In
mathematics, a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
defined on an
inner product space is said to have rotational invariance if its value does not change when arbitrary
rotations are applied to its argument.
Mathematics
Functions
For example, the function
:
is invariant under rotations of the plane around the origin, because for a rotated set of coordinates through any
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
''θ''
:
:
the function, after some cancellation of terms, takes exactly the same form
:
The rotation of coordinates can be expressed using
matrix form using the
rotation matrix,
:
or symbolically x′ = Rx. Symbolically, the rotation invariance of a real-valued function of two real variables is
:
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
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;
:
In all the above cases, the arguments (here called "coordinates" for concreteness) are rotated, not the function itself.
Operators
For a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
:
which maps elements from a
subset ''X'' of the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
ℝ to itself, rotational invariance may also mean that the function
commute
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
s with rotations of elements in ''X''. This also applies for an
operator
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another s ...
that acts on such functions. An example is the two-dimensional
Laplace operator
:
which acts on a function ''f'' to obtain another function ∇
2''f''. This operator is invariant under rotations.
If ''g'' is the function ''g''(''p'') = ''f''(''R''(''p'')), where ''R'' is any rotation, then (∇
2''g'')(''p'') = (∇
2''f'' )(''R''(''p'')); that is, rotating a function merely rotates its Laplacian.
Physics
In
physics, if a system behaves the same regardless of how it is oriented in space, then its
Lagrangian is rotationally invariant. According to
Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in ...
, if the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
(the integral over time of its Lagrangian) of a physical system is invariant under rotation, then
angular momentum is conserved.
Application to quantum mechanics
In
quantum mechanics, rotational invariance is the property that after a
rotation the new system still obeys
Schrödinger's equation. That is
:
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'', ''H''= 0.
For
infinitesimal rotation In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
:R = \begin
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end ...
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
:
then
:
thus
:
in other words
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syste ...
is conserved.
See also
*
Axial symmetry
Axial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around an axis.
*
Invariant measure In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping ...
*
Isotropy
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
*
Maxwell's theorem
In probability theory, Maxwell's theorem, named in honor of James Clerk Maxwell, states that if the probability distribution of a vector-valued random variable ''X'' = ( ''X''1, ..., ''X'n'' )''T'' is the same as the distribution of ''GX'' for ...
*
Rotational symmetry
References
*Stenger, Victor J. (2000). ''Timeless Reality''. Prometheus Books. Especially chpt. 12. Nontechnical.
Rotational symmetry
Conservation laws