Angular Velocity
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Angular velocity
From Wikipedia, the free encyclopedia
Classical mechanics
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Second law of motion
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In physics, the angular velocity is defined as the rate of change of angular
displacement and is a vector quantity (more precisely, a pseudovector) which
specifies the angular speed (rotational speed) of an object and the axis about
which the object is rotating. The SI unit of angular velocity is radians per
second, although it may be measured in other units such as degrees per
second, degrees per hour, etc. Angular velocity is usually represented by the
symbol omega (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 right-hand rule.[1]
Contents
1 Angular velocity of a particle
1.1 Particle in two dimensions
1.2 Particle in three dimensions
1.2.1 Addition of angular velocity vectors
2 Rotating frames
2.1 Angular velocity vector for a frame
2.1.1 Addition of angular velocity vectors in frames
2.1.2 Components from the vectors of the frame
2.1.3 Components from Euler angles
2.1.4 Components from infinitesimal rotation matrices
2.2 Angular velocity tensor
2.3 Properties of angular velocity tensors
2.3.1 Exponential of W
2.3.2 W is skew-symmetric
2.3.3 Duality with respect to the velocity vector
2.4 Coordinate-free description
2.5 Angular velocity as a vector field
3 Rigid body considerations
3.1 Consistency
4 Angular velocity symbol
5 See also
6 References
7 External links
Angular velocity of a particle
Particle in two dimensions
The angular velocity of the particle at P with respect to the origin O is
determined by the perpendicular component of the velocity vector v.
The angular velocity describes the speed of rotation and the orientation of
the instantaneous axis about which the rotation occurs. The direction of the
angular velocity pseudovector will be along the axis of rotation; in this case
(counter-clockwise rotation) the vector points up.
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 radians), 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 (cross-radial component, v⊥). If there
is no radial component, then the particle moves in a circle. On the other
hand, if there is no cross-radial component, then the particle moves along a
straight line from 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
\omega = \frac{d\phi}{dt}
This is related to the cross-radial (tangential) velocity by:[1]
\mathrm{v}_\perp=r\,\frac{d\phi}{dt}
An explicit formula for v⊥ in terms of v and θ is:
\mathrm{v}_\perp=|\mathrm{\mathbf{v}}|\,\sin(\theta)
Combining the above equations gives a formula for ω:
\omega=\frac{|\mathrm{\mathbf{v}}|\sin(\theta)}
{|\mathrm{\mathbf{r}}|}
In two dimensions the angular velocity is a single number that has no
direction, but it does have a sense or orientation. In two dimensions the
angular velocity is a pseudoscalar, a quantity that changes its sign under a
parity inversion (for example if one of the axes is inverted or if 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.
There are three types of angular velocity involved in the movement on an
ellipse corresponding to the three anomalies (true, eccentric and mean).
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 pseudovector. 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 right-hand rule indicates the positive direction of the
angular velocity pseudovector.
Let \vec u be a unitary vector along the instantaneous rotation axis, so that
from the top of the vector the rotation is counter-clock-wise. The angular
velocity vector \vec \omega is then defined as:
\vec\omega = \frac{d\phi}{dt}\vec u
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 rotation
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 2-dimensional case. Using the unit vector \vec u defined before,
the angular velocity vector may be written in a manner similar to that for two
dimensions:
\vec\omega=\frac{|\mathrm{\mathbf{v}}|\sin(\theta)}
{|\mathrm{\mathbf{r}}|}\,\vec u
which, by the definition of the cross product, can be written:
\vec\omega=\frac{\vec{r}\times\vec{v}}{|{\vec{r}}|^2}
Addition of angular velocity vectors
If a point rotates with \omega_2 in a frame F_2 which rotates itself with
angular speed \omega_1 with respect to an external frame F_1, we can define
the addition of \omega_1 + \omega_2 as the angular velocity vector of the
point with respect to F_1.
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. 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 skew-symmetric.
