4.2.1 · v1 · 75–90 min
State variables and phase space
Learning objectives
- Define state, state vector, phase space, trajectory, flow and initial condition
- convert an ODE to state form
- distinguish trajectory from phase portrait.
Worked examples
Worked example 1
System: T'=−.2(T−20). Goal: state representation. State/model: x=T, x'=−.2(x−20). Method/calculation: x(t)=20+(x₀−20)e^{−.2t}. Interpretation: each x₀ gives a different trajectory toward 20. Validation: substitution and units. Limit: ideal constant environment.
Worked examples
Worked example 1
System: oscillator q''+4q=0. State: (q,v), q'=v,v'=−4q. Calculation: with (1,0), q=cos2t,v=−2sin2t; v²+4q²=4. Phase interpretation: closed ellipse. Validation: differentiate and check invariant. Limit: no damping.
Guided practice
- choose a state for
y''+3y'+2y=0. - explain why one sensor output may not determine state. A/B/C/D: define; convert; correct “one trajectory is the portrait”; design an accessible state table. Error: confusing output with state.
Knowledge check and mastery evidence
4/4 checks including correct conversion and initial condition.
Related laboratory
Phase Portrait.
Next lesson
Equilibrium and stability.