In ^{−1}. Angular velocity is usually represented by the symbol

K.S.HEDRIH: Leonhard Euler (1707–1783) and rigid body dynamics

/ref> : $\backslash boldsymbol\backslash omega\; =\; \backslash dot\backslash alpha\backslash mathbf\; u\_1+\backslash dot\backslash beta\backslash mathbf\; u\_2+\backslash dot\backslash gamma\; \backslash mathbf\; u\_3$ 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: : $\backslash boldsymbol\backslash omega\; =\; (\backslash dot\backslash alpha\; \backslash sin\backslash beta\; \backslash sin\backslash gamma\; +\; \backslash dot\backslash beta\backslash cos\backslash gamma)\; \backslash hat\backslash mathbf\; i+\; (\backslash dot\backslash alpha\; \backslash sin\backslash beta\; \backslash cos\backslash gamma\; -\; \backslash dot\backslash beta\backslash sin\backslash gamma)\; \backslash hat\backslash mathbf\; j\; +\; (\backslash dot\backslash alpha\; \backslash cos\backslash beta\; +\; \backslash dot\backslash gamma)\; \backslash hat\backslash mathbf\; k$ where $\backslash hat\backslash mathbf\; i,\; \backslash hat\backslash mathbf\; j,\; \backslash hat\backslash mathbf\; k$ 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.

on the Classical Mechanics page o

especially Questions 1 and 2.

_{''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:
: $\backslash mathbf\_i=\backslash mathbf+\backslash mathbf\_i$
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 $\backslash mathbf\_i$ is unchanging. By _{''i''} is the velocity of the particle (in the lab frame) and V is the velocity of (the origin of the rigid body frame). Since $\backslash mathcal$ is a rotation matrix its inverse is its transpose. So we substitute $\backslash mathcal=\backslash mathcal^\backslash text\backslash mathcal$:
: $\backslash mathbf\_i\; =\; \backslash mathbf+\backslash frac\backslash mathcal\backslash mathbf\_$
: $\backslash mathbf\_i\; =\; \backslash mathbf+\backslash frac\backslash mathcal^\backslash text\backslash mathcal\backslash mathbf\_$
: $\backslash mathbf\_i\; =\; \backslash mathbf+\backslash frac\backslash mathcal^\backslash text\backslash mathbf\_$
or
: $\backslash mathbf\_i\; =\; \backslash mathbf+W\backslash mathbf\_$
where $W\; =\; \backslash frac\backslash mathcal^\backslash text$ is the previous angular velocity tensor.
It can be proved that this is a skew symmetric matrix, so we can take its

_{1} and ''O''_{2} are two fixed points on the rigid body, whose velocity is $\backslash mathbf\_1$ and $\backslash mathbf\_2$ respectively. Suppose the angular velocity with respect to ''O''_{1} and O_{2} is $\backslash boldsymbol\_1$ and $\backslash boldsymbol\_2$ respectively. Since point ''P'' and ''O''_{2} have only one velocity,
: $\backslash mathbf\_1\; +\; \backslash boldsymbol\_1\backslash times\backslash mathbf\_1\; =\; \backslash mathbf\_2\; +\; \backslash boldsymbol\_2\backslash times\backslash mathbf\_2$
: $\backslash mathbf\_2\; =\; \backslash mathbf\_1\; +\; \backslash boldsymbol\_1\backslash times\backslash mathbf\; =\; \backslash mathbf\_1\; +\; \backslash boldsymbol\_1\backslash times\; (\backslash mathbf\_1\; -\; \backslash mathbf\_2)$
The above two yields that
: $(\backslash boldsymbol\_2-\backslash boldsymbol\_1)\; \backslash times\; \backslash mathbf\_2=0$
Since the point ''P'' (and thus $\backslash mathbf\_2$) is arbitrary, it follows that
: $\backslash boldsymbol\_1\; =\; \backslash boldsymbol\_2$
If the reference point is the instantaneous axis of rotation the expression of the velocity of a point in the rigid body will have just the angular velocity term. This is because the velocity of the instantaneous axis of rotation is zero. An example of the 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.

