In classical mechanics, a particle is a single point in phase space. In Quantum Mechanics, the Uncertainty Principle \(\Delta x \Delta p \geq \hbar/2\) means a particle must occupy a minimum Area of phase space (a "Phase Cell"). You can never zoom in smaller than \(\hbar\). This granularity of phase space is what prevents atoms from collapsing and allows for the existence of stable matter.