# Rotation

##### Article

May 25, 2022

A rotation is the movement of a body following a circular path. In two dimensions, ie on the plane, a figure can rotate around a point called the center of instantaneous rotation; in three dimensions, the rotation occurs around a straight line called the axis of instantaneous rotation and more generally, a rotation in n dimensions occurs around a space with (n-2) dimensions. If the center or axis of rotation is completely external to the object that is rotating, the body is said to describe an orbit: the typical example is that of the earth's revolution around the Sun. Otherwise the object rotates on himself, as a top would do.

## In mathematics

In geometry, rotation means that rigid movement having as fixed points a point called the center (in two dimensions) or a line called the axis (in three dimensions) of rotation. This movement moves all points around the center, or axis, by a fixed angle. This formalizes the sensory perception according to which rotating a non-deformable object is equivalent to varying the angle of all its points by the same quantity, leaving the reciprocal angular relations unchanged. In particular, a rotation is a transformation that preserves the dot product. In order to distinguish the rotations from the reflections, their peculiarity is emphasized of preserving, in addition to the reciprocal angles, also the orientation of the figure (in formal terms, the sign of the scalar product as well as the module), i.e. that of mapping as they say triplets right-handed (left-handed) ordered in right-handed (left-handed) triples. This means that if three vectors orthogonal to each other obey the right-hand rule, then they will continue to do so even after being rotated, which does not happen after a reflection. The latter in fact sends right-handed triples to left-handed triples, and vice versa. Rotations, contrary to reflections, can be seen as the result of a continuous movement over time (as in the figure). This distinction between reflections and rotations, although intuitive, is mathematically expressed in a non-trivial way and requires some concepts typical of topology: a rotation is isotope to the identity function, while a reflection is not. In the formalism of linear algebra, a rotation in Euclidean space is equivalent to a special orthogonal matrix (i.e. a positive determinant). Finally, in this context it is customary to generalize the concept of rotation in Hilbert spaces of generic dimension, at the infinite limit (in which case we speak of unitary operators).

## In physics

The concept of rotation in physics is similar to the mathematical one, but contrary to the latter, its dynamic nature is emphasized by introducing a continuous dependence on time. The characteristic dynamic parameter of rotation thus becomes the angular velocity (or angular acceleration) which in turn determines, through the concept of mass, the angular momentum (or the kinetic energy of rotation). The latter is the physical quantity that appears in the equations of motion as a response to the application of a given mechanical moment.

## Earth's rotation

The Earth rotates on itself at the speed of 1 revolution in about 24 hours (see sidereal day). Its axis of rotation meets the earth's surface at two points, called the geographic north pole and the geographic south pole. This axis is inclined with respect to the ecliptic, that is, to the plane of the earth's orbit around the sun: this fact is the cause of the succession of the seasons. The inclination of the earth's axis fluctuates by about 2º with a period of about 40,000 years: currently it is about 23º27 '. According to some scholars, it was these oscillations that caused the ice ages by varying the amount of sunlight received by different parts of the Earth. The axis also rotates slowly along the plane of the ecliptic, c