nLab angular velocity

Idea

The analog of velocity for rotational movement.

For rotation in a plane inside a Cartesian space $\mathbb{R}^n$ the angular velocity is a bivector in $\wedge^2 \mathbb{R}^n$ of the form

$$\dot \omega \,\, e_1 \wedge e_2 \,,$$

where $e_1$ and $e_2$ are unit vector spanning the plane of rotation, and where $\dot \omega$ is the magnitude of the angular velocity.

Of $n = 3$ (and only then) can we identify bivectors with plain vectors (by the dual operation induced by the Hodge star operator). Often in the literature only this “angular velocity vector” in 3 dimensions is considered.

Related concepts

• rigid body dynamics

  • angular momentum

  • moment of inertia

• galaxy rotation curve

References

Standard discussion of angular velocity in $d \leq 3$ is for instance in

• wikipedia: angular velocity

The more general discussion in terms of bivectors is found for instance in Geometric Algebra-style documents, such as

• Chris Doran, Anthony Lasenby, Geometric Algebra for Physicists Cambridge University Press

  Physical applications of geometric algebra (pdf)