In
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, the
angular velocity is a vector quantity (more precisely, a
pseudovectorIn physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image...
) which specifies the angular speed of an object and the axis about which the object is rotating. The
SISi, si, or SI may refer to : Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per second,
revolutions per minuteRevolutions per minute is a measure of the frequency of a rotation. It annotates the number of full rotations completed in one minute around a fixed axis...
, degrees per hour, etc. It is sometimes also called the
rotational velocity and its magnitude the
rotational speedRotational speed tells how many complete rotations there are per time unit. It is therefore a cyclic frequency, measured in hertz in the SI System...
, typically measured in cycles or rotations per unit time (e.g.
revolutions per minuteRevolutions per minute is a measure of the frequency of a rotation. It annotates the number of full rotations completed in one minute around a fixed axis...
). Angular velocity is usually represented by the symbol
omegaOmega is the 24th and last letter of the Greek alphabet. In the Greek numeric system, it has a value of 800. The word literally means "great O" , as opposed to omicron, which means "little O"...
(
ω, rarely
Ω).
The direction of the angular velocity vector is perpendicular to the plane of rotation, in a direction which is usually specified by the
righthand ruleIn mathematics and physics, the righthand rule is a common mnemonic for understanding notation conventions for vectors in 3 dimensions. It was invented for use in electromagnetism by British physicist John Ambrose Fleming in the late 19th century....
.
Particle in two dimensions
The angular velocity of a particle is measured around or relative to a point, called the origin. As shown in the diagram (with angles
ɸ and
θ in
radianRadian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s), if a line is drawn from the origin (O) to the particle (P), then the velocity (
v) of the particle has a component along the radius (radial component,
v_{‖}) and a component perpendicular to the radius (crossradial component,
v_{⊥}). If there is no radial component the particle moves in a circle, while if there is no component perpendicular to the radius the particle moves along a straight line through the origin.
A radial motion produces no change in the direction of the particle relative to the origin, so for purposes of finding the angular velocity the radial component can be ignored. Therefore, the rotation is completely produced by the perpendicular motion around the origin, and the angular velocity is completely determined by this component.
In two dimensions the angular velocity
ω is given by

This is related to the crossradial (tangential) velocity by:

An explicit formula for
v_{⊥} in terms of
v and
θ is:

Combining the above equations gives a formula for
ω:

In two dimensions the angular velocity is a single number which has no direction, but it does have a sense or orientation. In two dimensions the angular velocity is a
pseudoscalarIn physics, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion such as improper rotations while a true scalar does not.The prototypical example of a pseudoscalar is the scalar triple product...
, a quantity which changes its sign under a
parity inversionIn physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...
(for example if one of the axes is inverted or they are swapped). The positive direction of rotation is taken, by convention, to be in the direction towards the
y axis from the
x axis. If parity is inverted, but the sense of a rotation does not, then the sign of the angular velocity changes.
Particle in three dimensions
In three dimensions, the angular velocity becomes a bit more complicated. The angular velocity in this case is generally thought of as a vector, or more precisely, a
pseudovectorIn physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image...
. It now has not only a magnitude, but a direction as well. The magnitude is the angular speed, and the direction describes the axis of rotation. The
righthand ruleIn mathematics and physics, the righthand rule is a common mnemonic for understanding notation conventions for vectors in 3 dimensions. It was invented for use in electromagnetism by British physicist John Ambrose Fleming in the late 19th century....
indicates the positive direction of the angular velocity pseudovector.
Being
an unitary vector over the instantaneous rotation axis, so that from the top of the vector the rotation is counterclockwise the angular velocity vector
can be defined as:

Just as in the two dimensional case, a particle will have a component of its velocity along the radius from the origin to the particle, and another component perpendicular to that radius. The combination of the origin point and the perpendicular component of the velocity defines a
plane of rotationIn geometry, a plane of rotation is an abstract object used to describe or visualise rotations in space. In three dimensions it is an alternative to the axis of rotation, but unlike the axis of rotation it can be used in other dimensions, such as two, four or more dimensions.Mathematically such...
in which the behavior of the particle (for that instant) appears just as it does in the two dimensional case. The axis of rotation is then a line normal to this plane, and this axis defined the direction of the angular velocity pseudovector, while the magnitude is the same as the pseudoscalar value found in the 2dimensional case. Using the unit vector
defined before, the angular velocity vector may be written in a manner similar to that for two dimensions:

which, by the definition of the
cross productIn mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in threedimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
, can be written:

Addition of angular velocity vectors
It is possible to define an addition operation for angular velocity vectors using composition of movements.
If a point rotates with
in a frame
which rotates itself with angular speed
respect an external frame
, we can define the addition of
like the angular velocity vector of the point respect
.
With this operation defined like this, angular velocity, which is a pseudovector, becomes also a real vector because it has two operations:
 An internal operation (addition) which is associative, commutative, distributive and with zero and unity elements
 An external operation (external product), with the normal properties for an external product.
This is the definition of a vector space. Therefore pseudovectors are a subset of the real vectors, despite their name suggesting the opposite. The only property that presents difficulties to prove is the commutativity of the addition. This can be proven from the fact that the velocity tensor W (see below) is skewsymmetric. Therefore
is a rotation matrix and in a time dt is an infinitesimal rotation matrix. Therefore it can be expanded as
The composition of rotations is not commutative, but when they are infinitesimal rotations the first order approximation of the previous series can be taken and
, and therefore
Rotating frames
Given a rotating frame composed by three unitary vectors, all the three must have the same angular speed in any instant. In such a frame each vector is a particular case of the previous case (moving particle), in which the module of the vector is constant.
Though it is just a particular case of the previous one, is a very important one for its relationship with the
rigid bodyIn physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...
study, and special tools have been developed for this case. There are two possible ways to describe the angular velocity of a rotating frame. The angular velocity vector and the angular velocity tensor. Both entities are related and they can be calculated from each other.
Angular velocity vector for a frame
It is defined as the angular velocity of each of the vectors of the frame, in a consistent way with the general definition.
It is known by the
Euler's rotation theoremIn geometry, Euler's rotation theorem states that, in threedimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two...
that for a rotating frame there exists an instantaneous axis of rotation in any instant. In the case of a frame, the angular velocity vector is over the instantaneous axis of rotation.
Any transversal section of a plane perpendicular to this axis has to behave as a two dimensional rotation. Thus, the magnitude of the angular velocity vector at a given time
t is consistent with the two dimensions case.
Angular velocity is a vector defining an addition operation. components can be calculated from the derivatives of the parameters defining the moving frame (Euler angles or rotation matrices)
Addition of angular velocity vectors in frames
As in the general case, the addition operation for angular velocity vectors can be defined using movement composition. In the case of rotating frames, the movement composition is simpler than the general case because the final matrix is always a product of rotation matrices.
As in the general case, addition is commutative
Components from the vectors of the frame
Substituting in the expression

any vector e of the frame we obtain
, and therefore
As the columns of the matrix of the frame are the components of its vectors, this allows also to calculate
from the matrix of the frame and its derivative.
Components from Euler angles
The components of the angular velocity pseudovector were first calculated by
Leonhard EulerLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
using his
Euler anglesThe Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3dimensional Euclidean space three parameters are required...
and an intermediate frame made out of the intermediate frames of the construction:
 One axis of the reference frame (the precession axis)
 The line of nodes of the moving frame respect the reference frame (nutation axis)
 One axis of the moving frame (the intrinsic rotation axis)
Euler proved that the projections of the angular velocity pseudovector over these three axes was the derivative of its associated angle (which is equivalent to decompose the instant rotation in three instantaneous Euler rotations). Therefore:

