Editing Rotational invariance

In [[mathematics]], a [[function (mathematics)|function]] defined on an [[inner product space]] is said to have '''rotational invariance''' if its value does not change when arbitrary [[rotation]]s are applied to its argument.

== Mathematics ==

=== Functions ===

For example, the function 
: <math>f(x,y) = x^2 + y^2 </math>
is invariant under rotations of the plane around the origin, because for a rotated set of coordinates through any [[angle]] ''θ''
: <math>x' = x \cos \theta  - y \sin \theta </math>
: <math>y' = x \sin \theta + y \cos \theta </math>
the function, after some cancellation of terms, takes exactly the same form 
: <math>f(x',y') = {x}^2 + {y}^2 </math>

The rotation of coordinates can be expressed using [[matrix (mathematics)|matrix]] form using the [[rotation matrix]],
: <math>\begin{bmatrix} x' \\ y' \\ \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix}\begin{bmatrix} x \\ y \\ \end{bmatrix} ,</math>
or symbolically, {{prime|'''x'''}} = '''Rx'''. Symbolically, the rotation invariance of a real-valued function of two real variables is
: <math>f(\mathbf{x}') = f(\mathbf{Rx}) = f(\mathbf{x}) </math>

In words, the function of the rotated coordinates takes exactly the same form as it did with the initial coordinates, the only difference is the rotated coordinates replace the initial ones. For a [[function of several real variables|real-valued function of three or more real variables]], this expression extends easily using appropriate rotation matrices. 

The concept also extends to a [[vector-valued function]] '''f''' of one or more variables;
: <math>\mathbf{f}(\mathbf{x}') = \mathbf{f}(\mathbf{Rx}) = \mathbf{f}(\mathbf{x}) .</math>

In all the above cases, the arguments (here called "coordinates" for concreteness) are rotated, not the function itself.

=== Operators ===

For a [[function (mathematics)|function]]
: <math>f : X \rightarrow X ,</math>
which maps elements from a [[subset]] ''X'' of the [[real line]] <math>\mathbb{R}</math> to itself, '''rotational invariance''' may also mean that the function [[commutative operation|commute]]s with rotations of elements in ''X''. This also applies for an [[Operator (mathematics)|operator]] that acts on such functions. An example is the two-dimensional [[Laplace operator]] 
: <math>\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} ,</math>
which acts on a function ''f'' to obtain another function ∇<sup>2</sup>''f''. This operator is invariant under rotations.

If ''g'' is the function ''g''(''p'') = ''f''(''R''(''p'')), where ''R'' is any rotation, then (∇<sup>2</sup>''g'')(''p'') = (∇<sup>2</sup>''f'' )(''R''(''p'')); that is, rotating a function merely rotates its Laplacian.  <!-- Should add the (classical) physics sense, and Computer Vision sense too -->

== Physics ==

In [[physics]], if a system behaves the same regardless of how it is oriented in space, then its [[Lagrangian mechanics|Lagrangian]] is rotationally invariant. According to [[Noether's theorem]], if the [[Action (physics)|action]] (the integral over time of its Lagrangian) of a physical system is invariant under rotation, then [[Conservation of angular momentum|angular momentum is conserved]].

=== Application to quantum mechanics ===

{{Further|Rotation operator (quantum mechanics)|Symmetry in quantum mechanics}}

In [[quantum mechanics]], '''rotational invariance''' is the property that after a [[rotation]] the new system still obeys the [[Schrödinger equation]].  That is
: <math>[R,E-H] = 0</math> 
for any rotation ''R''. Since the rotation does not depend explicitly on time, it commutes with the energy operator. Thus for rotational invariance we must have [''R'',&nbsp;''H''] = 0.

For [[infinitesimal rotation]]s (in the ''xy''-plane for this example; it may be done likewise for any plane) by an angle ''dθ'' the (infinitesimal) rotation operator is
: <math>R = 1 + J_z d\theta \,,</math>
then
: <math>\left[1 + J_z d\theta , \frac{d}{dt} \right] = 0 \,,</math>
thus
: <math>\frac{d}{dt}J_z = 0\,,</math>

in other words [[angular momentum]] is conserved.

== See also ==
* [[Axial symmetry]]
* [[Invariant measure]]
* [[Isotropy]]
* [[Maxwell's theorem]]
* [[Rotational symmetry]]

== References ==
* Stenger, Victor J. (2000). ''Timeless Reality''. Prometheus Books. Especially chpt. 12. Nontechnical.

[[Category:Rotational symmetry]]
[[Category:Conservation laws]]