# Rotation

Rotation is the movement of a body in such a way that the distance between a certain fixed point and any given point of that body remains constant.

The fixed point can be within the body, in which case the body is said to rotate upon itself, or spin. In 3-dimensional space, for a given rotational movement there is more than one fixed point: these form a line known as the axis of rotation.

An object may allow rotation with respect to an attached other object by means of one or more hinges (e.g. a door, scissors, a hinge joint).

In astronomy, rotation is a commonly observed phenomenon. Stars, planets and similar bodies all rotate around their central axes. The motion of the components of galaxies is more complex, but it usually includes a rotation component.

Mathematicians consider rotations to be norm-preserving linear transformations on a vector space with an inner product. These form a group of so called special orthogonal matrices (special means determinant equals 1; orthogonal means the transpose is the inverse) designated by the name SO(n) in case of an n-dimensional vector space.

One consequence of the rotation of a planet is the phenomenon of precession. Precession has the overall effect of introducing a long-term "wobble" in the movement of the axis of a planet. For example, the tilt of the Earth's axis to its orbital plane (obliquity of the ecliptic) is currently 66.5 degrees, but this angle has slowly changed over time due to the action of precession. One result of this motion is that the direction of the North Pole and South Pole with respect to the background stars has changed over time, such that the star currently appearing over the North Pole (Polaris) did not so appear ten thousand years ago, and will not ten thousand years from now. With respect to Earth, this phenomenon is called the precession of the equinoxes.

The speed of rotation is given by the angular frequency (rad/s) or frequency (cycles/s, cycles/min), or period (seconds, days, etc.). The time-rate of change of angular frequency is angular acceleration (rad/s²), This change is caused by torque. The ratio of the two (how heavy is it to start, stop, or otherwise change rotation) is given by the moment of inertia. The energy required for / released during rotation is the torque times the rotation angle, the energy stored in a rotating object is one half of the moment of inertia times the square of the angular frequency. The power required for angular acceleration is the torque times the angular frequency.

The angular velocity vector also describes the direction of the axis of rotation. Similarly the torque is a vector.

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