# Phase space

**Phase space**: Space of all possible states of a system, where each state has a unique point in the phase space

**Phase space trajectory**: path of system’s state evolving over time, useful for understanding patterns of the system’s progression

**Differentiating between phase and parameter space**: parameter space is the space of all possible parameter values of a model (e.g. with $n$ parameters, the parameter space is usually $\mathcal{P} \subseteq \mathcal{R}^n$). Phase spaces differ in that they include all possible *system states*, not strictly the parameters underlying the system. For example, in mechanical systems, the state is fully described by position and momentum variables, instead of perhaps the underlying parameters controlling the system (like gravitational forcecs, giving rise to *families* of phase spaces). Additionally, phase spaces/diagrams are commonly used to show the changing state of a system over time.