# Angular velocity

Angular velocity describes the speed of rotation. The direction of the angular velocity vector will be along the axis of rotation and In this case (counter-clockwise rotation) toward the viewer.

Angular velocity is the vector physical quantity that represents the rotation of a spinning body. It is usually represented by the symbol omega (Ω or ω). The magnitude of the angular velocity is the angular speed (or angular frequency) and is denoted by ω. The line of direction of the angular velocity is given by the axis of rotation, and the right hand rule indicates the positive direction, namely:

If you allow the fingers of your right hand to follow the direction of the rotation, then the direction of the angular velocity vector is indicated by your right thumb.

In SI units, angular velocity is measured in radians per second, (rad/s), although a direction must also be given. The dimensions of angular velocity are T -1, since radians are dimensionless.

With constant angular acceleration, the angular velocity conforms to the rotational equations of motion, equivalent to the standard linear equations of motion under constant linear acceleration.

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## The non-circular motion case

If the motion of a particle is described by a position vector-valued function r(t) — with respect to a fixed origin — then the angular velocity vector is

[itex] \vec\omega = {\mathbf{r} \times \mathbf{v} \over |\mathbf{r}|^2} \qquad \qquad (1) [itex]

where

[itex] \mathbf{v}(t) = \mathbf{r'}(t) [itex]

is the linear velocity vector. Equation (1) is applicable to non-circular motions, e.g. elliptic orbits.

### Derivation

Vector v can be resolved into a pair of components: [itex] \mathbf{v}_\perp [itex] which is perpendicular to r, and [itex] \mathbf{v}_\| [itex] which is parallel to r. The motion of the parallel component is completely linear and produces no rotation of the particle (with regard to the origin), so for purposes of finding the angular velocity it can be ignored. The motion of the perpendicular component is completely circular, since it is perpendicular to the radial vector, just like any tangent to a point on a circle.

The perpendicular component is

where the vector [itex] \mathbf{r} \times \mathbf{v} [itex] represents the area of the parallelogram two of whose sides are the vectors r and v. Dividing this area by the magnitude of r yields the height of this parallelogram between r and the side of the parallelogram parallel to r. This height is equal to the component of v which is perpendicular to r.

In the case of pure circular motion, the angular velocity is equal to linear velocity divided by the radius. In the case of generalized motion, the linear velocity is replaced by its component perpendicular to r, viz.

therefore, putting equations (2) and (3) together yields

[itex] \omega = {|\mathbf{r} \times \mathbf{v}| \over |\mathbf{r}|^2} = |\vec\omega|. \qquad \qquad (4)[itex]

Equation (4) gives the magnitude of the angular velocity vector. The vector's direction is given by its normalized version:

[itex] \hat\vec\omega = {\mathbf{r} \times \mathbf{v} \over |\mathbf{r} \times \mathbf{v}|}. \qquad \qquad (5)[itex]

Then the entire angular velocity vector is given by putting together its magnitude and its direction:

[itex] \vec\omega = \omega \hat\vec\omega [itex]

which, due to equations (4) and (5), is equal to

[itex] \vec\omega = {\mathbf{r} \times \mathbf{v} \over |\mathbf{r}|^2}, [itex]

which was to be demonstrated.

## The angular velocity of a rotation

### Introduction

We'll work in the more general setting of an [itex]n[itex]-dimensional real inner product space, [itex]V[itex]. A rotation is a smooth function [itex]\phi\colon[a,b]\to\mathsf{SO}(V)[itex], where [itex]a[itex] and [itex]b[itex] are real numbers with [itex]a

Differentiating the orthonormality relations [itex]\phi(t)^*\phi(t)=I[itex] gives

