Math 2.0

Ordinary differential equations Definition

First integral

Builds on Autonomous systemSolution of an ordinary differential equationTrajectory

Definition

Let UU be an open subset of ℝn\mathbb{R}^n and let f:U→ℝnf : U \to \mathbb{R}^n be a function. Let H:U→ℝH : U \to \mathbb{R} be a continuously differentiable function. It is a first integral of the autonomous system x′=f(x)x' = f(x) if ∇H(x)⋅f(x)=0for each x∈U,\nabla H(x) \cdot f(x) = 0 \quad \text{for each } x \in U, and if HH is not constant on any non-empty open subset of UU.

Remarks

The condition means that ddtH(γ(t))=0\frac{d}{dt} H(\gamma(t)) = 0 for each solution γ\gamma. So HH is constant along each solution. Each trajectory lies in a level set {x:H(x)=c}\{x : H(x) = c\}. In the plane, the level curves of HH give the phase portrait. For the harmonic oscillator x′=yx' = y, y′=−xy' = -x, the function H(x,y)=x2+y2H(x, y) = x^2 + y^2 is a first integral. The Hamiltonian of a time-independent Hamiltonian system is a first integral.

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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 levelTrajectorydefinitionFirst integral, definition, no evidence levelFirst integraldefinition
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. First integral definition

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Constant_of_motion, lead section
  2. Web https://en.wikipedia.org/wiki/Constant_of_motion, Methods for identifying constants of motion
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Kinddefinition
WritingW0 stub
MSC34A05, 34C14
DefinesFirst integral
UsesAutonomous system, Solution of an ordinary differential equation, Trajectory

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