A college text-book of physics

By Arthur Lalanne Kimball (''Angular Velocity of a particle'') * {{Authority control Angle Physical quantities Rotational symmetry Temporal rates Tensors Velocity

physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, angular velocity ($\backslash boldsymbol$ or $\backslash boldsymbol$), also known as angular frequency vector,(UP1) is a vector measure of rotation rate, that refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time.
There are two types of angular velocity. Orbital angular velocity refers to how fast a point object revolves about a fixed origin, i.e. the time rate of change of its angular position relative to the origin. Spin angular velocity refers to how fast a rigid body rotates with respect to its center of rotation and is independent of the choice of origin, in contrast to orbital angular velocity.
In general, angular velocity has dimension
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

of angle per unit time (angle replacing distance
Distance is a numerical measurement
'
Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

from linear velocity
The velocity of an object is the rate of change of its position with respect to a frame of reference
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical scie ...

with time in common). The unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* Unit (album), ...

of angular velocity is radians per second
The radian per second (symbol: rad⋅s−1 or rad/s) is the SI unit of angular velocity, commonly denoted by the Greek letter ''ω'' (omega). The radian per second is also the SI unit of angular frequency. The radian per second is defined as the ...

, with the radian being a dimensionless quantity
In dimensional analysis
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantity, base quantities (such as length, mass, time, and electric cur ...

, thus the SI units of angular velocity may be listed as somega
Omega (; capital
Capital most commonly refers to:
* Capital letter
Letter case (or just case) is the distinction between the letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (o ...

(''ω'', sometimes Ω). By convention, positive angular velocity indicates counter-clockwise rotation, while negative is clockwise.
For example, a geostationary
A geostationary orbit, also referred to as a geosynchronous equatorial orbit''Geostationary orbit'' and ''Geosynchronous (equatorial) orbit'' are used somewhat interchangeably in sources. (GEO), is a circular geosynchronous orbit
A geos ...

satellite completes one orbit per day above the equator, or 360 degrees per 24 hours, and has angular velocity ''ω'' = (360°)/(24 h) = 15°/h, or (2π rad)/(24 h) ≈ 0.26 rad/h. If angle is measured in radians
The radian, denoted by the symbol \text, is the SI unit
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sys ...

, the linear velocity is the radius times the angular velocity, $v\; =\; r\backslash omega$. With orbital radius 42,000 km from the earth's center, the satellite's speed through space is thus ''v'' = 42,000 km × 0.26/h ≈ 11,000 km/h. The angular velocity is positive since the satellite travels eastward with the Earth's rotation (counter-clockwise from above the north pole.)
Angular velocity is a pseudovector
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...

, with its magnitude measuring the ''angular speed Angular frequency ''ω'' (in radians per second), is larger than frequency ''ν'' (in cycles per second, also called Hertz, Hz), by a factor of . This figure uses the symbol ''ν'', rather than ''f'' to denote frequency.
In [ hysics, angular freque ...

'', the rate at which an object rotates or revolves, and its direction pointing perpendicular to the instantaneous plane of rotation or angular displacement. The orientation of angular velocity is conventionally specified by the right-hand rule
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

.(EM1)
Of a point particle

Particle in two dimensions

In the simplest case of circular motion at radius $r$, with position given by the angular displacement $\backslash phi(t)$ from the x-axis, the orbital angular velocity is the rate of change of angle with respect to time: $\backslash omega\; =\; \backslash frac$. If $\backslash phi$ is measured inradian
The radian, denoted by the symbol \text, is the SI unit
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

s, the arc-length from the positive x-axis around the circle to the particle is $\backslash ell=r\backslash phi$, and the linear velocity is $v(t)\; =\; \backslash frac\; =\; r\backslash omega(t)$, so that $\backslash omega\; =\; \backslash frac$.
In the general case of a particle moving in the plane, the orbital angular velocity is the rate at which the position vector relative to a chosen origin "sweeps out" angle. The diagram shows the position vector $\backslash mathbf$ from the origin $O$ to a particle $P$, with its polar coordinates
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

