Symmetry

Symmetry arises when a system looks the same in different coordinate systems; for example, the spherical symmetry of a central potential. Symmetry does not require that motion in a symmetrical system has to be symmetric, only that similar motion follows out from similar initial conditions. For example, the nodal patterns in the vibration of an isotropic square membrane do not have to have square symmetry. But it is true that for every normal mode of a given frequencey, there is another, rotated from the first, with that very same frequency; similarly for reflected modes.

The mathematical description of symmetry is that the dynamical matrix $A$ has a change of basis $S$ for which $S A = A S$ (an equivalence) which could of ourse also be written as

$$\begin{eqnarray*} S^{-1} A S & = & A. \end{eqnarray*}$$