# Angular displacement

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[math]\displaystyle{ \textbf{F} = \frac{d}{dt} (m\textbf{v}) }[/math] |

**Angular displacement** of a body is the angle (in radians, degrees or turns) through which a point revolves around a centre or a specified axis in a specified sense. When a body rotates about its axis, the motion cannot simply be analyzed as a particle, as in circular motion it undergoes a changing velocity and acceleration at any time (*t*). When dealing with the rotation of a body, it becomes simpler to consider the body itself rigid. A body is generally considered rigid when the separations between all the particles remains constant throughout the body's motion, so for example parts of its mass are not flying off. In a realistic sense, all things can be deformable, however this impact is minimal and negligible. Thus the rotation of a rigid body over a fixed axis is referred to as rotational motion.

## Example

In the example illustrated to the right (or above in some mobile versions), a particle or body P is at a fixed distance *r* from the origin, *O*, rotating counterclockwise. It becomes important to then represent the position of particle P in terms of its polar coordinates (*r*, *θ*). In this particular example, the value of *θ* is changing, while the value of the radius remains the same. (In rectangular coordinates (*x*, *y*) both *x* and *y* vary with time). As the particle moves along the circle, it travels an arc length *s*, which becomes related to the angular position through the relationship:

- [math]\displaystyle{ s = r\theta . }[/math]

## Measurements

Angular displacement may be measured in radians or degrees. Using radians provides a very simple relationship between distance traveled around the circle and the distance *r* from the centre.

- [math]\displaystyle{ \theta = \frac{s}{r} }[/math]

For example, if a body rotates 360° around a circle of radius *r*, the angular displacement is given by the distance traveled around the circumference - which is 2π*r* - divided by the radius: [math]\displaystyle{ \theta= \frac{2\pi r}r }[/math] which easily simplifies to: [math]\displaystyle{ \theta=2\pi }[/math]. Therefore, 1 revolution is [math]\displaystyle{ 2\pi }[/math] radians.

When a particle travels from point P to point Q over [math]\displaystyle{ \delta t }[/math], as it does in the illustration to the left, the radius of the circle goes through a change in angle [math]\displaystyle{ \Delta \theta = \theta_2 - \theta_1 }[/math] which equals the *angular displacement*.

## Three dimensions

In three dimensions, angular displacement is an entity with a direction and a magnitude. The direction specifies the axis of rotation, which always exists by virtue of the Euler's rotation theorem; the magnitude specifies the rotation in radians about that axis (using the right-hand rule to determine direction). This entity is called an axis-angle.

Despite having direction and magnitude, angular displacement is not a vector because it does not obey the commutative law for addition.^{[1]} Nevertheless, when dealing with infinitesimal rotations, second order infinitesimals can be discarded and in this case commutativity appears.

Several ways to describe **angular displacement** exist, like rotation matrices or Euler angles. See charts on SO(3) for others.

Given that any frame in the space can be described by a rotation matrix, the displacement among them can also be described by a rotation matrix. Being [math]\displaystyle{ A_0 }[/math] and [math]\displaystyle{ A_f }[/math] two matrices, the angular displacement matrix between them can be obtained as [math]\displaystyle{ \Delta A = A_f A_0^{-1} }[/math]. When this product is performed having a very small difference between both frames we will obtain a matrix close to the identity.

In the limit, we will have an infinitesimal rotation matrix.

## Infinitesimal rotation matrices

An infinitesimal rotation matrix or differential rotation matrix is a matrix representing an infinitely small rotation. While a rotation matrix is an orthogonal matrix [math]\displaystyle{ R^\mathsf{T} = R^{-1} }[/math] representing an element of [math]\displaystyle{ SO(n) }[/math] (the special orthogonal group), the differential of a rotation is a skew-symmetric matrix [math]\displaystyle{ A^\mathsf{T} = -A }[/math] in the tangent space [math]\displaystyle{ \mathfrak{so}(n) }[/math] (the special orthogonal Lie algebra), which is not itself a rotation matrix.

An infinitesimal rotation matrix has the form

- [math]\displaystyle{ I + d\theta \, A, }[/math]

where [math]\displaystyle{ I }[/math] is the identity matrix, [math]\displaystyle{ d\theta }[/math] is vanishingly small, and [math]\displaystyle{ A \in \mathfrak{so}(n). }[/math]

For example, if [math]\displaystyle{ A = L_x, }[/math] representing an infinitesimal three-dimensional rotation about the x-axis, a basis element of [math]\displaystyle{ \mathfrak{so}(3), }[/math]

- [math]\displaystyle{ dL_{x} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -d\theta \\ 0 & d\theta & 1 \end{bmatrix}. }[/math]

^{[2]}It turns out that

*the order in which infinitesimal rotations are applied is irrelevant*.

## See also

- Angular distance
- Angular position
- Angular velocity
- Infinitesimal rotation
- Linear elasticity
- Second moment of area

## Notes

## References

- ↑ Kleppner, Daniel; Kolenkow, Robert (1973).
*An Introduction to Mechanics*. McGraw-Hill. pp. 288–89. ISBN 9780070350489. https://archive.org/details/introductiontome00dani. - ↑ (Goldstein Poole)

### Sources

- Goldstein, Herbert; Poole, Charles P.; Safko, John L. (2002),
*Classical Mechanics*(third ed.), Addison Wesley, ISBN 978-0-201-65702-9 - Wedderburn, Joseph H. M. (1934),
*Lectures on Matrices*, AMS, ISBN 978-0-8218-3204-2, https://books.google.com/books?id=6eKVAwAAQBAJ

Original source: https://en.wikipedia.org/wiki/Angular displacement.
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