Introduction to Differential Equations

Symmetry - Terminology

Concepts of primary interest: Symmetric Anti-symmetric Asymmetric Even and odd Reflection Translation Inversion and Parity Rotation plus reflection example

SYMMETRY: An intrinsic property of a mathematical object or physical entity that causes it to remain invariant under certain classes of transformations (such as: rotation, reflection, inversion, or more abstract operations). The study of symmetry is systematized and formalized in the extremely powerful and beautiful field of mathematics called group theory.

Symmetry can be present in the form of coefficients of equations as well as in the physical arrangement of objects such as atoms in a crystal. The symmetry of a crystal or of a body may restrict its physical properties. Crystals with nonlinear dielectric response can be used to generate light with double the frequency of an incident beam, but only if those crystals lack inversion symmetry. A body with at least three-fold rotational symmetry about two axes must have a moment of inertia tensor that is a multiple of the identity (all principal moments must be equal).

In physics, the extremely powerful Noether's symmetry theorem states that each symmetry of a system leads to a conserved physical quantity. Symmetry under translation leads to momentum conservation, symmetry under rotation to angular momentum conservation, symmetry with respect to translations in time to energy conservation, etc.

Some Terminology: Symmetric: A function, distribution or physical entity is symmetric if there is a symmetry operation or transformation that maps it into an identical copy of itself.

Contact: [email protected] Symmetry operations include inversion, rotation, translation, reflection, etc. A matrix is symmetric under the transpose operation if the transpose operation maps that matrix into + .

Anti-symmetric: A function, distribution or physical entity is anti-symmetric if there is a transformation such as inversion or reflection that maps it into the negative of itself. A matrix is anti-symmetric if the transpose operation maps that matrix into - . (also called skew-symmetric)

Asymmetric: A function, distribution or physical entity is asymmetric if it lacks symmetry under the operations of interest. The function x2 + x is neither even nor odd about zero. It is asymmetric with respect to one-dimensional inversion about the origin. A function, distribution or physical entity may be symmetric with respect to one or more symmetry operations and asymmetric with respect to other symmetry operations.

The terms symmetric and anti-symmetric are also used to describe even and odd functions. Understand that even and odd are less rich as concepts than symmetric and anti-symmetric.

Even: A function that is even about the origin is often called a symmetric function. More precisely: A function is even if it is symmetric under a one-dimensional inversion of coordinates about the origin. Evenness of a function is only one of a multitude of possible symmetries. The term even is a special, limited case of symmetric behavior.

Odd: A function that is odd about the origin is often called an anti-symmetric function. The term odd is a special, limited case of anti-symmetric behavior.

Symmetries with Examples Reflection: Reflection maps a point in space to its mirror image point with respect to a plane (that we will assume contains the origin). One computes the projection of an arbitrary position r into the reflection plane as rrnr=−ˆ ⋅ nˆ and its mirror image point as r = r − 2 nˆ ⋅ r nˆ where nˆ is the P () M () normal to the reflection plane. A function is symmetric with respect to the reflection if f ()rM = f ()r and anti-symmetric if f (rM ) = − f (r ).

4/12/2007 Physics Handout Series.Tank: Symmetry/Terms SymTerm-2 − nˆ ⋅r nˆ − nˆ ⋅r nˆ ( ) ( ) r r M r P

n ˆ

An ideal football has two reflection planes. One is perpendicular to the long axis at its mid-point. The other includes the long axis and the seam that splits the laces. The ball is asymmetric for reflections about any other plane that includes the long axis because only one seam has laces.

Translation: A periodic function is symmetric under a translation of one period. A crystal is symmetric under a translation by a lattice displacement vector. A chunk of glass lacks the regular order that a crystal possesses and is asymmetric with respect to translations.

Coordinate Inversion: r →−r : A function of position is symmetric with respect to inversion of coordinates through the origin if f ()r = f (−r ). If instead, the function has the property that

4/12/2007 Physics Handout Series.Tank: Symmetry/Terms SymTerm-3 f ()r =−f ()−r then f ()r is anti-symmetric under inversion. A function that fails to possess either behavior is asymmetric under inversion. In Quantum Mechanics, functions that are symmetric under inversion are classed as functions of even parity while those that are anti-symmetric are of odd parity.

Rotation: A body has rotational symmetry if a rotation of the body by a set angle Δφ about an axis 2 π transforms the body into itself. A square has four-fold (or Δ φ = 4 ) rotational symmetry about an axis perpendicular to the intersection of its diagonals. A sphere has full rotation symmetry about any axis through its center.

−r 2 Example: In spherical coordinates, the function f (r ) = re sin θ cos (3 φ) has a six-fold rotational symmetry about the polar axis, and it is symmetric under a reflection about the plane ⎛ 2 π ⎞ θ = π . One also finds the family of reflection planes φ = m for φ = 0, 1, and 2. [Note that 2 ⎝ 3 ⎠ ⎛ 2 π ⎞ the coordinate surfaces φ = m are half-infinite planes with the polar axis as the position finite edge. ⎝ 3 ⎠ The reflection planes are the fully infinite extensions of these coordinates surfaces. Each reflection plane is ⎛ 2 π ⎞ ⎛ 2 π ⎞ the union of the φ = m surface and the φ = m + π surface.] Study the figure below. ⎝ 3 ⎠ ⎝ 3 ⎠ Are there other reflection planes?

4/12/2007 Physics Handout Series.Tank: Symmetry/Terms SymTerm-4

SphericalPlot3D[ Sin[theta] (Cos[3 * phi])^2, {theta,0 , Pi}, {phi,0, 2 Pi}, PlotRange-> All, PlotPoints -> 64]

Problems 1.)

References:

1. The Wolfram web site: mathworld.wolfram.com/

2. Noether's Theorem may be discussed in your intermediate mechanics textbook.

2. Template Only: H. F. Weinberger, Partial Differential Equations, Blaisdell, Watham, MA (1965).

4/12/2007 Physics Handout Series.Tank: Symmetry/Terms SymTerm-5