Phase Space
Core Idea
Phase space is a space in which each point represents one complete possible state of a dynamical system.
对 pendulum，一个 state 由 angle 与 angular velocity 决定，所以 phase space 可以用 平面表示。这里的 point 不是 bob 在 physical space 中的位置。
Reading the Picture
- point: one complete state;
- trajectory: one system history from one initial condition;
- Vector Field: instantaneous direction and speed of state change;
- fixed point: a state where the field is zero;
- phase flow: the collection of all trajectories.
高维系统遵循同一思想，即使无法直接 visualization。
Terminology Note
数学语境常宽泛地使用 phase space；Hamiltonian mechanics 中，物理学家可能把它更严格地保留给 position-momentum space。具体含义要看 context。
Connections
- Differential Equations generate trajectories.
- Fixed-Point Stability studies flow near equilibrium states.
- 12 - The Other Way to Visualize Derivatives supplies contraction/expansion intuition.