[itex]\phi(t)^*\dot\phi(t)+\dot\phi(t)^*\phi(t)=0;[itex]

since [itex](\phi(t)^*\dot\phi(t))^*=\dot\phi(t)^*\phi(t)[itex], this says that [itex]\phi(t)^*\dot\phi(t)=\sigma(t)[itex] is a skew-adjoint linear transformation of [itex]V[itex]. The function [itex]\sigma[itex] is called the angular velocity of the rotation. Conversely [1], given a fixed skew-adjoint linear transformation [itex]S[itex] of [itex]V[itex], there is a rotation having [itex]S[itex] as its constant angular velocity. Therefore [2] the set [itex]\mathsf{so}(V)[itex] of possible values of the angular velocity is the [itex]{n(n-1)\over2}[itex]-dimensional space of skew-adjoint linear transformations of [itex]V[itex]. (The set [itex]\mathsf{so}(V)[itex] is the Lie algebra of the Lie group [itex]\mathsf{SO}(V)[itex], that is, the tangent space to [itex]\mathsf{SO}(V)[itex] at the identity [see 1].)

### Motivation

The name, 'rotation', comes from imagining a rigid [itex]n[itex]-dimensional solid object able to turn freely about a fixed point [itex]\mathrm{O}[itex] during the time interval [itex][a,b][itex]. (In three dimensions, 'freely' can be taken to mean that both the speed and the axis of the turning are allowed to vary.) At each time [itex]t\in[a,b][itex], we define a map [itex]\phi_t\colon V\to V[itex] according to the orientation of the object, as follows. We choose a particular orientation of the object as the 'reference' orientation. Any pencil-mark on the object occupies a certain point in space when the object is in its reference orientation: then [itex]\phi_t[itex] maps that point to the point occupied by the same mark at time [itex]t[itex].

How many marks do we need to draw in order to determine [itex]\phi_t[itex]? Because the object is rigid, the distance between two marks is the same at time [itex]t[itex] as it is with the object in the reference position, which means that [itex]\phi_t[itex] is an isometry. In finite dimensional real inner product spaces, every isometry that fixes the origin is an orthogonal linear transformation [3]. In particular, since [itex]\phi_t[itex] is a linear transformation, provided the positions of the marks span the whole of [itex]V[itex] (and if they span it when the object is in any one orientation, they will always span it), the marks' positions at time [itex]t[itex] determine [itex]\phi_t[itex]. Minimal spanning sets for [itex]V[itex] contain [itex]n[itex] points, so we need to draw at least [itex]n[itex] marks on the object in order to determine [itex]\phi_t[itex].

Now we have defined the special orthogonal linear transformation [itex]\phi_t[itex] that describes the orientation of the object at time [itex]t[itex], we can take a step back and define the map [itex] \phi\colon[a,b]\to\mathsf{SO}(V) [itex] by [itex] \phi(t) = \phi_t [itex] for all [itex] t\in[a,b] [itex]. We call [itex]\phi[itex] a rotation, a rotary motion or rotatative motion, or 'a motion which fixes [itex]\mathrm{O}[itex]'; [itex]\phi[itex] is a smooth trajectory in the space of special orthogonal linear transformations of [itex]V[itex].

The formulation in the introduction deals with these legal niceties by jumping straight to [itex]\mathsf{SO}(V)[itex]. Also note that nothing in the definitions prohibits us from hypothetically ascribing angular velocity to rotating point particles, as well as to rigid, extended objects.

### Notes

[1] Rotations and Angular Momentum (http://math.ucr.edu/home/baez/classical/galilei2.pdf) on the Classical Mechanics page of the website of John Baez (http://math.ucr.edu/home/baez/README.html), and especially Questions 1 and 2.

[2] Peter M. Neumann; Gabrielle A. Stoy; Edward C. Thompson. Groups and Geometry, Oxford 1994, ISBN 01798534515. See p. 165.

[3] ibid., pp. 108-110, trivially extended to real inner product spaces of arbitrary finite dimension. The norm associated with our inner-product works in place of the Euclidean distance function. Exercise 10.5 shows that all distance-preserving maps are isometries as we claimed.

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