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For an increment $h$, direct expansion and the chain rule giveFor example, $\Vert x+h{\Vert }^2-\Vert x{\Vert }^2=2x^{T}h+\Vert h{\Vert }^2$, so the quadratic remainder divided by $\Vert h\Vert$ tends to zero. The corresponding gradients are $\nabla g(x)=2x$ and $\nabla k(x)=2A^{T}Ax$; the derivative of $F$ is the constant matrix $A$.

Differentiating orthonormality gives ${\dot{e}}_i\cdot e_j=-e_i\cdot {\dot{e}}_j$. With $\omega =\frac{1}{2}{\sum }_i{e}_i\times {\dot{e}}_i$, the vector triple product yieldsThe body coordinates $x_i$ of a fixed material point are constant. Differentiating its position $R_G+{\sum }_i{x}_i{e}_i$ therefore givesThis identifies $\omega$ as the angular velocity of the rigid body.

Because $G$ is the center of mass, $\int \rho rdV=0$, so the translational contribution to the angular momentum about $G$ vanishes. The remaining vector triple product givesThis symmetric matrix is the inertia tensor about $G$. The angular-momentum balance about the center of mass is $dL/dt={\tau }_{\mathrm{ext},G}$ in an inertial frame. With no external forces, that torque is zero, so the inertial vector $L$ is constant. Its components in rotating body axes need not individually be constant.

For the tool, let the disc lie at $z=0$ and the rod extend to $z=l=\sqrt{3}a$. The total mass is $3M$ and its center is at $z_G=l/3$. Axial symmetry makes the inertia tensor diagonal with $I_x=I_y$.

The standard disc moments are $M{R}^2/2$ about its normal and $M{R}^2/4$ about a diameter. A thin rod has moment $m{l}^2/12$ about a transverse axis through its center. The parallel axis theorem adds $m{d}^2$ for a parallel shift of distance $d$. ThereforeandThe rod contributes zero about its own longitudinal axis. Thus $\boxed{J=2M{a}^2{I}_3}$, an isotropic inertia tensor even though the shape is not spherical. This gives a nonspherical rigid body with isotropic inertia. Consequently $L=2M{a}^2\omega$. Conservation of $L$ in torque-free motion proves the angular velocity is constant for every initial orientation and spin.

A real square matrix is orthogonal when $Q^{T}Q=I$, equivalently when its columns form an orthonormal basis. Taking determinants gives $(\det Q{)}^2=1$, soIn particular it is invertible with inverse $Q^T$ and preserves the Euclidean norm.

Comparing the $j$th columns of $A=QR$ gives $a_j=Q{r}_j$. ThusTaking nonnegative square roots proves $\boxed{\Vert a_j\Vert =\Vert r_j\Vert }$. This is simply norm preservation by an orthogonal matrix.

If $F:U\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^m$ is differentiable at $x$, and $G$ is defined on a neighborhood of $F(x)$ and differentiable there, the chain rule statesIn standard coordinates the same statement isThe evaluation point in the first factor is $F(x)$, and the matrix order follows the order in which the linear maps act on increments.

The total derivative of $F$ at $x$ is the linear map $D{F}_x:{\mathbb{R}}^n\to {\mathbb{R}}^m$ such thatThe Jacobian matrix is the $m\times n$ matrix $J_F(x)=(\partial F_i/\partial x_j)(x)$. When the total derivative exists, this is its matrix in standard bases: $D{F}_x(h)=J_F(x)h$. Existence of partial derivatives alone need not imply existence of the total derivative.

For an increment $h$, direct expansion and the chain rule giveFor example, $\Vert x+h{\Vert }^2-\Vert x{\Vert }^2=2x^{T}h+\Vert h{\Vert }^2$, so the quadratic remainder divided by $\Vert h\Vert$ tends to zero. The corresponding gradients are $\nabla g(x)=2x$ and $\nabla k(x)=2A^{T}Ax$; the derivative of $F$ is the constant matrix $A$.

