Math 2.0

Mechanics of particles and systems Definition

Precession

Builds on Rigid bodyEuler anglesAngular velocity

Definition

Let a rigid body turn about a fixed point OO. Let e3e_3 be a unit vector fixed in the body, such as a symmetry axis. Let uu be a fixed unit vector in space, such as the upward vertical. Describe the orientation of the body by Euler angles (φ,θ,ψ)(\varphi, \theta, \psi) with respect to uu. Here θ∈[0,π]\theta \in [0, \pi] is the angle between uu and e3e_3, and φ\varphi is the azimuth of e3e_3 about uu. The angle ψ\psi is the angle of rotation of the body about e3e_3. The precession of the body is the change of the angle φ\varphi in time, that is, the rotation of the axis e3e_3 about uu. Its rate φ˙\dot\varphi is the precession rate.

Remarks

The angular velocity of the body is φ˙ u+θ˙ n+ψ˙ e3\dot\varphi\, u + \dot\theta\, n + \dot\psi\, e_3, where nn is the unit vector along the line of nodes, perpendicular to uu and e3e_3. So the motion splits into precession φ˙\dot\varphi, nutation θ˙\dot\theta and spin ψ˙\dot\psi. The angle φ\varphi is defined only when e3≠±ue_3 \ne \pm u.

Example: a symmetric top with mass mm, centre of mass at distance ℓ\ell from OO on its axis, and moment I3I_3 about the axis, spinning fast with rate ω3\omega_3. For a fast top in steady motion the precession rate is approximately φ˙≈mgℓ/(I3ω3)\dot\varphi \approx m g \ell / (I_3 \omega_3).

Non-example: a body spinning about a fixed axis e3e_3 has constant φ\varphi, so it does not precess.

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Cross product, concept, no evidence levelCross productconceptDerivative, concept, no evidence levelDerivativeconceptEuclidean vector, concept, no evidence levelEuclidean vectorconceptEuler angles, concept, no evidence levelEuler anglesconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionRigid body, concept, no evidence levelRigid bodyconceptVector space, definition, no evidence levelVector spacedefinitionVelocity, concept, no evidence levelVelocityconceptSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionOrthogonal group, object, no evidence levelOrthogonal groupobjectDeterminant, definition, no evidence levelDeterminantdefinitionSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectAngular velocity, concept, no evidence levelAngular velocityconceptPrecession, definition, no evidence levelPrecessiondefinition
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. Cross product concept
  2. Derivative concept
  3. Euclidean vector concept
  4. Euler angles concept
  5. Identity matrix object
  6. Matrix definition
  7. Rigid body concept
  8. Vector space definition
  9. Velocity concept
  10. Square matrix definition
  11. Transpose definition
  12. Linear map definition
  13. Multilinear map definition
  14. Orthogonal matrix definition
  15. Alternating multilinear map definition
  16. Orthogonal group object
  17. Determinant definition
  18. Special orthogonal group object
  19. Angular velocity concept
  20. Precession definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Precession
About this page
TagQCGAG68T5Z
URIm2://QCGAG68T5Z
Kinddefinition
WritingW0 stub
MSC70E05, 70E17
DefinesPrecession
UsesRigid body, Euler angles, Angular velocity
Changed3 November 2026

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