Therefore R=e^{Wt} is a rotation matrix and in a time dt is an infinitesimal
rotation matrix. Therefore it can be expanded as R = I + W\cdot dt + {1 \over
2} (W \cdot dt)^2 + ...
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 (I+W_1\cdot dt)(I+W_2 \cdot dt)=(I+W_2 \cdot dt)(I+W_1\cdot
dt) and therefore \omega_1 + \omega_2 = \omega_2 + \omega_1
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 just a particular case of a moving particle, this is a very important
one for its relationship with the rigid body 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 theorem 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
Schematic construction for addition of angular velocity vectors for rotating
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 \omega_1 + \omega_2 =
\omega_2 + \omega_1
Components from the vectors of the frame
Substituting in the expression
\boldsymbol\omega=\frac{\mathbf{r}\times\mathbf{v}}
{|\mathrm{\mathbf{r}}|^2}
any vector e of the frame we obtain \vec \omega=\frac{\vec {e}\times
\dot{\vec{e}}}{|{\vec{e}}|^2}, and therefore \vec \omega = \vec
{e}_1\times \dot{\vec{e}}_1 = \vec {e}_2\times \dot{\vec{e}}_2 = \vec
{e}_3\times \dot{\vec{e}}_3.
As the columns of the matrix of the frame are the components of its vectors,
this allows also to calculate \omega from the matrix of the frame and its
derivative.
Components from Euler angles
Diagram showing Euler frame in green
The components of the angular velocity pseudovector were first calculated by
Leonhard Euler using his Euler angles 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:[2]
\vec \omega = \dot\alpha \bold u_1 +\dot\beta \bold u_2 +\dot\gamma
\bold u_3
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:
\vec \omega = (\dot\alpha\sin\beta\sin\gamma+\dot\beta\cos\gamma)
{\bold I} +(\dot\alpha\sin\beta\cos\gamma-\dot\beta\sin\gamma){\bold J} +
(\dot\alpha\cos\beta+\dot\gamma){\bold K}
where IJK are unit vectors for the frame fixed in the moving body. This
example has been made using the Z-X-Z convention for Euler angles.[3]
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
See also: Skew-symmetric matrix
It can be introduced from rotation matrices. Any vector \vec r that rotates
around an axis with an angular speed vector \vec \omega (as defined before)
satisfies:
\frac {d \vec r(t)} {dt} = \vec{\omega} \times\vec{r}
We can introduce here the angular velocity tensor associated to the angular
speed \omega:
W(t) = \begin{pmatrix} 0 & -\omega_z(t) & \omega_y(t) \\ \omega_z(t) & 0
& -\omega_x(t) \\ -\omega_y(t) & \omega_x(t) & 0 \\ \end{pmatrix}
This tensor W(t) will act as if it were a (\vec \omega \times) operator :
\vec \omega(t) \times \vec{r}(t) = W(t) \vec{r}(t)
Given the orientation matrix A(t) of a frame, we can obtain its instant angular
velocity tensor W as follows. We know that:
\frac {d \vec r(t)} {dt} = W \cdot \vec{r}
As angular speed must be the same for the three vectors of a rotating frame,
if we have a matrix A(t) whose columns are the vectors of the frame, we can
write for the three vectors as a whole:
\frac {dA(t)} {dt} = W \cdot A (t)
And therefore the angular velocity tensor we are looking for is:
W = \frac {dA(t)} {dt} \cdot A^{-1}(t)
Properties of angular velocity tensors
See also: Infinitesimal rotation
In general, the angular velocity in an n-dimensional space is the time
derivative of the angular displacement tensor which is a second rank skewsymmetric tensor.
This tensor W will have n(n-1)/2 independent components and this number is
the dimension of the Lie algebra of the Lie group of rotations of an ndimensional inner product space.[4]
Exponential of W
In three dimensions angular velocity can be represented by a pseudovector
because second rank tensors are dual to pseudovectors in three dimensions.
As \frac {dA(t)} {dt} = W\cdot A(t). This can be read as a differential
equation that defines A(t) knowing W(t).
