Symmetry (coupled oscillators)

A physical system has symmetry if there is some coordinate transformation that can be made which gives an equivalent system. For example, the systems shown below have mirror symmetry because mirroring the system results in an equivalent system.

Say we have a coordinate system $x_1 \ldots x_n$, we want to transform each $x_k \to \tilde x_k$ such that the system remains equivalent.

In the two examples above, if we had horizontal displacements $x_1$, $x_2$, then our mirrored coordinates would be $\tilde x_1 = -x_2$, $\tilde x_2 = -x_1$.

We can encode this transformation as a symmetry matrix such that $\tilde{\mathbf x} = \mathbf S \mathbf x$ (where $\tilde{\mathbf x}$ and $\mathbf x$ are the vectors containing the original and mirrored coordinates)

Then, since the new system is physically equivalent to the old system, we can say that the new coordinates are proportional to the old ones: $\tilde{\mathbf x} \propto \mathbf x$. Writing this out with an explicit constant of proportionality we get $\mathbf S \mathbf x = \beta \mathbf x$ for some constant $\beta$.

Now we guess a solution $\mathbf x = \mathbf A_1 \cos(\omega_1 t)$. Plugging in our ansatz, $\mathbf S \mathbf A_1 \cos(\omega_1 t) = \beta \mathbf A_1 \cos(\omega_1 t)$. The cosines cancel to give us $\mathbf S \mathbf A_1 = \beta \mathbf A_1$. This means that $\beta$ is an eigenvalue of $\mathbf S$ and $\mathbf A_1$ is its corresponding eigenvector.