For a single-photon transition, angular momentum addition requires $|J_i-J_f|\le L\le J_i+J_f$, $L\ge 1$, together with the electric/magnetic parity rules above. Thus a known final state and a measured multipole restrict both the spin difference and parity of the emitting level. For example, an E1 line reverses parity and allows $\Delta J=0,1$ except $0\to 0$; a mixed M1+E2 line preserves parity and must satisfy both angular triangles. An E2 line preserves parity and allows spin differences up to two.

Line energies identify connected levels, while calibrated intensities and coincidences test cascade flow and branching. If a proposed assignment permits a fast low-order transition to a lower state, an experimentally stringent absence of that line argues against the assignment. Nevertheless, an allowed transition can have a small or vanishing nuclear matrix element. Pure E2 radiation alone does not prove that M1 is forbidden by angular momentum. Detection sensitivity, efficiency and overlap must be considered, and in real nuclei internal conversion can remove photons; this question explicitly excludes that competing channel. Lifetimes or reduced-strength estimates make a missing-line argument substantially stronger than a bare selection-rule statement.

The lepton mixing matrix relates charged-current neutrino flavour fields to mass eigenfields: ${\nu }_{\alpha L}={\sum }_i{U}_{\alpha i}{\nu }_{iL}$. Nondegenerate masses then give relative propagation phases and flavour oscillations. This is the three-massive-neutrino extension of the minimal massless-neutrino Standard Model.

A unitary three-by-three matrix has nine real parameters: three rotation angles and six phases. For Dirac leptons, the independent field rephasings remove five phases, because one common phase acts trivially. Three angles and one physical Dirac phase remain. For Majorana neutrinos two additional phases cannot be removed, but they cancel from ordinary oscillation probabilities.

The phases printed in the lower two rows do not give a unitary matrix for general $\delta$. A consistent convention is $U=R_{23}{U}_{13}(\delta )R_{12}$, where the real rotations use the indicated mixing angles andThe lower-row phase factors in the resulting full matrix are $e^{+i\delta }$, opposite to the $e^{-i\delta }$ in $U_{e3}$. For the source's all-minus assignment, the scalar product of the electron row with the conjugated muon row is $-2i{s}_{23}{s}_{13}{c}_{13}\sin \delta$, which is nonzero at $\delta =\pi /2$ and nonzero mixing. This supplies an explicit counterexample. The electron-row moduli used below are unaffected by this correction.^[ref]

The dominant semileptonic decay is $t\to b{W}^{+}\to b{\ell }^{+}{\nu }_{\ell }$. Replacing $b$ by $s$ or $d$ weights the amplitude by $V_{ts}$ or $V_{td}$ rather than $V_{tb}$. Neglecting the down-type masses,with the source's rounded matrix entries. For the $W$, three leptonic widths and two open up-type quark rows with three colours give nine common width units. Thus, ignoring QCD and final-mass corrections,Summing all three flavours gives approximately one third; summing only electron and muon modes gives approximately $22.2\%$. These are per-top branching ratios, not the separate branching ratio of a specified two-top event topology.

An isolated high-transverse-momentum charged lepton, missing transverse momentum from the neutrino, and a tagged bottom jet form a much cleaner signature than an entirely hadronic final state amid QCD jets. Electrons and muons are the easiest charged-lepton flavours to identify cleanly. Tau decays add neutrinos and often hadrons, making identification and reconstruction more difficult. The figure contains the quark/lepton-level charged-current diagram as well as the rare-loop examples used below.

Use dimensionless physical helicity $h=\mathbf{S}\cdot \hat{\mathbf{p}}$, with spin eigenvalues $\pm 1/2$ in natural units. In the Dirac representation,The entries $1$ in the supplied transformation denote two-by-two identities. Since its matrix acts only on the upper/lower spinor blocks, it commutes with the repeated Pauli block. Therefore $u^{-1}=u$ andThe eigenvalues remain the physical spin projection along the direction of motion, $+1/2$ or $-1/2$. A convention omitting the factor one half instead labels them $\pm 1$; this changes no measurement probabilities. A change of gamma-matrix representation cannot change the spin observable.