\frac {dA(t)} {A} = W \cdot {dt}
And if the angular speed is constant then W is also constant and the equation
can be integrated. The result is:
A(t) = e^{W \cdot t}
which shows a connection with the Lie group of rotations.
W is skew-symmetric
It is possible to prove that angular velocity tensor are skew symmetric
matrices which means that a W = \frac {dR(t)}{dt}\cdot {R^T} satisfies
W^T= -W.
To prove it we start taking the time derivative of \mathcal{R}\mathcal{R}^T
being R(t) a rotation matrix:
\mathcal{I}=\mathcal{R}\mathcal{R}^T because R(t) is a rotation matrix
0=\frac{d\mathcal{R}}
{dt}\mathcal{R}^T+\mathcal{R}\frac{d\mathcal{R}^T}{dt}
Applying the formula (AB)T = BTAT:
0 = \frac{d\mathcal{R}}{dt}\mathcal{R}^T+\left(\frac{d\mathcal{R}}
{dt}\mathcal{R}^T\right)^T = W + W^T
Thus, W is the negative of its transpose, which implies it is a skew symmetric
matrix.
Duality with respect to the velocity vector
The tensor is a matrix with this structure:
W(t) = \begin{pmatrix} 0 & -\omega_z(t) & \omega_y(t) \\ \omega_z(t) & 0
& -\omega_x(t) \\ -\omega_y(t) & \omega_x(t) & 0 \\ \end{pmatrix}
As it is a skew symmetric matrix it has a Hodge dual vector which is precisely
the previous angular velocity vector \vec \omega:
\boldsymbol\omega=[\omega_x,\omega_y,\omega_z]
Coordinate-free description
At any instant, t, the angular velocity tensor represents a linear map between
the position vectors \mathbf{r}(t) and their velocity vectors \mathbf{v}(t) of
a rigid body rotating around the origin:
\mathbf{v} = W\mathbf{r}
where we omitted the t parameter, and regard \mathbf{v} and \mathbf{r} as
elements of the same 3-dimensional Euclidean vector space V.
The relation between this linear map and the angular velocity
pseudovector \omega is the following.
Because of W is the derivative of an orthogonal transformation, the
B(\mathbf{r},\mathbf{s}) = (W\mathbf{r}) \cdot \mathbf{s}
bilinear form is skew-symmetric[disambiguation needed]. (Here \cdot stands
for the scalar product). So we can apply the fact of exterior algebra that there
is a unique linear form L on \Lambda^2 V that
L(\mathbf{r}\wedge \mathbf{s}) = B(\mathbf{r},\mathbf{s})
where \mathbf{r}\wedge \mathbf{s} \in \Lambda^2 V is the wedge product
of \mathbf{r} and \mathbf{s}.
Taking the dual vector L* of L we get
(W\mathbf{r})\cdot \mathbf{s} = L^* \cdot (\mathbf{r}\wedge
\mathbf{s})
Introducing \omega := *L^* , as the Hodge dual of L*, and apply further
Hodge dual identities we arrive at
(W\mathbf{r}) \cdot \mathbf{s} = * ( *L^* \wedge \mathbf{r} \wedge
\mathbf{s}) = * (\omega \wedge \mathbf{r} \wedge \mathbf{s}) = *(\omega
\wedge \mathbf{r}) \cdot \mathbf{s} = (\omega \times \mathbf{r}) \cdot
\mathbf{s}
where
\omega \times \mathbf{r} := *(\omega \wedge \mathbf{r})
by definition.
Because \mathbf{s} is an arbitrary vector, from nondegeneracy of scalar
product follows
W\mathbf{r} = \omega \times \mathbf{r}
Angular velocity as a vector field
For angular velocity tensor maps velocities to positions, it is a vector field. In
particular, this vector field is a Killing vector field belonging to an element of
the Lie algebra so(3) of the 3-dimensional rotation group SO(3). This element
of so(3) can also be regarded as the angular velocity vector.