$(r,\; \backslash phi)$. (All variables are functions of time $t$.) The particle has linear velocity splitting as $\backslash mathbf\; =\; \backslash mathbf\_\backslash ,\; +\backslash mathbf\_\backslash perp$, with the radial component $\backslash mathbf\_\backslash ,$ parallel to the radius, and the cross-radial (or tangential) component $\backslash mathbf\_\backslash perp$ perpendicular to the radius. When there is no radial component, the particle moves around the origin in a circle; but when there is no cross-radial component, it moves in a straight line from the origin. Since radial motion leaves the angle unchanged, only the cross-radial component of linear velocity contributes to angular velocity.
The angular velocity ''ω'' is the rate of change of angular position with respect to time, which can be computed from the cross-radial velocity as:
: $\backslash omega\; =\; \backslash frac\; =\; \backslash frac.$
Here the cross-radial speed $v\_\backslash perp$ is the signed magnitude of $\backslash mathbf\_\backslash perp$, positive for counter-clockwise motion, negative for clockwise. Taking polar coordinates for the linear velocity $\backslash mathbf$ gives magnitude $v$ (linear speed) and angle $\backslash theta$ relative to the radius vector; in these terms, $v\_\backslash perp\; =\; v\backslash sin(\backslash theta)$, so that
: $\backslash omega\; =\; \backslash frac.$
These formulas may be derived doing $\backslash mathbf=(r\backslash cos(\backslash varphi),r\backslash sin(\backslash varphi))$, being $r$ a function of the distance to the origin with respect to time, and $\backslash varphi$ a function of the angle between the vector and the x axis. Then $\backslash frac\; =\; (\backslash dot\backslash cos(\backslash varphi)\; -\; r\backslash dot\backslash sin(\backslash varphi),\; \backslash dot\backslash sin(\backslash varphi)\; +\; r\backslash dot\backslash cos(\backslash varphi))$. Which is equal to $\backslash dot(\backslash cos(\backslash varphi),\; \backslash sin(\backslash varphi))\; +\; r\backslash dot(-\backslash sin(\backslash varphi),\; \backslash cos(\backslash varphi))\; =\; \backslash dot\backslash hat\; +\; r\backslash dot\backslash hat$. (See Unit vector
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

in cylindrical coordinates). Knowing $\backslash frac=\backslash mathbf$, we conclude that the radial component of the velocity is given by $\backslash dot$, because $\backslash hat$ is a radial unit vector; and the perpendicular component is given by $r\backslash dot$ because $\backslash hat$ is a perpendicular unit vector.
In two dimensions, angular velocity is a number with plus or minus sign indicating orientation, but not pointing in a direction. The sign is conventionally taken to be positive if the radius vector turns counter-clockwise, and negative if clockwise. Angular velocity then may be termed a pseudoscalar
In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not.
Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. ...

, a numerical quantity which changes sign under a parity inversion, such as inverting one axis or switching the two axes.
Particle in three dimensions

Inthree-dimensional space
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the ...

