Math 2.0

Mechanics of particles and systems Definition

Angular velocity

Builds on Special orthogonal groupSkew-symmetric matrixCross productDerivative

Definition

Let I⊆ℝI \subseteq \R be an open interval and let R ⁣:I→SO(3)R \colon I \to SO(3) be a differentiable curve in the special orthogonal group. The matrix R(t)R(t) gives the orientation at time tt of a rotating body or frame relative to a fixed frame. For each t∈It \in I the matrix R˙(t)R(t)T\dot R(t) R(t)^T is skew-symmetric. The angular velocity at time tt is the unique vector ω(t)∈ℝ3\omega(t) \in \R^3 such that R˙(t)R(t)Ty=ω(t)×yfor all y∈ℝ3,\dot R(t) R(t)^T y = \omega(t) \times y \quad \text{for all } y \in \R^3, where ×\times is the cross product. Its length ∣ω(t)∣\abs{\omega(t)} is the angular speed.

Remarks

The matrix is skew-symmetric because differentiating RRT=IR R^T = I gives R˙RT+(R˙RT)T=0\dot R R^T + (\dot R R^T)^T = 0, and every skew-symmetric 3×33 \times 3 matrix acts as y↦ω×yy \mapsto \omega \times y for exactly one vector ω\omega. Example: rotation about the zz-axis by the angle θ(t)\theta(t) gives R˙RTy=θ˙ (−y2,y1,0)\dot R R^T y = \dot\theta\, (-y_2, y_1, 0), so ω=(0,0,θ˙)\omega = (0, 0, \dot\theta). Non-example: the vector (0,0,θ(t))(0, 0, \theta(t)) is not the angular velocity; the angle is a position, and only its rate θ˙\dot\theta enters. A point fixed in the body at R(t)XR(t) X has velocity ω(t)×R(t)X\omega(t) \times R(t) X. For a single particle at position r≠0r \ne 0 with velocity vv, its angular velocity about the origin is often defined as r×v/∣r∣2r \times v / \abs{r}^2.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

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Cross product, concept, no evidence levelCross productconceptDerivative, concept, no evidence levelDerivativeconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionSkew-symmetric matrix, concept, no evidence levelSkew-symmetric matrixconceptVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal 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, definition, no evidence levelAngular velocitydefinition
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. Identity matrix object
  4. Matrix definition
  5. Skew-symmetric matrix concept
  6. Vector space definition
  7. Square matrix definition
  8. Transpose definition
  9. Linear map definition
  10. Orthogonal matrix definition
  11. Multilinear map definition
  12. Orthogonal group object
  13. Alternating multilinear map definition
  14. Determinant definition
  15. Special orthogonal group object
  16. Angular velocity definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Angular_velocity
About this page
Tag5D6DHEP157
URIm2://5D6DHEP157
Kinddefinition
WritingW0 stub
MSC70B05, 70B10
DefinesAngular velocity
UsesSpecial orthogonal group, Skew-symmetric matrix, Cross product, Derivative
Changed3 November 2026

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