= Solution
\b[A <normal mode> is a motion in which all coordinates have a fixed relative pattern and oscillate at one frequency.] Near a stable equilibrium, let the quadratic kinetic and potential energies be $T=\dot{\mathbf q}^{\mathsf T}\mathbf M\dot{\mathbf q}/2$ and $V=\mathbf q^{\mathsf T}\mathbf K\mathbf q/2$. With positive-definite <mass matrix> $\mathbf M$, the equations and harmonic ansatz give
$$
\mathbf M\ddot{\mathbf q}+\mathbf K\mathbf q=0,\qquad\boxed{\mathbf K\mathbf a_s=\omega_s^2\mathbf M\mathbf a_s.}
$$
The stiffness and mass matrices are symmetric. For distinct frequencies, their symmetry gives $(\omega_s^2-\omega_r^2)\mathbf a_r^{\mathsf T}\mathbf M\mathbf a_s=0$, so the mode vectors are orthogonal in the mass inner product. A basis can also be chosen in a degenerate eigenspace. Expanding $\mathbf q=\sum_s Q_s\mathbf a_s$ decouples the equations into $\ddot Q_s+\omega_s^2Q_s=0$; initial displacements and velocities determine the coefficients. Positive stiffness gives real frequencies, a zero <eigenvalue> can describe a free symmetry motion, and a negative one signals instability rather than an oscillatory mode.<Classical Dynamics by David Tong>[Section 2.6: small oscillations and generalized normal-mode eigenvalue equations]{ref}
For \b[two equal masses between fixed walls with three equal springs], the <stiffness matrix> is $k\begin{pmatrix}2&-1\\-1&2\end{pmatrix}$ and the <mass matrix> is $m\mathbf1$. The patterns $(1,1)$ and $(1,-1)$ have frequencies $\sqrt{k/m}$ and $\sqrt{3k/m}$. An initial displacement of one mass excites both modes and can produce <beats>.
For \b[a uniform taut string fixed at both ends], the <wave equation> $\partial_t^2y=c_s^2\partial_x^2y$, with $c_s=\sqrt{\mathcal T/\mu_\ell}$, has patterns and angular frequencies
$$
\boxed{y_n(x)\propto\sin(n\pi x/l),\qquad\omega_n=n\pi c_s/l,\quad n=1,2,\ldots.}
$$
The <boundary conditions> select the spectrum. <Fourier expansion> of arbitrary admissible initial data provides the mode amplitudes, illustrating the extension from a finite set of coordinates to a field.
For the <normal modes of inductively coupled LC circuits>, take two identical circuits with self-inductance $L$, capacitance $C$ and <mutual inductance> $\mathcal M$, where $|\mathcal M|<L$. The <capacitor> charges obey $L\ddot q_1+\mathcal M\ddot q_2+q_1/C=0$ and the exchanged equation. In-phase and opposite-phase charge patterns have
$$
\boxed{\omega_+=\frac1{\sqrt{C(L+\mathcal M)}},\qquad\omega_-=\frac1{\sqrt{C(L-\mathcal M)}}.}
$$
The <positive inductance matrix for two identical coils> is the physical counterpart of positive mass. Here <inductance> plays the role of the <mass matrix>. Mechanical, wave and electrical examples therefore share the same <eigenvalue> structure. Weak forcing excites modes according to its spatial overlap with each pattern; <damping> broadens the <resonances>, while strong nonlinearity can invalidate a fixed linear-mode decomposition.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2015/paper-2-normal-modes.png]
{title=Normal modes range from two-mass displacement patterns to the standing-wave harmonics of a fixed string}
{source=paper-2-normal-modes.py}
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