Small oscillation

Near a stable equilibrium, retaining quadratic Lagrangian terms gives a linear system $\mathsf{M}\ddot{z}+\mathsf{K}z=0$.

At an equilibrium, a positive-definite Hessian matrix of the potential gives stable small oscillations. If a continuous symmetry supplies zero modes, positive definiteness on the directions transverse to the symmetry gives stability of the relative configuration.

Angular frequency measures phase advance per unit time. For $x(t)=A\cos (\omega t+\phi )$ it is $\omega =2\pi /T$, where $T$ is the period.

For two masses hung in series from springs of natural length $\ell$, the lower equilibrium spring tension is $m_{2}g$ and the upper tension is $(m_1+m_2)g$. Hence their equilibrium lengths are

For equal masses $m$ and spring constants $k$, vertical displacements from equilibrium have stiffness and mass matricesThe two squared frequencies and displacement ratios areGravity shifts the equilibrium but does not enter these vertical frequencies.

The Hessians of kinetic and potential energy at equilibrium give the symmetric mass matrix $\mathsf{M}$ and stiffness matrix $\mathsf{K}$.

Normal frequencies satisfy $\det (\mathsf{K}-{\Omega }^2\mathsf{M})=0$, and each null vector gives the fixed amplitude ratio of a mode.

For symmetric stiffness matrix $K$ and positive-definite mass matrix $M$, eigenvectors $a_i,a_j$ ofwith distinct squared frequencies satisfyThus normal modes are orthogonal in the kinetic-energy inner product.

For equal lengths and mass ratio $\mu =m/M$, the squared frequencies are ${\omega }_0^2(1+\mu \pm \sqrt{\mu (1+\mu )})$.

A weakly coupled degenerate pair can split by order $\sqrt{\mu }$ when the small parameter enters a singular mass or stiffness matrix.

If the mass and stiffness matrices commute with a permutation representation, each symmetry subspace is invariant under the generalized eigenvalue problem for small oscillations. On an irreducible subspace, Schur lemma forces a commuting operator to be scalar, producing a degenerate normal mode multiplet.

Let three equal masses move on a circle of radius $r$, and let the potential be $U(\alpha )+U(\beta )+U(\gamma )$ with $\alpha +\beta +\gamma =2\pi$. At equal spacing $s_0=2\pi /3$, angular displacements have a zero-frequency rigid-rotation mode $(1,1,1)$ and a two-dimensional sum-zero eigenspace withThe two vibrational modes are degenerate by the permutation symmetry.

For three unit masses with unit springs between every pair and one unit spring anchoring each mass to the origin,where $J$ is the all-ones matrix. The collective mode $(1,1,1)$ has angular frequency $1$, while the two-dimensional sum-zero subspace has angular frequency $2$.

For equal masses $m$, wall-spring constants $k_1,k_2$, and coupling-spring constant $k_3$, the small-oscillation mass and stiffness matrices areIf $k_1=k(1+\epsilon \delta )$, $k_2=k(1-\epsilon \delta )$, and $k_3=k\epsilon$, the squared normal frequencies are