# Translational symmetry

In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by a: Ta(p) = p + a. For translational invariant functions f : R 2 → R {\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} } it is f ( A ) = f ( A + t ) {\displaystyle f(A)=f(A+t)} . The Lebesgue measure is an example for such a function.

In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation. Discrete translational symmetry is invariant under discrete translation.

Analogously an operator A on functions is said to be translationally invariant with respect to a translation operator $T_{\delta }$ if the result after applying A doesn't change if the argument function is translated. More precisely it must hold that

$\forall \delta \ Af=A(T_{\delta }f).$ Laws of physics are translationally invariant under a spatial translation if they do not distinguish different points in space. According to Noether's theorem, space translational symmetry of a physical system is equivalent to the momentum conservation law.

Translational symmetry of an object means that a particular translation does not change the object. For a given object, the translations for which this applies form a group, the symmetry group of the object, or, if the object has more kinds of symmetry, a subgroup of the symmetry group.