Rigid body considerations
See also: axes conventions
Position of point P located in the rigid body (shown in blue). Ri is the position
with respect to the lab frame, centered at O and ri is the position with
respect to the rigid body frame, centered at O' . The origin of the rigid body
frame is at vector position R from the lab frame.
The same equations for the angular speed can be obtained reasoning over a
rotating rigid body. 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
Ri in the lab frame, and at position ri in the body frame. It is seen that the
position of the particle can be written:
\mathbf{R}_i=\mathbf{R}+\mathbf{r}_i
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 \mathbf{r}_i is unchanging. By Euler's rotation theorem, we
may replace the vector \mathbf{r}_i with \mathcal{R}\mathbf{r}_{io} where
\mathcal{R} is a 3x3 rotation matrix and \mathbf{r}_{io} 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 \mathcal{R} which is changing in
time and not the reference vector \mathbf{r}_{io}, 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
\mathbf{r}_i 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:
\mathbf{R}_i=\mathbf{R}+\mathcal{R}\mathbf{r}_{io}
Taking the time derivative yields the velocity of the particle:
\mathbf{V}_i=\mathbf{V}+\frac{d\mathcal{R}}{dt}\mathbf{r}_{io}
where Vi 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 \mathcal{R} is a rotation
matrix its inverse is its transpose. So we substitute
\mathcal{I}=\mathcal{R}^T\mathcal{R}:
\mathbf{V}_i = \mathbf{V}+\frac{d\mathcal{R}}
{dt}\mathcal{I}\mathbf{r}_{io}
\mathbf{V}_i = \mathbf{V}+\frac{d\mathcal{R}}
{dt}\mathcal{R}^T\mathcal{R}\mathbf{r}_{io}
\mathbf{V}_i = \mathbf{V}+\frac{d\mathcal{R}}
{dt}\mathcal{R}^T\mathbf{r}_{i}
or
\mathbf{V}_i = \mathbf{V}+W\mathbf{r}_{i}
where W = \frac{d\mathcal{R}}{dt}\mathcal{R}^T is the previous angular
velocity tensor.
It can be proved that this is a skew symmetric matrix, so we can take its dual
to get a 3 dimensional pseudovector which is precisely the previous angular
velocity vector \vec \omega:
\boldsymbol\omega=[\omega_x,\omega_y,\omega_z]
Substituting ω for W into the above velocity expression, and replacing matrix
multiplication by an equivalent cross product:
\mathbf{V}_i=\mathbf{V}+\boldsymbol\omega\times\mathbf{r}_i
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.
Proving the independence of angular velocity from choice of origin
See the graph to the right: The origin of lab frame is O, while O1 and O2 are
two fixed points on the rigid body, whose velocity is \mathbf{v}_1 and
\mathbf{v}_2 respectively. Suppose the angular velocity with respect to O1
and O2 is \boldsymbol{\omega}_1 and \boldsymbol{\omega}_2 respectively.
Since point P and O2 have only one velocity,
\mathbf{v}_1 + \boldsymbol{\omega}_1\times\mathbf{r}_1 =
\mathbf{v}_2 + \boldsymbol{\omega}_2\times\mathbf{r}_2
\mathbf{v}_2 = \mathbf{v}_1 + \boldsymbol{\omega}_1\times\mathbf{r}
= \mathbf{v}_1 + \boldsymbol{\omega}_1\times (\mathbf{r}_1 \mathbf{r}_2)
The above two yields that
(\boldsymbol{\omega}_1-\boldsymbol{\omega}_2) \times \mathbf{r}_2=0
Since the point P (and thus \mathbf{r}_2 ) is arbitrary, it follows that
\boldsymbol{\omega}_1 = \boldsymbol{\omega}_2
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 purely rolling spherical (or, more
generally, convex) rigid body.
Angular velocity symbol
When preparing electronic documents, some document editing software will
attempt to use the Symbol typeface to render the ω character. Where the
font is not supported, a w is displayed instead ("v=rw" instead of "v=rω", for
instance). As w represents weight, not angular velocity, this can lead to
confusion.
