Math 2.0

Mechanics of particles and systems Example

Pendulum equation

Builds on Harmonic oscillatorFirst integralHyperbolic equilibrium point

Example

Let g>0g > 0 be the acceleration of gravity and ℓ>0\ell > 0 the length of a rigid massless rod, with a point mass at its end. Let θ(t)\theta(t) be the angle of the rod from the downward vertical at time tt. Without friction, the motion satisfies the pendulum equation θ′′(t)+gℓsin⁡θ(t)=0.\theta''(t) + \frac{g}{\ell} \sin \theta(t) = 0. As a first-order system, it is θ′=ω\theta' = \omega, ω′=−gℓsin⁡θ\omega' = -\frac{g}{\ell} \sin \theta. The function E(θ,ω)=12ω2−gℓcos⁡θE(\theta, \omega) = \frac{1}{2} \omega^2 - \frac{g}{\ell} \cos \theta is a first integral.

Remarks

For small angles, sin⁡θ≈θ\sin \theta \approx \theta. This gives the harmonic oscillator θ′′+gℓθ=0\theta'' + \frac{g}{\ell} \theta = 0, with the solution θ(t)=θ0cos⁡(g/ℓ t)\theta(t) = \theta_0 \cos\left(\sqrt{g/\ell}\, t\right) for θ(0)=θ0\theta(0) = \theta_0, θ′(0)=0\theta'(0) = 0, and the period 2πℓ/g2 \pi \sqrt{\ell / g}. For the nonlinear equation, the period depends on the amplitude. It is given by an elliptic integral and it is longer than 2πℓ/g2 \pi \sqrt{\ell / g}. The equilibrium (0,0)(0, 0), hanging down, is stable. The equilibrium (π,0)(\pi, 0), standing up, is a hyperbolic saddle, so it is unstable.

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Complex numbers, object, no evidence levelComplex numbersobjectDifferential equation, concept, no evidence levelDifferential equationconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptEuler's formula, statement, no evidence levelEuler's formulastatementOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionGrönwall's inequality, statement, no evidence levelGrönwall's inequalitystatementOrder of a differential equation, definition, no evidence levelOrder of a differenti…definitionSolution 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 problemdefinitionLinear differential equation of order n, definition, no evidence levelLinear differential e…definitionEquilibrium point, definition, no evidence levelEquilibrium pointdefinitionSystem of linear differential equations, definition, no evidence levelSystem of linear diff…definitionUniqueness of solutions for Lipschitz vector fields, statement, no evidence levelUniqueness of solutio…statementHomogeneous linear equation, definition, no evidence levelHomogeneous linear eq…definitionLinear equation with constant coefficients, definition, no evidence levelLinear equation with …definitionHomogeneous linear system with constant coefficients, definition, no evidence levelHomogeneous linear sy…definitionMaximal solution, definition, no evidence levelMaximal solutiondefinitionSuperposition principle, statement, no evidence levelSuperposition princip…statementCharacteristic equation, definition, no evidence levelCharacteristic equati…definitionMatrix exponential as the solution of a linear system, statement, no evidence levelMatrix exponential as…statementFlow of a vector field, definition, no evidence levelFlow of a vector fielddefinitionSolution space of a homogeneous linear equation has dimension n, statement, no evidence levelSolution space of a h…statementLyapunov stability, definition, no evidence levelLyapunov stabilitydefinitionTrajectory, definition, no evidence levelTrajectorydefinitionFundamental system of solutions, definition, no evidence levelFundamental system of…definitionAsymptotic stability, definition, no evidence levelAsymptotic stabilitydefinitionFirst integral, definition, no evidence levelFirst integraldefinitionSolutions for distinct real roots, statement, no evidence levelSolutions for distinc…statementStability of linear systems via eigenvalues, statement, no evidence levelStability of linear s…statementSolutions for repeated roots, statement, no evidence levelSolutions for repeate…statementLinearisation at an equilibrium, concept, no evidence levelLinearisation at an e…conceptSolutions for complex conjugate roots, statement, no evidence levelSolutions for complex…statementHyperbolic equilibrium point, definition, no evidence levelHyperbolic equilibriu…definitionHarmonic oscillator, example, no evidence levelHarmonic oscillatorexamplePendulum equation, example, no evidence levelPendulum equationexample
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  1. Complex numbers object
  2. Differential equation concept
  3. Function concept
  4. Interval concept
  5. Euler's formula statement
  6. Ordinary differential equation definition
  7. Lipschitz condition definition
  8. Grönwall's inequality statement
  9. Order of a differential equation definition
  10. Solution of an ordinary differential equation definition
  11. Autonomous system definition
  12. Initial value problem definition
  13. Linear differential equation of order n definition
  14. Equilibrium point definition
  15. System of linear differential equations definition
  16. Uniqueness of solutions for Lipschitz vector fields statement
  17. Homogeneous linear equation definition
  18. Linear equation with constant coefficients definition
  19. Homogeneous linear system with constant coefficients definition
  20. Maximal solution definition
  21. Superposition principle statement
  22. Characteristic equation definition
  23. Matrix exponential as the solution of a linear system statement
  24. Flow of a vector field definition
  25. Solution space of a homogeneous linear equation has dimension n statement
  26. Lyapunov stability definition
  27. Trajectory definition
  28. Fundamental system of solutions definition
  29. Asymptotic stability definition
  30. First integral definition
  31. Solutions for distinct real roots statement
  32. Stability of linear systems via eigenvalues statement
  33. Solutions for repeated roots statement
  34. Linearisation at an equilibrium concept
  35. Solutions for complex conjugate roots statement
  36. Hyperbolic equilibrium point definition
  37. Harmonic oscillator example
  38. Pendulum equation example

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Pendulum_(mechanics), Simple gravity pendulum
  2. Web https://en.wikipedia.org/wiki/Pendulum_(mechanics), Small-angle approximation
About this page
Tag0SHPXM2HWH
URIm2://0SHPXM2HWH
Kindexample
WritingW0 stub
MSC70K05, 34C15
Instance ofHamiltonian system, Autonomous system
UsesHarmonic oscillator, First integral, Hyperbolic equilibrium point

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