This basis is not orthonormal and it is difficult to use, but now the velocity vector can be changed to the fixed frame or to the moving frame with just a change of bases. For example, changing to the mobile frame:

where IJK are unit vectors for the frame fixed in the moving body.
Components from infinitesimal rotation matrices
The components of the angular velocity vector can be calculated from infinitesimal rotations (if available) as follows:
 As any rotation matrix has a single real eigenvalue, which is +1, this eigenvalue shows the rotation axis.
 Its module can be deduced from the value of the infinitesimal rotation.
Angular velocity tensor
It can be introduced from rotation matrices. Any vector that rotates around an axis with an angular speed vector (as defined before) satisfies:
We can introduce here the angular velocity tensor associated to the angular speed :

This tensor W(t) will act as if it were a operator :

Given the orientation matrix A(t) of a frame, we can obtain its instant angular velocity tensor W as follows. We know that:

As angular speed must be the same for the three vectors of a rotating frame A(t), we can write for all the three:

And therefore the angular velocity tensor we are looking for is:

Properties of angular velocity tensors
In general, the angular velocity in an ndimensional space is the time derivative of the angular displacement tensor which is a second rank skewsymmetric tensor.
This tensor W will have n(n1)/2 independent components and this number is the dimension of the Lie algebraIn mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
of the Lie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
of rotations of an ndimensional inner product space.
Exponential of W
In three dimensions angular velocity can be represented by a pseudovector because second rank tensors are dualIn mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finitedimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
to pseudovectors in three dimensions.
As . This can be read as a differential equation that defines A(t) knowing W(t).


And if the angular speed is constant then is also constant and the equation can be integrated. The result is:


which shows a connection with the Lie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
of rotations.
W is skewsymmetric
It is possible to prove that angular velocity tensor are skew symmetric matricesIn mathematics, and in particular linear algebra, a skewsymmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation If the entry in the and is aij, i.e...
. Being , with R(t) a rotation matrix, and taking the time derivative of :
 , because R(t) is a rotation matrix

Applying the formula (AB)^{t} = B^{t}A^{t}:

Thus, W is the negative of its transpose, which implies it is a skew symmetric matrix.
Duality respect the velocity vector
The tensor is a matrix with this structure:

As it is a skew symmetric matrixIn mathematics, and in particular linear algebra, a skewsymmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation If the entry in the and is aij, i.e...
it has a Hodge dual vectorIn mathematics, the Hodge star operator or Hodge dual is a significant linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finitedimensional oriented inner product space.Dimensions and algebra:...
which is precisely the previous angular velocity vector :

Coordinatefree description
At any instant, , the angular velocity tensor is a linear map between the position vectors
and their velocity vectors of a rigid body rotating around the origin:

where we omitted the parameter, and regard and as elements of the same 3dimensional Euclidean vector space .
The relation between this linear map and the angular velocity pseudovectorIn physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image...
is the following.
Because of W is the derivative of an orthogonal transformation, the

bilinear form is skewsymmetric. (Here stands for the scalar product). So we can apply the fact of exterior algebraIn mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higherdimensional analogs...
that there is a unique linear form on that
 ,
where is the wedge product of and .
Taking the dual vector L* of L we get

Introducing , as the Hodge dualIn mathematics, the Hodge star operator or Hodge dual is a significant linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finitedimensional oriented inner product space.Dimensions and algebra:...
of L*, and apply further Hodge dual identities we arrive at

where

by definition.
Because is an arbitrary vector, from nondegeneracy of scalar product follows

Angular velocity as a vector field
For angular velocity tensor maps velocities to positions, it is a vector fieldIn vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
. In particular, this vector field is a Killing vector fieldIn mathematics, a Killing vector field , named after Wilhelm Killing, is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold...
belonging to an element of the Lie algebraIn mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
so(3) of the 3dimensional rotation groupIn mechanics and geometry, the rotation group is the group of all rotations about the origin of threedimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation ...
, SO(3). This element of so(3) can also be regarded as the angular velocity vector.
Rigid body considerations
The same equations for the angular speed can be obtained reasoning over a rotating rigid bodyIn physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...
. Here is not assumed that the rigid body rotates around the origin. Instead it can be supposed rotating around an arbitrary point which is moving with a linear velocity V(t) in each instant.
To obtain the equations it is convenient to imagine a rigid body attached to the frames and consider a coordinate system that is fixed with respect to the rigid body. Then we will study the coordinate transformations between this coordinate and the fixed "laboratory" system.
As shown in the figure on the right, the lab system's origin is at point O, the rigid body system origin is at O' and the vector from O to O' is R. A particle (i) in the rigid body is located at point P and the vector position of this particle is R_{i} in the lab frame, and at position r_{i} in the body frame. It is seen that the position of the particle can be written:

The defining characteristic of a rigid body is that the distance between any two points in a rigid body is unchanging in time. This means that the length of the vector is unchanging. By Euler's rotation theoremIn geometry, Euler's rotation theorem states that, in threedimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two...
, we may replace the vector with where is a 3x3 rotation matrix and is the position of the particle at some fixed point in time, say t=0. This replacement is useful, because now it is only the rotation matrix which is changing in time and not the reference vector , as the rigid body rotates about point O'. Also, since the three columns of the rotation matrix represent the three versors of a reference frame rotating together with the rigid body, any rotation about any axis becomes now visible, while the vector would not rotate if the rotation axis were parallel to it, and hence it would only describe a rotation about an axis perpendicular to it (i.e., it would not see the component of the angular velocity pseudovector parallel to it, and would only allow the computation of the component perpendicular to it). The position of the particle is now written as:

Taking the time derivative yields the velocity of the particle:

where V_{i} is the velocity of the particle (in the lab frame) and V is the velocity of O' (the origin of the rigid body frame). Since is a rotation matrix its inverse is its transpose. So we substitute :



or

where is the previous angular velocity tensor.
It can be proved that this is skew symmetric matrixIn mathematics, and in particular linear algebra, a skewsymmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation If the entry in the and is aij, i.e...
, so we can take its dualIn mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finitedimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
to get a 3 dimensional pseudovector which is precisely the previous angular velocity vector :

Substituting ω for W into the above velocity expression, and replacing matrix multiplication by an equivalent cross product:

It can be seen that the velocity of a point in a rigid body can be divided into two terms – the velocity of a reference point fixed in the rigid body plus the cross product term involving the angular velocity of the particle with respect to the reference point. This angular velocity is the "spin" angular velocity of the rigid body as opposed to the angular velocity of the reference point O' about the origin O.
Consistency
We have supposed that the rigid body rotates around an arbitrary point. We should prove that the angular velocity previously defined is independent from the choice of origin, which means that the angular velocity is an intrinsic property of the spinning rigid body.
See the graph to the right: The origin of lab frame is O, while O_{1} and O_{2} are two fixed points on the rigid body, whose velocity is and respectively. Suppose the angular velocity with respect to O_{1} and O_{2} is and respectively. Since point P and O_{2} have only one velocity,


The above two yields that

Since the point P (and thus ) is arbitrary, it follows that

If the reference point is the instantaneous axis of rotation the expression of velocity of a point in the rigid body will have just the angular velocity term. This is because the velocity of instantaneous axis of rotation is zero. An example of instantaneous axis of rotation is the hinge of a door. Another example is the point of contact of a pure rolling spherical rigid body.
See also
 Angular displacement
Angular displacement of a body is the angle in radians through which a point or line has been rotated in a specified sense about a specified axis....
 Angular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
 Angular acceleration
Angular acceleration is the rate of change of angular velocity over time. In SI units, it is measured in radians per second squared , and is usually denoted by the Greek letter alpha . Mathematical definition :...
 Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
 Areal velocity
Areal velocity is the rate at which area is swept out by a particle as it moves along a curve. In many applications, the curve lies in a plane, but in others, it is a space curve....
 Isometry
In mathematics, an isometry is a distancepreserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
 Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
 Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
 Rigid body dynamics
In physics, rigid body dynamics is the study of the motion of rigid bodies. Unlike particles, which move only in three degrees of freedom , rigid bodies occupy space and have geometrical properties, such as a center of mass, moments of inertia, etc., that characterize motion in six degrees of...
 Rotation group
In mechanics and geometry, the rotation group is the group of all rotations about the origin of threedimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation ...
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