From Wikipedia, the free encyclopedia
Classical mechanics
\vec{F} = m\vec{a}
Second law of motion
History
Timeline
Branches[show]
Fundamentals[show]
Formulations[show]
Core topics[show]
Rotation[show]
Scientists[show]
v
t
e
In physics, the angular velocity is defined as the rate of change of angular
displacement and is a vector quantity (more precisely, a pseudovector) which
specifies the angular speed (rotational speed) of an object and the axis about
which the object is rotating. The SI unit of angular velocity is radians per
second, although it may be measured in other units such as degrees per
second, degrees per hour, etc. Angular velocity is usually represented by the
symbol omega (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 right-hand rule.[1]
Contents
1 Angular velocity of a particle
1.1 Particle in two dimensions
1.2 Particle in three dimensions
1.2.1 Addition of angular velocity vectors
2 Rotating frames
2.1 Angular velocity vector for a frame
2.1.1 Addition of angular velocity vectors in frames
2.1.2 Components from the vectors of the frame
2.1.3 Components from Euler angles
2.1.4 Components from infinitesimal rotation matrices
2.2 Angular velocity tensor
2.3 Properties of angular velocity tensors
2.3.1 Exponential of W
2.3.2 W is skew-symmetric
2.3.3 Duality with respect to the velocity vector
2.4 Coordinate-free description
2.5 Angular velocity as a vector field
3 Rigid body considerations
3.1 Consistency
4 Angular velocity symbol
5 See also
6 References
7 External links
Angular velocity of a particle
Particle in two dimensions
The angular velocity of the particle at P with respect to the origin O is
determined by the perpendicular component of the velocity vector v.
The angular velocity describes the speed of rotation and the orientation of
the instantaneous axis about which the rotation occurs. The direction of the
angular velocity pseudovector will be along the axis of rotation; in this case
(counter-clockwise rotation) the vector points up.
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 radians), 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 (cross-radial component, v⊥). If there
is no radial component, then the particle moves in a circle. On the other
hand, if there is no cross-radial component, then the particle moves along a
straight line from 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
\omega = \frac{d\phi}{dt}
This is related to the cross-radial (tangential) velocity by:[1]
\mathrm{v}_\perp=r\,\frac{d\phi}{dt}
An explicit formula for v⊥ in terms of v and θ is:
\mathrm{v}_\perp=|\mathrm{\mathbf{v}}|\,\sin(\theta)
Combining the above equations gives a formula for ω:
\omega=\frac{|\mathrm{\mathbf{v}}|\sin(\theta)}
{|\mathrm{\mathbf{r}}|}
In two dimensions the angular velocity is a single number that has no
direction, but it does have a sense or orientation. In two dimensions the
angular velocity is a pseudoscalar, a quantity that changes its sign under a
parity inversion (for example if one of the axes is inverted or if 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.
There are three types of angular velocity involved in the movement on an
ellipse corresponding to the three anomalies (true, eccentric and mean).
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 pseudovector. 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 right-hand rule indicates the positive direction of the
angular velocity pseudovector.
Let \vec u be a unitary vector along the instantaneous rotation axis, so that
from the top of the vector the rotation is counter-clock-wise. The angular
velocity vector \vec \omega is then defined as:
\vec\omega = \frac{d\phi}{dt}\vec u
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 rotation
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 2-dimensional case. Using the unit vector \vec u defined before,
the angular velocity vector may be written in a manner similar to that for two
dimensions:
\vec\omega=\frac{|\mathrm{\mathbf{v}}|\sin(\theta)}
{|\mathrm{\mathbf{r}}|}\,\vec u
which, by the definition of the cross product, can be written:
\vec\omega=\frac{\vec{r}\times\vec{v}}{|{\vec{r}}|^2}
Addition of angular velocity vectors
If a point rotates with \omega_2 in a frame F_2 which rotates itself with
angular speed \omega_1 with respect to an external frame F_1, we can define
the addition of \omega_1 + \omega_2 as the angular velocity vector of the
point with respect to F_1.
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. 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 skew-symmetric.
