= Solution
\b[Definition and equations.] A <normal mode> is a motion in which every component of a <linear system> has the same time dependence with fixed amplitude ratios and phases. For a conservative system near equilibrium, the most useful form is
$$
\mathbf q(t)=\operatorname{Re}\{a\,\mathbf u\,e^{-i\omega t}\}.
$$
The <vector> $\mathbf u$ describes the <mode shape>, while $\omega$ is its <angular frequency>. Linear combinations of modes describe arbitrary sufficiently small motions when the modes form a complete <basis>.
Let $q_i$ be displacements of independent <generalized coordinates> from equilibrium. Expand the <Lagrangian> to quadratic order:
$$
L=\frac12\dot{\mathbf q}^{\mathsf T}M\dot{\mathbf q}
-\frac12\mathbf q^{\mathsf T}K\mathbf q,
\qquad M\ddot{\mathbf q}+K\mathbf q=0.
$$
The <mass matrix> $M$ defines a <positive-definite quadratic form> on the independent coordinates, while the <stiffness matrix> $K$ is the <Hessian> of the <potential energy>. Substituting a mode gives the <generalized eigenvalue problem for small oscillations>
$$
\boxed{K\mathbf u=\omega^2M\mathbf u,
\qquad \det(K-\omega^2M)=0.}
$$
A practical calculation identifies the equilibrium, chooses independent coordinates, computes the quadratic kinetic and potential <energies>, solves for the <eigenvalues> and then obtains their <eigenvectors>. One can equivalently diagonalize the real symmetric <matrix> $M^{-1/2}KM^{-1/2}$. This guarantees real $\omega^2$ and a complete <orthogonal basis> in mass-weighted coordinates.
For distinct <eigenvalues>, the fact that $K$ and $M$ are real <symmetric matrices> implies
$$
(\omega_i^2-\omega_j^2)\mathbf u_j^{\mathsf T}M\mathbf u_i=0.
$$
Thus distinct modes are <orthogonal> in the <inner product> weighted by the <mass matrix>. Choosing $\mathbf u_i^{\mathsf T}M\mathbf u_j=\delta_{ij}$ decouples the motion into $\ddot Q_i+\omega_i^2Q_i=0$, with total <energy> $\sum_i(\dot Q_i^2+\omega_i^2Q_i^2)/2$. A <positive-definite matrix> $K$ gives positive <eigenvalues> and oscillatory stability. Zero modes can describe rigid translations or <rotations> without restoring <force>; a negative <eigenvalue> gives exponential instability rather than an oscillatory mode.
\b[<Symmetry>.] A <symmetry> preserving both quadratic <energies> maps modes into modes of the same <frequency>. Coordinates adapted to <symmetry> divide the <eigenvalue> problem into smaller independent blocks. For two identical subsystems, exchanging them gives symmetric and antisymmetric modes. The <degeneracy of normal modes> means that independent modes share one <frequency>. Higher-dimensional <group representations> of a <symmetry> can enforce such degeneracies; <symmetry> does not require every pair of modes to be degenerate, and accidental degeneracies can also occur. Within a degenerate <eigenspace>, <orthogonal> linear combinations are equally valid <mode shapes>.
For two equal masses $m$, each attached to a fixed support by a spring of stiffness $k$ and coupled to the other by a spring of stiffness $k_c$,
$$
K=\begin{pmatrix}k+k_c&-k_c\\-k_c&k+k_c\end{pmatrix},\qquad M=m\mathsf I.
$$
The symmetric mode $\mathbf u_+\propto(1,1)$ leaves the coupling spring unstretched and has $\omega_+^2=k/m$. The antisymmetric mode $\mathbf u_-\propto(1,-1)$ stretches it and has $\omega_-^2=(k+2k_c)/m$. Superposing the two gives <beats> when their <frequencies> are close.
\b[Continuous systems.] The same idea replaces matrices by <differential operators> and mode <vectors> by functions satisfying <boundary conditions>. For a taut string of length $L$, <tension> $\mathcal T$ and <mass> per unit length $\mu_\ell$, the <wave equation> and fixed endpoints give
$$
y_n(x,t)=A_n\sin(n\pi x/L)\cos(\omega_nt+\phi_n),
\qquad \omega_n=\frac{n\pi}{L}\sqrt{\frac{\mathcal T}{\mu_\ell}},\quad n\geq1.
$$
<Orthogonality> of the <sine> functions allows an arbitrary initial <displacement> to be decomposed into modes. Changing the endpoints changes the spectrum: <boundary conditions> are part of the physical mode problem.
A uniform chain of equal masses $m$ joined by springs $k_s$, with spacing $a$, has travelling <normal modes> $q_j\propto e^{i(kja-\omega t)}$ with
$$
\omega^2(k)=\frac{4k_s}{m}\sin^2(ka/2).
$$
Their quantization gives <phonons>; the long-wavelength limit is a <sound> wave. <Rectangular electromagnetic cavity> modes satisfy <Maxwell's equations> and conductor <boundary conditions>, with $\omega=c|\mathbf k|$ in <vacuum>. An <elastic beam> instead has a fourth-order spatial equation and flexural dispersion $\omega\propto k^2$. These examples show how restoring <forces> and geometry determine the <frequency spectrum>.
\b[Real systems.] Small-amplitude motion is often approximately linear even when the full equations are nonlinear. Large amplitudes produce <frequency> shifts, mode coupling and transfer of <energy> between modes. Friction broadens resonances and makes free oscillations decay. A <damping> <matrix> proportional to $M$ or $K$ can retain the undamped <mode shapes>; general <damping> need not, and the appropriate eigenfrequencies and modes can be complex. External periodic forcing predominantly excites modes near <resonance>, with amplitude set by coupling to the forcing and dissipation. Imperfections split ideal degeneracies, while radiation or coupling to a surrounding fluid can remove <energy> from the chosen subsystem. \b[<Normal modes> remain useful when <linearization>, the <boundary conditions> and the treatment of <damping> accurately describe the motion being measured.]
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