Math 2.0

Mechanics of particles and systems Definition

Euler free nutation

Builds on Rigid bodyInertia tensor and principal axesEuler's equations of rigid body motionAngular velocity

Definition

Let a rigid body have principal moments of inertia A=B<CA = B < C about body axes e1,e2,e3e_1, e_2, e_3, and let no external torque act on it. Write its angular velocity in the body frame as ω=(ω1,ω2,ω3)\omega = (\omega_1, \omega_2, \omega_3). By Euler's equations, ω3\omega_3 is constant and ω1(t)+i ω2(t)=m eiσEt,σE=C−AA ω3,\omega_1(t) + i\,\omega_2(t) = m\, e^{i \sigma_E t}, \qquad \sigma_E = \frac{C - A}{A}\,\omega_3, for a complex constant mm. If m≠0m \ne 0, this motion of ω\omega around e3e_3 in the body frame is the Euler free nutation, and 2π/σE2\pi/\sigma_E is the Euler period.

Remarks

Check: Euler's equations with A=BA = B read Aω˙1=−(C−A)ω2ω3A\dot\omega_1 = -(C - A)\omega_2\omega_3, Aω˙2=(C−A)ω3ω1A\dot\omega_2 = (C - A)\omega_3\omega_1 and Cω˙3=0C\dot\omega_3 = 0, so ddt(ω1+iω2)=iσE(ω1+iω2)\frac{d}{dt}(\omega_1 + i\omega_2) = i\sigma_E(\omega_1 + i\omega_2). Example: for the Earth (C−A)/A≈1/305(C - A)/A \approx 1/305, so the Euler period is about 305305 sidereal days. Non-example: if m=0m = 0, the body spins steadily about its symmetry axis and there is no nutation. The word "nutation" here refers to a motion relative to the body. Seen from space, the same motion is a small free precession of the symmetry axis around the fixed angular momentum vector.

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Cross product, concept, no evidence levelCross productconceptDerivative, concept, no evidence levelDerivativeconceptEuclidean space, object, no evidence levelEuclidean spaceobjectEuclidean vector, concept, no evidence levelEuclidean vectorconceptIdentity matrix, object, no evidence levelIdentity matrixobjectInertia tensor and principal axes, concept, no evidence levelInertia tensor and pr…conceptMatrix, definition, no evidence levelMatrixdefinitionOrdinary differential equation, concept, no evidence levelOrdinary differential…conceptRigid body, concept, no evidence levelRigid bodyconceptTorque, concept, no evidence levelTorqueconceptVector space, definition, no evidence levelVector spacedefinitionVelocity, concept, no evidence levelVelocityconceptClassical mechanics, concept, no evidence levelClassical mechanicsconceptSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionMomentum, concept, no evidence levelMomentumconceptOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAngular momentum, concept, no evidence levelAngular momentumconceptOrthogonal group, object, no evidence levelOrthogonal groupobjectAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectAngular velocity, concept, no evidence levelAngular velocityconceptEuler's equations of rigid body motion, statement, no evidence levelEuler's equations of …statementEuler free nutation, definition, no evidence levelEuler free nutationdefinition
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.
  • no evidence level
  1. Cross product concept
  2. Derivative concept
  3. Euclidean space object
  4. Euclidean vector concept
  5. Identity matrix object
  6. Inertia tensor and principal axes concept
  7. Matrix definition
  8. Ordinary differential equation concept
  9. Rigid body concept
  10. Torque concept
  11. Vector space definition
  12. Velocity concept
  13. Classical mechanics concept
  14. Square matrix definition
  15. Transpose definition
  16. Linear map definition
  17. Momentum concept
  18. Orthogonal matrix definition
  19. Multilinear map definition
  20. Angular momentum concept
  21. Orthogonal group object
  22. Alternating multilinear map definition
  23. Determinant definition
  24. Special orthogonal group object
  25. Angular velocity concept
  26. Euler's equations of rigid body motion statement
  27. Euler free nutation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Chandler_wobble
About this page
TagYV2XRKXHEV
URIm2://YV2XRKXHEV
Kinddefinition
WritingW0 stub
MSC70E15, 86A30
DefinesEuler free nutation
UsesRigid body, Inertia tensor and principal axes, Euler's equations of rigid body motion, Angular velocity
Changed3 November 2026

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