, we again have the position vector r of a moving particle. Here, orbital angular velocity is a pseudovector
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...

whose magnitude is the rate at which r sweeps out angle, and whose direction is perpendicular to the instantaneous plane in which r sweeps out angle (i.e. the plane spanned by r and v). However, as there are ''two'' directions perpendicular to any plane, an additional condition is necessary to uniquely specify the direction of the angular velocity; conventionally, the right-hand rule
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

is used.
Let the pseudovector $\backslash mathbf$ be the unit vector perpendicular to the plane spanned by r and v, so that the right-hand rule is satisfied (i.e. the instantaneous direction of angular displacement is counter-clockwise looking from the top of $\backslash mathbf$). Taking polar coordinates $(r,\backslash phi)$ in this plane, as in the two-dimensional case above, one may define the orbital angular velocity vector as:
: $\backslash boldsymbol\backslash omega\; =\backslash omega\; \backslash mathbf\; u\; =\; \backslash frac\backslash mathbf\; u=\backslash frac\backslash mathbf\; u,$
where ''θ'' is the angle between r and v. In terms of the cross product, this is:
: $\backslash boldsymbol\backslash omega\; =\backslash frac.$
From the above equation, one can recover the tangential velocity as:
:$\backslash mathbf\_\; =\backslash boldsymbol\; \backslash times\backslash mathbf$
Note that the above expression for $$ is only valid if $$ is in the same plane as the motion.
Addition of angular velocity vectors

Schematic construction for addition of angular velocity vectors for rotating frames If a point rotates with orbital angular velocity $\backslash omega\_1$ about its center of rotation in a coordinate frame $F\_1$ which itself rotates with a spin angular velocity $\backslash omega\_2$ with respect to an external frame $F\_2$, we can define $\backslash omega\_1\; +\; \backslash omega\_2$ to be the composite orbital angular velocity vector of the point about its center of rotation with respect to $F\_2$. This operation coincides with usual addition of vectors, and it gives angular velocity the algebraic structure of a truevector
Vector may refer to:
Biology
*Vector (epidemiology)
In epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...

, rather than just a pseudo-vector.
The only non-obvious property of the above addition is commutativity
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. This can be proven from the fact that the velocity tensor ''W'' (see below) is skew-symmetric, so that $R=e^$ is a rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation (mathematics), rotation in Euclidean space. For example, using the convention below, the matrix
:R = \begin
\cos \theta & -\sin \theta \\
\sin \th ...

which can be expanded as $R\; =\; I\; +\; W\backslash cdot\; dt\; +\; \backslash tfrac\; 12\; (W\; \backslash cdot\; dt)^2\; +\; \backslash cdots$. The composition of rotations is not commutative, but $(I+W\_1\backslash cdot\; dt)(I+W\_2\; \backslash cdot\; dt)=(I+W\_2\; \backslash cdot\; dt)(I+W\_1\backslash cdot\; dt)$ is commutative to first order, and therefore $\backslash omega\_1\; +\; \backslash omega\_2\; =\; \backslash omega\_2\; +\; \backslash omega\_1$.
Notice that this also defines the subtraction as the addition of a negative vector.
For a rigid body or reference frame

Given a rotating frame of three unit coordinate vectors, all the three must have the same angular speed at each instant. In such a frame, each vector may be considered as a moving particle with constant scalar radius. The rotating frame appears in the context ofrigid bodies
In physics, a rigid body (also known as a rigid object ) is a solid body
Body may refer to:
In science
* Physical body, an object in physics that represents a large amount, has mass or takes up space
* Body (biology), the physical material of ...

, and special tools have been developed for it: the spin angular velocity may be described as a vector or equivalently as a tensor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

.
Consistent with the general definition, the spin angular velocity of a frame is defined as the orbital angular velocity of any of the three vectors (same for all) with respect to its own center of rotation. The addition of angular velocity vectors for frames is also defined by the usual vector addition (composition of linear movements), and can be useful to decompose the rotation as in a gimbal
A gimbal is a pivoted support that permits rotation of an object about an axis. A set of three gimbals, one mounted on the other with orthogonal pivot axes, may be used to allow an object mounted on the innermost gimbal to remain independent of ...

