TheInfoList

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. "Physical scie ...

, angular velocity ($\boldsymbol$ or $\boldsymbol$), 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 s−1. Angular velocity is usually represented by the symbol
omega 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, $v = r\omega$. 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 $\phi\left(t\right)$ from the x-axis, the orbital angular velocity is the rate of change of angle with respect to time: $\omega = \frac$. If $\phi$ is measured in
radian 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 $\ell=r\phi$, and the linear velocity is $v(t) = \frac = r\omega(t)$, so that $\omega = \frac$. 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 $\mathbf$ 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 ...

$\left(r, \phi\right)$. (All variables are functions of time $t$.) The particle has linear velocity splitting as $\mathbf = \mathbf_\, +\mathbf_\perp$, with the radial component $\mathbf_\,$ parallel to the radius, and the cross-radial (or tangential) component $\mathbf_\perp$ 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: : $\omega = \frac = \frac.$ Here the cross-radial speed $v_\perp$ is the signed magnitude of $\mathbf_\perp$, positive for counter-clockwise motion, negative for clockwise. Taking polar coordinates for the linear velocity $\mathbf$ gives magnitude $v$ (linear speed) and angle $\theta$ relative to the radius vector; in these terms, $v_\perp = v\sin\left(\theta\right)$, so that : $\omega = \frac.$ These formulas may be derived doing $\mathbf=\left(r\cos\left(\varphi\right),r\sin\left(\varphi\right)\right)$, being $r$ a function of the distance to the origin with respect to time, and $\varphi$ a function of the angle between the vector and the x axis. Then $\frac = \left(\dot\cos\left(\varphi\right) - r\dot\sin\left(\varphi\right), \dot\sin\left(\varphi\right) + r\dot\cos\left(\varphi\right)\right)$. Which is equal to $\dot\left(\cos\left(\varphi\right), \sin\left(\varphi\right)\right) + r\dot\left(-\sin\left(\varphi\right), \cos\left(\varphi\right)\right) = \dot\hat + r\dot\hat$. (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 $\frac=\mathbf$, we conclude that the radial component of the velocity is given by $\dot$, because $\hat$ is a radial unit vector; and the perpendicular component is given by $r\dot$ because $\hat$ 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

In
three-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 $\mathbf$ 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 $\mathbf$). Taking polar coordinates $\left(r,\phi\right)$ in this plane, as in the two-dimensional case above, one may define the orbital angular velocity vector as: : $\boldsymbol\omega =\omega \mathbf u = \frac\mathbf u=\frac\mathbf u,$ where ''θ'' is the angle between r and v. In terms of the cross product, this is: : $\boldsymbol\omega =\frac.$ From the above equation, one can recover the tangential velocity as: :$\mathbf_ =\boldsymbol \times\mathbf$ 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 $\omega_1$ about its center of rotation in a coordinate frame $F_1$ which itself rotates with a spin angular velocity $\omega_2$ with respect to an external frame $F_2$, we can define $\omega_1 + \omega_2$ 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 true
vector 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 $R = I + W\cdot dt + \tfrac 12 \left(W \cdot dt\right)^2 + \cdots$. The composition of rotations is not commutative, but $\left(I+W_1\cdot dt\right)\left(I+W_2 \cdot dt\right)=\left(I+W_2 \cdot dt\right)\left(I+W_1\cdot dt\right)$ is commutative to first order, and therefore $\omega_1 + \omega_2 = \omega_2 + \omega_1$. 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 of
rigid 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: $\omega_1 + \omega_2 = \omega_2 + \omega_1$. 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 $\dot$ of any point in the body is given by :$\dot = \dot +\left(-\right)\times$

## 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 $\mathbf_1, \mathbf_2, \mathbf_3$ fixed to the body and with their common origin at O. The angular velocity vector of both frame and body about O is then : $\boldsymbol\omega = \left\left(\dot \mathbf_1\cdot\mathbf_2\right\right) \mathbf_3 + \left\left(\dot \mathbf_2\cdot\mathbf_3\right\right) \mathbf_1 + \left\left(\dot \mathbf_3\cdot\mathbf_1\right\right) \mathbf_2,$ Here :$\dot \mathbf_i= \frac$ is the time rate of change of the frame vector $\mathbf_i, i=1,2,3,$ due to the rotation. Note that this formula is incompatible with the expression : $\boldsymbol\omega =\frac.$ 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'' $\boldsymbol\omega$ 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 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 (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:K.S.HEDRIH: Leonhard Euler (1707–1783) and rigid body dynamics
/ref> : $\boldsymbol\omega = \dot\alpha\mathbf u_1+\dot\beta\mathbf u_2+\dot\gamma \mathbf u_3$ 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: : $\boldsymbol\omega = \left(\dot\alpha \sin\beta \sin\gamma + \dot\beta\cos\gamma\right) \hat\mathbf i+ \left(\dot\alpha \sin\beta \cos\gamma - \dot\beta\sin\gamma\right) \hat\mathbf j + \left(\dot\alpha \cos\beta + \dot\gamma\right) \hat\mathbf k$ where $\hat\mathbf i, \hat\mathbf j, \hat\mathbf k$ 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 $\boldsymbol\omega=\left(\omega_x,\omega_y,\omega_z\right)$ defined above may be equivalently expressed as an angular velocity tensor, the matrix (or linear mapping) ''W'' = ''W''(''t'') defined by: : $W = \begin 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \\ \end$ This is an infinitesimal rotation matrix. The linear mapping ''W'' acts as $\left(\boldsymbol\omega \times\right)$: : $\boldsymbol\omega \times \mathbf = W \cdot\mathbf.$

