Angular velocity

In physics, angular velocity (formula_1 or formula_2), also known as angular frequency vector, 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 of angle per unit time (angle replacing distance from linear velocity with time in common). The SI unit of angular velocity is radians per second, with the radian being a dimensionless quantity, thus the SI units of angular velocity may be listed as s−1. Angular velocity is usually represented by the symbol omega (ω, sometimes Ω). By convention, positive angular velocity indicates counter-clockwise rotation, while negative is clockwise.
For example, a geostationary 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 linear velocity is the radius times the angular velocity, formula_3. 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, with its magnitude measuring the "angular speed", 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.
Of a point particle.
Particle in two dimensions.
In the simplest case of circular motion at radius formula_4, with position given by the angular displacement formula_5 from the x-axis, the orbital angular velocity is the rate of change of angle with respect to time: formula_6. If formula_7 is measured in radians, the arc-length from the positive x-axis around the circle to the particle is formula_8, and the linear velocity is formula_9, so that formula_10.
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 formula_11 from the origin formula_12 to a particle formula_13, with its polar coordinates formula_14. (All variables are functions of time formula_15.) The particle has linear velocity splitting as formula_16, with the radial component formula_17 parallel to the radius, and the cross-radial (or tangential) component formula_18 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:
Here the cross-radial speed formula_20 is the signed magnitude of formula_18, positive for counter-clockwise motion, negative for clockwise. Taking polar coordinates for the linear velocity formula_22 gives magnitude formula_23 (linear speed) and angle formula_24 relative to the radius vector; in these terms, formula_25, so that
These formulas may be derived doing formula_27, being formula_4 a function of the distance to the origin with respect to time, and formula_29 a function of the angle between the vector and the x axis. Then formula_30. Which is equal to formula_31. (See Unit vector in cylindrical coordinates). Knowing formula_32, we conclude that the radial component of the velocity is given by formula_33, because formula_34 is a radial unit vector; and the perpendicular component is given by formula_35 because formula_36 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, 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, we again have the position vector r of a moving particle. Here, orbital angular velocity is a pseudovector 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 is used.
Let the pseudovector formula_37 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 formula_37). Taking polar coordinates formula_39 in this plane, as in the two-dimensional case above, one may define the orbital angular velocity vector as:
where "θ" is the angle between r and v. In terms of the cross product, this is:
From the above equation, one can recover the tangential velocity as:
Note that the above expression for formula_43 is only valid if formula_44 is in the same plane as the motion.
Addition of angular velocity vectors.
If a point rotates with orbital angular velocity formula_45 about its center of rotation in a coordinate frame formula_46 which itself rotates with a spin angular velocity formula_47 with respect to an external frame formula_48, we can define formula_49 to be the composite orbital angular velocity vector of the point about its center of rotation with respect to formula_48. This operation coincides with usual addition of vectors, and it gives angular velocity the algebraic structure of a true vector, rather than just a pseudo-vector.
The only non-obvious property of the above addition is commutativity. This can be proven from the fact that the velocity tensor "W" (see below) is skew-symmetric, so that formula_51 is a rotation matrix which can be expanded as formula_52. The composition of rotations is not commutative, but formula_53 is commutative to first order, and therefore formula_54.
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, and special tools have been developed for it: the spin angular velocity may be described as a vector or equivalently as a tensor.
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. 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: formula_54.
By Euler's rotation theorem, 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 formula_56 fixed in the rigid body, the velocity formula_57 of any point in the body is given by
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 formula_59 fixed to the body and with their common origin at O. The angular velocity vector of both frame and body about O is then
Here 
Note that this formula is incompatible with the expression
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" formula_64 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 using his Euler angles and the use of an intermediate frame:
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:
This basis is not orthonormal and it is difficult to use, but now the velocity vector can be changed to the fixed frame or to the moving frame with just a change of bases. For example, changing to the mobile frame:
where formula_67 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 formula_68 defined above may be equivalently expressed as an angular velocity tensor, the matrix (or linear mapping) "W" = "W"("t") defined by:
This is an infinitesimal rotation matrix. The linear mapping "W" acts as formula_70:
Calculation from the orientation matrix.
