∞

Phase Space

Undergraduate

Definition

Phase space represents all possible states of a dynamical system as coordinates. System evolution can be visualized as trajectories.

(q₁, ..., qₙ, p₁, ..., pₙ) ∈ ℝ^2n

Phase space coordinates for n-degree system

\dotqᵢ = (∂ H)/(∂ pᵢ), \dotpᵢ = -(∂ H)/(∂ qᵢ)

Hamilton's equations

Example 1

Draw phase space for simple pendulum.

Example 2

What is the phase space trajectory of harmonic oscillator?

Discovered by: Joseph Liouville, William Hamilton (19th century)

Geometric analysis of system states developed alongside analytical mechanics.

Planetary motion, pendulum

Ensemble theory

System stability analysis