Therefore R=e^{Wt} is a rotation matrix and in a time dt is an infinitesimal
rotation matrix. Therefore it can be expanded as R = I + W\cdot dt + {1 \over
2} (W \cdot dt)^2 + ...
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 (I+W_1\cdot dt)(I+W_2 \cdot dt)=(I+W_2 \cdot dt)(I+W_1\cdot
dt) and therefore \omega_1 + \omega_2 = \omega_2 + \omega_1
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 just a particular case of a moving particle, this is a very important
one for its relationship with the rigid body 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 theorem 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
Schematic construction for addition of angular velocity vectors for rotating
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 \omega_1 + \omega_2 =
\omega_2 + \omega_1
Components from the vectors of the frame
Substituting in the expression
\boldsymbol\omega=\frac{\mathbf{r}\times\mathbf{v}}
{|\mathrm{\mathbf{r}}|^2}
any vector e of the frame we obtain \vec \omega=\frac{\vec {e}\times
\dot{\vec{e}}}{|{\vec{e}}|^2}, and therefore \vec \omega = \vec
{e}_1\times \dot{\vec{e}}_1 = \vec {e}_2\times \dot{\vec{e}}_2 = \vec
{e}_3\times \dot{\vec{e}}_3.
As the columns of the matrix of the frame are the components of its vectors,
this allows also to calculate \omega from the matrix of the frame and its
derivative.
Components from Euler angles
Diagram showing Euler frame in green
The components of the angular velocity pseudovector were first calculated by
Leonhard Euler using his Euler angles 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:[2]
\vec \omega = \dot\alpha \bold u_1 +\dot\beta \bold u_2 +\dot\gamma
\bold u_3
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:
\vec \omega = (\dot\alpha\sin\beta\sin\gamma+\dot\beta\cos\gamma)
{\bold I} +(\dot\alpha\sin\beta\cos\gamma-\dot\beta\sin\gamma){\bold J} +
(\dot\alpha\cos\beta+\dot\gamma){\bold K}
where IJK are unit vectors for the frame fixed in the moving body. This
example has been made using the Z-X-Z convention for Euler angles.[3]
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
See also: Skew-symmetric matrix
It can be introduced from rotation matrices. Any vector \vec r that rotates
around an axis with an angular speed vector \vec \omega (as defined before)
satisfies:
\frac {d \vec r(t)} {dt} = \vec{\omega} \times\vec{r}
We can introduce here the angular velocity tensor associated to the angular
speed \omega:
W(t) = \begin{pmatrix} 0 & -\omega_z(t) & \omega_y(t) \\ \omega_z(t) & 0
& -\omega_x(t) \\ -\omega_y(t) & \omega_x(t) & 0 \\ \end{pmatrix}
This tensor W(t) will act as if it were a (\vec \omega \times) operator :
\vec \omega(t) \times \vec{r}(t) = W(t) \vec{r}(t)
Given the orientation matrix A(t) of a frame, we can obtain its instant angular
velocity tensor W as follows. We know that:
\frac {d \vec r(t)} {dt} = W \cdot \vec{r}
As angular speed must be the same for the three vectors of a rotating frame,
if we have a matrix A(t) whose columns are the vectors of the frame, we can
write for the three vectors as a whole:
\frac {dA(t)} {dt} = W \cdot A (t)
And therefore the angular velocity tensor we are looking for is:
W = \frac {dA(t)} {dt} \cdot A^{-1}(t)
Properties of angular velocity tensors
See also: Infinitesimal rotation
In general, the angular velocity in an n-dimensional space is the time
derivative of the angular displacement tensor which is a second rank skewsymmetric tensor.
This tensor W will have n(n-1)/2 independent components and this number is
the dimension of the Lie algebra of the Lie group of rotations of an ndimensional inner product space.[4]
Exponential of W
In three dimensions angular velocity can be represented by a pseudovector
because second rank tensors are dual to pseudovectors in three dimensions.
As \frac {dA(t)} {dt} = W\cdot A(t). This can be read as a differential
equation that defines A(t) knowing W(t).
