Math 2.0

Ordinary differential equations Definition

Phase space

Builds on Autonomous systemTrajectoryOrdinary differential equation

Definition

Let U⊆ℝnU \subseteq \R^n be an open set and let f:U→ℝnf : U \to \R^n be a continuous map. The phase space of the autonomous system x′=f(x)x' = f(x) is the set UU. Its points are called states, and the curve t↦x(t)t \mapsto x(t) of a solution in UU is its trajectory. For an ordinary differential equation y(n)=g(y,y′,…,y(n−1))y^{(n)} = g(y, y', \dots, y^{(n-1)}) of order nn, the phase space is the phase space of the equivalent first-order system for the state (y,y′,…,y(n−1))(y, y', \dots, y^{(n-1)}).

Remarks

Example: the equation x′′=−xx'' = -x becomes the system x′=vx' = v, v′=−xv' = -x on U=ℝ2U = \R^2. Its phase space is the plane with coordinates (x,v)(x, v). The solution x=cos⁡tx = \cos t has the trajectory (cos⁡t,−sin⁡t)(\cos t, -\sin t), which runs on the unit circle, since cos⁡2t+sin⁡2t=1\cos^2 t + \sin^2 t = 1. Non-example: for a pendulum, the set of angles alone is not the phase space. The angle at one time does not determine the motion; the angular velocity is also needed. For a non-autonomous system x′=f(t,x)x' = f(t, x) on an open set D⊆ℝ×ℝnD \subseteq \R \times \R^n, the set DD is called the extended phase space. For a Hamiltonian system with nn degrees of freedom, the phase space has the 2n2n coordinates (q1,…,qn,p1,…,pn)(q_1, \dots, q_n, p_1, \dots, p_n) of positions and momenta.

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Differential equation, concept, no evidence levelDifferential equationconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionGrönwall's inequality, statement, no evidence levelGrönwall's inequalitystatementSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionAutonomous system, definition, no evidence levelAutonomous systemdefinitionInitial value problem, definition, no evidence levelInitial value problemdefinitionUniqueness of solutions for Lipschitz vector fields, statement, no evidence levelUniqueness of solutio…statementMaximal solution, definition, no evidence levelMaximal solutiondefinitionFlow of a vector field, definition, no evidence levelFlow of a vector fielddefinitionTrajectory, definition, no evidence levelTrajectorydefinitionPhase space, definition, no evidence levelPhase spacedefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Differential equation concept
  2. Function concept
  3. Interval concept
  4. Ordinary differential equation definition
  5. Lipschitz condition definition
  6. Grönwall's inequality statement
  7. Solution of an ordinary differential equation definition
  8. Autonomous system definition
  9. Initial value problem definition
  10. Uniqueness of solutions for Lipschitz vector fields statement
  11. Maximal solution definition
  12. Flow of a vector field definition
  13. Trajectory definition
  14. Phase space definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Phase_space
About this page
TagRY2RAH3802
URIm2://RY2RAH3802
Kinddefinition
WritingW0 stub
MSC34C99, 37C10, 70H05
DefinesPhase space
UsesAutonomous system, Trajectory, Ordinary differential equation
Changed2 November 2026

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