A vector formula_72 undergoing uniform circular motion around a fixed axis satisfies:
Given the orientation matrix "A"("t") of a frame, whose columns are the moving orthonormal coordinate vectors formula_74, we can obtain its angular velocity tensor "W"("t") as follows. Angular velocity must be the same for the three vectors formula_75, so arranging the three vector equations into columns of a matrix, we have:
(This holds even if "A"("t") does not rotate uniformly.) Therefore the angular velocity tensor is:
since the inverse of the orthogonal matrix formula_78 is its transpose formula_79.
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 of the Lie group of rotations of an "n"-dimensional inner product space.
Duality with respect to the velocity vector.
In three dimensions, angular velocity can be represented by a pseudovector because second rank tensors are dual to pseudovectors in three dimensions. Since the angular velocity tensor "W" = "W"("t") is a skew-symmetric matrix:
its Hodge dual is a vector, which is precisely the previous angular velocity vector formula_81.
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:
This equation can be integrated to give:
which shows a connection with the Lie group of rotations.
"W" is skew-symmetric.
We prove that angular velocity tensor is skew symmetric, i.e. formula_84 satisfies formula_85.
A rotation matrix "A" is orthogonal, inverse to its transpose, so we have formula_86. For formula_87 a frame matrix, taking the time derivative of the equation gives:
Applying the formula formula_89,
Thus, "W" is the negative of its transpose, which implies it is skew symmetric.
Coordinate-free description.
At any instant formula_15, the angular velocity tensor represents a linear map between the position vector formula_92 and the velocity vectors formula_93 of a point on a rigid body rotating around the origin:
The relation between this linear map and the angular velocity pseudovector formula_95 is the following.
Because "W" is the derivative of an orthogonal transformation, the bilinear form
is skew-symmetric. Thus we can apply the fact of exterior algebra that there is a unique linear form formula_97 on formula_98 that
where formula_100 is the exterior product of formula_11 and formula_102.
Taking the sharp "L" of "L" we get
Introducing formula_104, as the Hodge dual of "L", and applying the definition of the Hodge dual twice supposing that the preferred unit 3-vector is formula_105
where
by definition.
Because formula_102 is an arbitrary vector, from nondegeneracy of scalar product follows
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 particular, the spin angular velocity is a Killing vector field belonging to an element of the Lie algebra SO(3) of the 3-dimensional rotation group SO(3).
Also, it can be shown that the spin angular velocity vector field is exactly half of the curl of the linear velocity vector field v(r) of the rigid body. In symbols,
Rigid body considerations.
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 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:
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 formula_112 is unchanging. By Euler's rotation theorem, we may replace the vector formula_112 with formula_114 where formula_115 is a 3×3 rotation matrix and formula_116 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 formula_115 that is changing in time and not the reference vector formula_116, as the rigid body rotates about point . 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 formula_112 would not rotate if the rotation axis were parallel to it, and hence it would only describe a rotation about an axis perpendicular to it (i.e., it would not see the component of the angular velocity pseudovector parallel to it, and would only allow the computation of the component perpendicular to it). The position of the particle is now written as:
Taking the time derivative yields the velocity of the particle:
where V"i" is the velocity of the particle (in the lab frame) and V is the velocity of (the origin of the rigid body frame). Since formula_115 is a rotation matrix its inverse is its transpose. So we substitute formula_123:
or
where formula_128 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 that is precisely the previous angular velocity vector formula_129:
Substituting "ω" for "W" into the above velocity expression, and replacing matrix multiplication by an equivalent cross product:
It can be seen that the velocity of a point in a rigid body can be divided into two terms – the velocity of a reference point fixed in the rigid body plus the cross product term involving the orbital angular velocity of the particle with respect to the reference point. This angular velocity is what physicists call the "spin angular velocity" of the rigid body, as opposed to the "orbital" angular velocity of the reference point about the origin "O".
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.)
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 formula_132 and formula_133 respectively. Suppose the angular velocity with respect to "O"1 and O2 is formula_134 and formula_135 respectively. Since point "P" and "O"2 have only one velocity,
The above two yields that
Since the point "P" (and thus formula_139) is arbitrary, it follows that
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.