\frac {dA(t)} {A} = W \cdot {dt}
And if the angular speed is constant then W is also constant and the equation
can be integrated. The result is:
A(t) = e^{W \cdot t}
which shows a connection with the Lie group of rotations.
W is skew-symmetric
It is possible to prove that angular velocity tensor are skew symmetric
matrices which means that a W = \frac {dR(t)}{dt}\cdot {R^T} satisfies
W^T= -W.
To prove it we start taking the time derivative of \mathcal{R}\mathcal{R}^T
being R(t) a rotation matrix:
\mathcal{I}=\mathcal{R}\mathcal{R}^T because R(t) is a rotation matrix
0=\frac{d\mathcal{R}}
{dt}\mathcal{R}^T+\mathcal{R}\frac{d\mathcal{R}^T}{dt}
Applying the formula (AB)T = BTAT:
0 = \frac{d\mathcal{R}}{dt}\mathcal{R}^T+\left(\frac{d\mathcal{R}}
{dt}\mathcal{R}^T\right)^T = W + W^T
Thus, W is the negative of its transpose, which implies it is a skew symmetric
matrix.
Duality with respect to the velocity vector
The tensor is a matrix with this structure:
W(t) = \begin{pmatrix} 0 & -\omega_z(t) & \omega_y(t) \\ \omega_z(t) & 0
& -\omega_x(t) \\ -\omega_y(t) & \omega_x(t) & 0 \\ \end{pmatrix}
As it is a skew symmetric matrix it has a Hodge dual vector which is precisely
the previous angular velocity vector \vec \omega:
\boldsymbol\omega=[\omega_x,\omega_y,\omega_z]
Coordinate-free description
At any instant, t, the angular velocity tensor represents a linear map between
the position vectors \mathbf{r}(t) and their velocity vectors \mathbf{v}(t) of
a rigid body rotating around the origin:
\mathbf{v} = W\mathbf{r}
where we omitted the t parameter, and regard \mathbf{v} and \mathbf{r} as
elements of the same 3-dimensional Euclidean vector space V.
The relation between this linear map and the angular velocity
pseudovector \omega is the following.
Because of W is the derivative of an orthogonal transformation, the
B(\mathbf{r},\mathbf{s}) = (W\mathbf{r}) \cdot \mathbf{s}
bilinear form is skew-symmetric[disambiguation needed]. (Here \cdot stands
for the scalar product). So we can apply the fact of exterior algebra that there
is a unique linear form L on \Lambda^2 V that
L(\mathbf{r}\wedge \mathbf{s}) = B(\mathbf{r},\mathbf{s})
where \mathbf{r}\wedge \mathbf{s} \in \Lambda^2 V is the wedge product
of \mathbf{r} and \mathbf{s}.
Taking the dual vector L* of L we get
(W\mathbf{r})\cdot \mathbf{s} = L^* \cdot (\mathbf{r}\wedge
\mathbf{s})
Introducing \omega := *L^* , as the Hodge dual of L*, and apply further
Hodge dual identities we arrive at
(W\mathbf{r}) \cdot \mathbf{s} = * ( *L^* \wedge \mathbf{r} \wedge
\mathbf{s}) = * (\omega \wedge \mathbf{r} \wedge \mathbf{s}) = *(\omega
\wedge \mathbf{r}) \cdot \mathbf{s} = (\omega \times \mathbf{r}) \cdot
\mathbf{s}
where
\omega \times \mathbf{r} := *(\omega \wedge \mathbf{r})
by definition.
Because \mathbf{s} is an arbitrary vector, from nondegeneracy of scalar
product follows
W\mathbf{r} = \omega \times \mathbf{r}
Angular velocity as a vector field
For angular velocity tensor maps velocities to positions, it is a vector field. In
particular, this vector field is a Killing vector field belonging to an element of
the Lie algebra so(3) of the 3-dimensional rotation group SO(3). This element
of so(3) can also be regarded as the angular velocity vector.
