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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

예제 1

Draw phase space for simple pendulum.

예제 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

#위상공간#상공간#phase space#trajectory