. All components of the vector can be calculated as derivatives of the parameters defining the moving frames (Euler angles or rotation matrices). As in the general case, addition is commutative: $\backslash omega\_1\; +\; \backslash omega\_2\; =\; \backslash omega\_2\; +\; \backslash omega\_1$.
By Euler's rotation theorem
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

, any rotating frame possesses an instantaneous axis of rotation, which is the direction of the angular velocity vector, and the magnitude of the angular velocity is consistent with the two-dimensional case.
If we choose a reference point $$ fixed in the rigid body, the velocity $\backslash dot$ of any point in the body is given by
:$\backslash dot\; =\; \backslash dot\; +(-)\backslash times$
Components from the basis vectors of a body-fixed frame

Consider a rigid body rotating about a fixed point O. Construct a reference frame in the body consisting of an orthonormal set of vectors $\backslash mathbf\_1,\; \backslash mathbf\_2,\; \backslash mathbf\_3$ fixed to the body and with their common origin at O. The angular velocity vector of both frame and body about O is then : $\backslash boldsymbol\backslash omega\; =\; \backslash left(\backslash dot\; \backslash mathbf\_1\backslash cdot\backslash mathbf\_2\backslash right)\; \backslash mathbf\_3\; +\; \backslash left(\backslash dot\; \backslash mathbf\_2\backslash cdot\backslash mathbf\_3\backslash right)\; \backslash mathbf\_1\; +\; \backslash left(\backslash dot\; \backslash mathbf\_3\backslash cdot\backslash mathbf\_1\backslash right)\; \backslash mathbf\_2,$ Here :$\backslash dot\; \backslash mathbf\_i=\; \backslash frac$ is the time rate of change of the frame vector $\backslash mathbf\_i,\; i=1,2,3,$ due to the rotation. Note that this formula is incompatible with the expression : $\backslash boldsymbol\backslash omega\; =\backslash frac.$ as that formula defines only the angular velocity of a ''single point'' about O, while the formula in this section applies to a frame or rigid body. In the case of a rigid body a ''single'' $\backslash boldsymbol\backslash omega$ has to account for the motion of ''all'' particles in the body.Components from Euler angles

The components of the spin angular velocity pseudovector were first calculated byLeonhard Euler
Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

using his Euler angles
The Euler angles are three angles introduced by Leonhard Euler
Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics ( ...

and the use of an intermediate frame:
* One axis of the reference frame (the precession axis)
* The line of nodes of the moving frame with respect to 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 on each of these three axes is the derivative of its associated angle (which is equivalent to decomposing the instantaneous rotation into three instantaneous Euler rotations). Therefore:/ref> : $\backslash boldsymbol\backslash omega\; =\; \backslash dot\backslash alpha\backslash mathbf\; u\_1+\backslash dot\backslash beta\backslash mathbf\; u\_2+\backslash dot\backslash gamma\; \backslash mathbf\; u\_3$ 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: : $\backslash boldsymbol\backslash omega\; =\; (\backslash dot\backslash alpha\; \backslash sin\backslash beta\; \backslash sin\backslash gamma\; +\; \backslash dot\backslash beta\backslash cos\backslash gamma)\; \backslash hat\backslash mathbf\; i+\; (\backslash dot\backslash alpha\; \backslash sin\backslash beta\; \backslash cos\backslash gamma\; -\; \backslash dot\backslash beta\backslash sin\backslash gamma)\; \backslash hat\backslash mathbf\; j\; +\; (\backslash dot\backslash alpha\; \backslash cos\backslash beta\; +\; \backslash dot\backslash gamma)\; \backslash hat\backslash mathbf\; k$ where $\backslash hat\backslash mathbf\; i,\; \backslash hat\backslash mathbf\; j,\; \backslash hat\backslash mathbf\; k$ 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.

