Math 2.0

Mechanics of particles and systems Definition

Euler force

Builds on Rotating reference frameFictitious forceAngular accelerationCross product

Definition

Let a reference frame turn about a fixed origin relative to an inertial frame. Its angular velocity ω(t)∈ℝ3\omega(t) \in \R^3 is a differentiable function of time tt. Let ω˙=dω/dt\dot\omega = d\omega/dt be its angular acceleration. Let a particle of mass m>0m > 0 have position r∈ℝ3r \in \R^3 in the rotating frame, measured from the origin. The Euler force on the particle is the fictitious force FE=−m ω˙×r,F_{\mathrm{E}} = -m\, \dot\omega \times r, where ×\times is the cross product.

Remarks

Example: let the frame turn about the zz-axis with ω(t)=(0,0,t)\omega(t) = (0, 0, t), so ω˙=(0,0,1)\dot\omega = (0, 0, 1). For m=2m = 2 and r=(3,0,0)r = (3, 0, 0) one gets FE=−2 (0,0,1)×(3,0,0)=(0,−6,0)F_{\mathrm{E}} = -2\, (0, 0, 1) \times (3, 0, 0) = (0, -6, 0). This points against the direction in which the frame speeds up its turning.

Non-example: if ω\omega is constant, then ω˙=0\dot\omega = 0 and the Euler force is zero, even though the centrifugal force is not. The Euler force is also called the azimuthal force or transverse force.

Reading path

To understand this page from the ground up, read these 30 pages in order. The foundations come first.

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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 matrixobjectMatrix, definition, no evidence levelMatrixdefinitionOrdinary differential equation, concept, no evidence levelOrdinary differential…conceptSecond derivative, concept, no evidence levelSecond derivativeconceptVector space, definition, no evidence levelVector spacedefinitionVelocity, concept, no evidence levelVelocityconceptCross product, definition, no evidence levelCross productdefinitionClassical 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 levelMomentumconceptInertial frame of reference, concept, no evidence levelInertial frame of ref…conceptAcceleration, concept, no evidence levelAccelerationconceptOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionNewton's laws of motion, statement, no evidence levelNewton's laws of moti…statementOrthogonal 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 velocityconceptRotating reference frame, concept, no evidence levelRotating reference fr…conceptAngular acceleration, concept, no evidence levelAngular accelerationconceptFictitious force, concept, no evidence levelFictitious forceconceptEuler force, definition, no evidence levelEuler forcedefinition
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. Matrix definition
  7. Ordinary differential equation concept
  8. Second derivative concept
  9. Vector space definition
  10. Velocity concept
  11. Cross product definition
  12. Classical mechanics concept
  13. Square matrix definition
  14. Transpose definition
  15. Linear map definition
  16. Momentum concept
  17. Inertial frame of reference concept
  18. Acceleration concept
  19. Orthogonal matrix definition
  20. Multilinear map definition
  21. Newton's laws of motion statement
  22. Orthogonal group object
  23. Alternating multilinear map definition
  24. Determinant definition
  25. Special orthogonal group object
  26. Angular velocity concept
  27. Rotating reference frame concept
  28. Angular acceleration concept
  29. Fictitious force concept
  30. Euler force definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Euler_force
About this page
TagNN0219TYZ1
URIm2://NN0219TYZ1
Kinddefinition
WritingW0 stub
MSC70B05, 70A05
DefinesEuler force
UsesRotating reference frame, Fictitious force, Angular acceleration, Cross product
Changed3 November 2026

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