## Calculation from the orientation matrix

A vector $\mathbf r$ undergoing uniform circular motion around a fixed axis satisfies: :$\frac = \boldsymbol \times\mathbf = W \cdot \mathbf$ Given the orientation matrix ''A''(''t'') of a frame, whose columns are the moving orthonormal coordinate vectors $\mathbf e_1,\mathbf e_2,\mathbf e_3$, we can obtain its angular velocity tensor ''W''(''t'') as follows. Angular velocity must be the same for the three vectors $\mathbf r = \mathbf e_i$, so arranging the three vector equations into columns of a matrix, we have: : $\frac = W \cdot A.$ (This holds even if ''A''(''t'') does not rotate uniformly.) Therefore the angular velocity tensor is: : $W = \frac \cdot A^ = \frac \cdot A^,$ 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 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 ...
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 Momentum
on 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 are
dual 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 ...
: : $W = \begin 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0\\ \end,$ 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

## 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: : $\frac = W \cdot A .$ This equation can be integrated to give: : $A\left(t\right) = e^A\left(0\right) ,$ which shows a connection with 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.

## ''W'' is skew-symmetric

We prove that angular velocity tensor is skew symmetric, i.e. $W = \frac \cdot A^\text$ satisfies $W^\text = -W$. A rotation matrix ''A'' is orthogonal, inverse to its transpose, so we have $I=A\cdot A^\text$. For $A=A\left(t\right)$ a frame matrix, taking the time derivative of the equation gives: : $0=\fracA^\text+A\frac$ Applying the formula $\left(A B\right)^\text=B^\textA^\text$, : $0 = \fracA^\text+\left\left(\frac A^\text\right\right)^\text = W + W^\text$ 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 $\mathbf\left(t\right)$ and the velocity vectors $\mathbf\left(t\right)$ of a point on a rigid body rotating around the origin: : $\mathbf = W\mathbf .$ The relation between this linear map and the angular velocity
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 ...

$\boldsymbol\omega$ 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 : $B\left(\mathbf,\mathbf\right) = \left(W\mathbf\right) \cdot \mathbf$ 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 $\Lambda^2 V$ that : $L\left(\mathbf\wedge \mathbf\right) = B\left(\mathbf,\mathbf\right)$ where $\mathbf\wedge \mathbf \in \Lambda^2 V$ 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 $\mathbf$ and $\mathbf$. 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 *Sexual Harassment/Assault Response & Prevention, a US Army program dealing with sexual ha ...
''L'' of ''L'' we get : $\left(W\mathbf\right)\cdot \mathbf = L^\sharp \cdot \left(\mathbf\wedge \mathbf\right)$ Introducing $\boldsymbol\omega := \left(L^\sharp\right)$, 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 $\star 1$ : $\left(W\mathbf\right) \cdot \mathbf = \left( \left( L^\sharp \right) \wedge \mathbf \wedge \mathbf\right) = \left(\boldsymbol\omega \wedge \mathbf \wedge \mathbf\right) = \left(\boldsymbol\omega \wedge \mathbf \right) \cdot \mathbf = \left(\boldsymbol\omega \times \mathbf \right) \cdot \mathbf ,$ where : $\boldsymbol\omega \times \mathbf := \left(\boldsymbol\omega \wedge \mathbf\right)$ by definition. Because $\mathbf$ 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 : $W\mathbf = \boldsymbol\omega \times \mathbf$

## 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 constant
vector 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, : $\boldsymbol = \frac \nabla\times\mathbf$

# Rigid body considerations

The same equations for the angular speed can be obtained reasoning over a rotating
rigid 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 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: : $\mathbf_i=\mathbf+\mathbf_i$ 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 $\mathbf_i$ is unchanging. 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 ...
, we may replace the vector $\mathbf_i$ with $\mathcal\mathbf_$ where $\mathcal$ 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 $\mathbf_$ 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 $\mathcal$ that is changing in time and not the reference vector $\mathbf_$, 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 $\mathbf_i$ 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: : $\mathbf_i=\mathbf+\mathcal\mathbf_$ Taking the time derivative yields the velocity of the particle: : $\mathbf_i=\mathbf+\frac\mathbf_$ where V''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 $\mathcal$ is a rotation matrix its inverse is its transpose. So we substitute $\mathcal=\mathcal^\text\mathcal$: : $\mathbf_i = \mathbf+\frac\mathcal\mathbf_$ : $\mathbf_i = \mathbf+\frac\mathcal^\text\mathcal\mathbf_$ : $\mathbf_i = \mathbf+\frac\mathcal^\text\mathbf_$ or : $\mathbf_i = \mathbf+W\mathbf_$ where $W = \frac\mathcal^\text$ is the previous angular velocity tensor. It can be proved that this is a skew symmetric matrix, so we can take its
dual 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 $\boldsymbol \omega$: :

## 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''1 and ''O''2 are two fixed points on the rigid body, whose velocity is $\mathbf_1$ and $\mathbf_2$ respectively. Suppose the angular velocity with respect to ''O''1 and O2 is $\boldsymbol_1$ and $\boldsymbol_2$ respectively. Since point ''P'' and ''O''2 have only one velocity, : $\mathbf_1 + \boldsymbol_1\times\mathbf_1 = \mathbf_2 + \boldsymbol_2\times\mathbf_2$ : $\mathbf_2 = \mathbf_1 + \boldsymbol_1\times\mathbf = \mathbf_1 + \boldsymbol_1\times \left(\mathbf_1 - \mathbf_2\right)$ The above two yields that : $\left(\boldsymbol_2-\boldsymbol_1\right) \times \mathbf_2=0$ Since the point ''P'' (and thus $\mathbf_2$) is arbitrary, it follows that : $\boldsymbol_1 = \boldsymbol_2$ 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.

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