Tensor

The angular velocity vector $\backslash boldsymbol\backslash omega=(\backslash omega\_x,\backslash omega\_y,\backslash omega\_z)$ defined above may be equivalently expressed as an angular velocity tensor, the matrix (or linear mapping) ''W'' = ''W''(''t'') defined by: : $W\; =\; \backslash begin\; 0\; \&\; -\backslash omega\_z\; \&\; \backslash omega\_y\; \backslash \backslash \; \backslash omega\_z\; \&\; 0\; \&\; -\backslash omega\_x\; \backslash \backslash \; -\backslash omega\_y\; \&\; \backslash omega\_x\; \&\; 0\; \backslash \backslash \; \backslash end$ This is an infinitesimal rotation matrix. The linear mapping ''W'' acts as $(\backslash boldsymbol\backslash omega\; \backslash times)$: : $\backslash boldsymbol\backslash omega\; \backslash times\; \backslash mathbf\; =\; W\; \backslash cdot\backslash mathbf.$Calculation from the orientation matrix

A vector $\backslash mathbf\; r$ undergoing uniform circular motion around a fixed axis satisfies: :$\backslash frac\; =\; \backslash boldsymbol\; \backslash times\backslash mathbf\; =\; W\; \backslash cdot\; \backslash mathbf$ Given the orientation matrix ''A''(''t'') of a frame, whose columns are the moving orthonormal coordinate vectors $\backslash mathbf\; e\_1,\backslash mathbf\; e\_2,\backslash mathbf\; e\_3$, we can obtain its angular velocity tensor ''W''(''t'') as follows. Angular velocity must be the same for the three vectors $\backslash mathbf\; r\; =\; \backslash mathbf\; e\_i$, so arranging the three vector equations into columns of a matrix, we have: : $\backslash frac\; =\; W\; \backslash cdot\; A.$ (This holds even if ''A''(''t'') does not rotate uniformly.) Therefore the angular velocity tensor is: : $W\; =\; \backslash frac\; \backslash cdot\; A^\; =\; \backslash frac\; \backslash cdot\; A^,$ since the inverse of the orthogonal matrix $A$ is its transpose $A^$.Properties

In general, the angular velocity in an ''n''-dimensional space is the time derivative of the angular displacement tensor, which is a second rank skew-symmetric tensor. This tensor ''W'' will have independent components, which is the dimension of theLie algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of the Lie group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of rotations
A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...

of an ''n''-dimensional inner product space.Rotations and Angular Momentumon the Classical Mechanics page o

especially Questions 1 and 2.

Duality with respect to the velocity vector

In three dimensions, angular velocity can be represented by a pseudovector because second rank tensors aredual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
** . . . see more cases in :Duality theories
* Dual ...

to pseudovectors in three dimensions. Since the angular velocity tensor ''W'' = ''W''(''t'') is a skew-symmetric matrix
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

:
: $W\; =\; \backslash begin\; 0\; \&\; -\backslash omega\_z\; \&\; \backslash omega\_y\; \backslash \backslash \; \backslash omega\_z\; \&\; 0\; \&\; -\backslash omega\_x\; \backslash \backslash \; -\backslash omega\_y\; \&\; \backslash omega\_x\; \&\; 0\backslash \backslash \; \backslash end,$
its Hodge dual
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

is a vector, which is precisely the previous angular velocity vector $\backslash boldsymbol\backslash omega=;\; href="/html/ALL/s/omega\_x,\backslash omega\_y,\backslash omega\_z.html"\; ;"title="omega\_x,\backslash omega\_y,\backslash omega\_z">omega\_x,\backslash omega\_y,\backslash omega\_z$Exponential of ''W''

If we know an initial frame ''A''(0) and we are given a ''constant'' angular velocity tensor ''W'', we can obtain ''A''(''t'') for any given ''t''. Recall the matrix differential equation: : $\backslash frac\; =\; W\; \backslash cdot\; A\; .$ This equation can be integrated to give: : $A(t)\; =\; e^A(0)\; ,$ which shows a connection with theLie group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of rotations.
''W'' is skew-symmetric