Rigid body considerations
See also: axes conventions
Position of point P located in the rigid body (shown in blue). Ri is the position
with respect to the lab frame, centered at O and ri is the position with
respect to the rigid body frame, centered at O' . The origin of the rigid body
frame is at vector position R from the lab frame.
The same equations for the angular speed can be obtained reasoning over a
rotating rigid body. 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
Ri in the lab frame, and at position ri in the body frame. It is seen that the
position of the particle can be written:
\mathbf{R}_i=\mathbf{R}+\mathbf{r}_i
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 \mathbf{r}_i is unchanging. By Euler's rotation theorem, we
may replace the vector \mathbf{r}_i with \mathcal{R}\mathbf{r}_{io} where
\mathcal{R} is a 3x3 rotation matrix and \mathbf{r}_{io} 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 \mathcal{R} which is changing in
time and not the reference vector \mathbf{r}_{io}, 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
\mathbf{r}_i 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:
\mathbf{R}_i=\mathbf{R}+\mathcal{R}\mathbf{r}_{io}
Taking the time derivative yields the velocity of the particle:
\mathbf{V}_i=\mathbf{V}+\frac{d\mathcal{R}}{dt}\mathbf{r}_{io}
where Vi 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 \mathcal{R} is a rotation
matrix its inverse is its transpose. So we substitute
\mathcal{I}=\mathcal{R}^T\mathcal{R}:
\mathbf{V}_i = \mathbf{V}+\frac{d\mathcal{R}}
{dt}\mathcal{I}\mathbf{r}_{io}
\mathbf{V}_i = \mathbf{V}+\frac{d\mathcal{R}}
{dt}\mathcal{R}^T\mathcal{R}\mathbf{r}_{io}
\mathbf{V}_i = \mathbf{V}+\frac{d\mathcal{R}}
{dt}\mathcal{R}^T\mathbf{r}_{i}
or
\mathbf{V}_i = \mathbf{V}+W\mathbf{r}_{i}
where W = \frac{d\mathcal{R}}{dt}\mathcal{R}^T is the previous angular
velocity tensor.
It can be proved that this is a skew symmetric matrix, so we can take its dual
to get a 3 dimensional pseudovector which is precisely the previous angular
velocity vector \vec \omega:
\boldsymbol\omega=[\omega_x,\omega_y,\omega_z]
Substituting ω for W into the above velocity expression, and replacing matrix
multiplication by an equivalent cross product:
\mathbf{V}_i=\mathbf{V}+\boldsymbol\omega\times\mathbf{r}_i
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.
Proving the independence of angular velocity from choice of origin
See the graph to the right: The origin of lab frame is O, while O1 and O2 are
two fixed points on the rigid body, whose velocity is \mathbf{v}_1 and
\mathbf{v}_2 respectively. Suppose the angular velocity with respect to O1
and O2 is \boldsymbol{\omega}_1 and \boldsymbol{\omega}_2 respectively.
Since point P and O2 have only one velocity,
\mathbf{v}_1 + \boldsymbol{\omega}_1\times\mathbf{r}_1 =
\mathbf{v}_2 + \boldsymbol{\omega}_2\times\mathbf{r}_2
\mathbf{v}_2 = \mathbf{v}_1 + \boldsymbol{\omega}_1\times\mathbf{r}
= \mathbf{v}_1 + \boldsymbol{\omega}_1\times (\mathbf{r}_1 \mathbf{r}_2)
The above two yields that
(\boldsymbol{\omega}_1-\boldsymbol{\omega}_2) \times \mathbf{r}_2=0
Since the point P (and thus \mathbf{r}_2 ) is arbitrary, it follows that
\boldsymbol{\omega}_1 = \boldsymbol{\omega}_2
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 purely rolling spherical (or, more
generally, convex) rigid body.
Angular velocity symbol
When preparing electronic documents, some document editing software will
attempt to use the Symbol typeface to render the ω character. Where the
font is not supported, a w is displayed instead ("v=rw" instead of "v=rω", for
instance). As w represents weight, not angular velocity, this can lead to
confusion.