We prove that angular velocity tensor is skew symmetric, i.e. $W\; =\; \backslash frac\; \backslash cdot\; A^\backslash text$ satisfies $W^\backslash text\; =\; -W$. A rotation matrix ''A'' is orthogonal, inverse to its transpose, so we have $I=A\backslash cdot\; A^\backslash text$. For $A=A(t)$ a frame matrix, taking the time derivative of the equation gives: : $0=\backslash fracA^\backslash text+A\backslash frac$ Applying the formula $(A\; B)^\backslash text=B^\backslash textA^\backslash text$, : $0\; =\; \backslash fracA^\backslash text+\backslash left(\backslash frac\; A^\backslash text\backslash right)^\backslash text\; =\; W\; +\; W^\backslash text$ Thus, ''W'' is the negative of its transpose, which implies it is skew symmetric.Coordinate-free description

At any instant $t$, the angular velocity tensor represents a linear map between the position vector $\backslash mathbf(t)$ and the velocity vectors $\backslash mathbf(t)$ of a point on a rigid body rotating around the origin: : $\backslash mathbf\; =\; W\backslash mathbf\; .$ The relation between this linear map and the angular velocitypseudovector
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...

$\backslash boldsymbol\backslash omega$ is the following.
Because ''W'' is the derivative of an orthogonal transformation In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and t ...

, the bilinear form
: $B(\backslash mathbf,\backslash mathbf)\; =\; (W\backslash mathbf)\; \backslash cdot\; \backslash mathbf$
is skew-symmetric. Thus we can apply the fact of exterior algebra
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

that there is a unique linear form
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

$L$ on $\backslash Lambda^2\; V$ that
: $L(\backslash mathbf\backslash wedge\; \backslash mathbf)\; =\; B(\backslash mathbf,\backslash mathbf)$
where $\backslash mathbf\backslash wedge\; \backslash mathbf\; \backslash in\; \backslash Lambda^2\; V$ is the exterior product
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...

of $\backslash mathbf$ and $\backslash mathbf$.
Taking the sharp
Sharp or SHARP may refer to:
Short for
*Self Help Addiction Recovery Program, a charitable organisation founded in 1991 by Barbara Bach and Pattie Boyd
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''L'' of ''L'' we get
: $(W\backslash mathbf)\backslash cdot\; \backslash mathbf\; =\; L^\backslash sharp\; \backslash cdot\; (\backslash mathbf\backslash wedge\; \backslash mathbf)$
Introducing $\backslash boldsymbol\backslash omega\; :=\; (L^\backslash sharp)$, as the Hodge dual
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

of ''L'', and applying the definition of the Hodge dual twice supposing that the preferred unit 3-vector is $\backslash star\; 1$
: $(W\backslash mathbf)\; \backslash cdot\; \backslash mathbf\; =\; (\; (\; L^\backslash sharp\; )\; \backslash wedge\; \backslash mathbf\; \backslash wedge\; \backslash mathbf)\; =\; (\backslash boldsymbol\backslash omega\; \backslash wedge\; \backslash mathbf\; \backslash wedge\; \backslash mathbf)\; =\; (\backslash boldsymbol\backslash omega\; \backslash wedge\; \backslash mathbf\; )\; \backslash cdot\; \backslash mathbf\; =\; (\backslash boldsymbol\backslash omega\; \backslash times\; \backslash mathbf\; )\; \backslash cdot\; \backslash mathbf\; ,$
where
: $\backslash boldsymbol\backslash omega\; \backslash times\; \backslash mathbf\; :=\; (\backslash boldsymbol\backslash omega\; \backslash wedge\; \backslash mathbf)$
by definition.
Because $\backslash mathbf$ is an arbitrary vector, from nondegeneracy of scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

follows
: $W\backslash mathbf\; =\; \backslash boldsymbol\backslash omega\; \backslash times\; \backslash mathbf$
Angular velocity as a vector field

Since the spin angular velocity tensor of a rigid body (in its rest frame) is a linear transformation that maps positions to velocities (within the rigid body), it can be regarded as a constantvector field
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product ...

. In particular, the spin angular velocity is a Killing vector fieldIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

belonging to an element of the Lie algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

SO(3) of the 3-dimensional rotation group SO(3)
In mechanics
Mechanics (Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximatel ...

.
Also, it can be shown that the spin angular velocity vector field is exactly half of the curl
Curl or CURL may refer to:
Science and technology
* Curl (mathematics)
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the p ...

of the linear velocity vector field v(r) of the rigid body. In symbols,
: $\backslash boldsymbol\; =\; \backslash frac\; \backslash nabla\backslash times\backslash mathbf$
Rigid body considerations

The same equations for the angular speed can be obtained reasoning over a rotatingrigid body
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...

. Here is not assumed that the rigid body rotates around the origin. Instead, it can be supposed rotating around an arbitrary point that 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 and the vector from ''O'' to is R. A particle (''i'') in the rigid body is located at point P and the vector position of this particle is REuler's rotation theorem
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

, we may replace the vector $\backslash mathbf\_i$ with $\backslash mathcal\backslash mathbf\_$ where $\backslash mathcal$ is a 3×3 rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation (mathematics), rotation in Euclidean space. For example, using the convention below, the matrix
:R = \begin
\cos \theta & -\sin \theta \\
\sin \th ...

and $\backslash mathbf\_$ is the position of the particle at some fixed point in time, say . This replacement is useful, because now it is only the rotation matrix $\backslash mathcal$ that is changing in time and not the reference vector $\backslash mathbf\_$, as the rigid body rotates about point . Also, since the three columns of the rotation matrix represent the three versor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

s of a reference frame rotating together with the rigid body, any rotation about any axis becomes now visible, while the vector $\backslash mathbf\_i$ 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:
: $\backslash mathbf\_i=\backslash mathbf+\backslash mathcal\backslash mathbf\_$
Taking the time derivative yields the velocity of the particle:
: $\backslash mathbf\_i=\backslash mathbf+\backslash frac\backslash mathbf\_$
where Vdual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
** . . . see more cases in :Duality theories
* Dual ...

to get a 3 dimensional pseudovector that is precisely the previous angular velocity vector $\backslash boldsymbol\; \backslash omega$:
: $\backslash boldsymbol\backslash omega=;\; href="/html/ALL/s/omega\_x,\backslash omega\_y,\backslash omega\_z.html"\; ;"title="omega\_x,\backslash omega\_y,\backslash omega\_z">omega\_x,\backslash omega\_y,\backslash omega\_z$Consistency

We have supposed that the rigid body rotates around an arbitrary point. We should prove that the spin angular velocity previously defined is independent of the choice of origin, which means that the spin angular velocity is an intrinsic property of the spinning rigid body. (Note the marked contrast of this with the ''orbital'' angular velocity of a point particle, which certainly ''does'' depend on the choice of origin.) 320 px, Proving the independence of spin angular velocity from choice of origin See the graph to the right: The origin of lab frame is ''O'', while ''O''See also

*Angular acceleration
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...

* Angular frequency
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...

* Angular momentum
In , angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of . It is an important quantity in physics because it is a —the total angular momentum of a closed system remains constant.
In three , the ...

* Areal velocity
In classical mechanics, areal velocity (also called sector velocity or sectorial velocity) is the rate at which area
Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surfac ...

* Isometry
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

* Orthogonal group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

* Rigid body dynamics
In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external force
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (e ...

* Vorticity
In continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as point particle, discrete particles. The French mathematician Augustin-Louis C ...

References

* *External links

A college text-book of physics

By Arthur Lalanne Kimball (''Angular Velocity of a particle'') * {{Authority control Angle Physical quantities Rotational symmetry Temporal rates Tensors Velocity