The order-${ϵ}^2$ equation isUsing $\cos t\cos 2t=(\cos 3t+\cos t)/2$ and $\cos t\sin 2t=(\sin 3t+\sin t)/2$, its resonance coefficients aresecular term are removed by setting these to zero:The eigenvalues of this two-by-two system satisfyThey are real with opposite signs for $-1/12<a_2<5/12$. Generic initial amplitudes then contain an exponentially growing slow-time component. At either leading endpoint the matrix is nonzero nilpotent matrix, so suitable initial amplitudes give linear slow-time growth; higher-order detuning decides the precise corrected boundary. Outside this interval the leading eigenvalues are purely imaginary and the slow amplitudes remain oscillatory.

Substitution into $a=1+a_2{ϵ}^2+\cdots$ gives the requested leading instability interval“Can become unbounded” permits special initial amplitudes without growth. The calculation identifies the second instability tongue of a weak Mathieu oscillator and its slow amplitudes for the second Mathieu instability tongue; it is not an assertion that every solution grows throughout the closed interval.

The angular-momentum commutators giveFor example $[L_x+i{L}_y,L_z]=-i\hslash L_y+i(i\hslash L_x)=-\hslash L_{+}$. In the displayed basis $|1,1⟩,|1,0⟩,|1,-1⟩$, the only nonzero upper entries of $L_{+}$ connect adjacent magnetic quantum numbers. Subtracting their column and row $L_z$ eigenvalues multiplies each by $-\hslash$, proving the first matrix commutator; the lower matrix similarly gives the second. Direct multiplication giveswhose difference is $2\hslash L_z$. Also $L^2=L_z^2+(L_{+}{L}_{-}+L_{-}{L}_{+})/2=2{\hslash }^{2}I$, confirming $\ell =1$.

A rotation changes the quantization axis without changing the spectrum, so $L_{\mathbf{n}}$ has eigenvalues $+\hslash ,0,-\hslash$. Its dimensionless matrix isSolving $(L_{\mathbf{n}}-\hslash I)v=0$ and normalizing gives the spin-one maximum-projection eigenvectorFor instance the first equation is $(\cos \theta -1){\cos }^2(\theta /2)+{\sin }^2\theta /2=0$, and the squared components sum to one. An overall phase is immaterial.

At $\theta =\pi /3$, $v_{+}=(3/4,\sqrt{6}/4,1/4{)}^{\mathsf{T}}$. The Born rule therefore predicts, in the requested order,This treats the Stern-Gerlach experiment as the stipulated ideal measurement of orbital projection. In an actual atom, other magnetic moments and spin-orbit coupling can change which angular momentum the apparatus resolves.

Write $\gamma =ge/(2m)$, retaining the printed magnetic moment convention. The displayed two-dimensional matrices are Pauli matrices, and the spin Hamiltonian operator isFor an x-directed field let $\omega =\gamma B_x/2$. The Schrödinger equation gives $i{\dot{c}}_{+}=-\omega c_{-}$ and $i{\dot{c}}_{-}=-\omega c_{+}$. With $c_{+}(0)=1,c_{-}(0)=0$, these equations yieldEquivalently, $U(t)=e^{i\omega t{\sigma }_x}=I\cos \omega t+i{\sigma }_x\sin \omega t$. If $e$ denotes a positive elementary charge magnitude, a free electron is conventionally assigned a minus sign in its magnetic moment; that alternative reverses the rotation sense. The expressions here consistently use the moment specified in the paper.

matrix multiplication gives ${\sigma }_i{\sigma }_j={\delta }_{ij}I+i{ϵ}_{ijk}{\sigma }_k$, hence $[{\sigma }_i,{\sigma }_j]=2i{ϵ}_{ijk}{\sigma }_k$. For physical neutron spin, use $\mathbf{S}=(\hslash /2)\mathbf{\sigma }$, so $[S_i,S_j]=i\hslash {ϵ}_{ijk}{S}_k$, $S^2=3{\hslash }^2/4$, and measurement along any axis gives $\pm \hslash /2$.

For the specified field, $H=(\alpha B\hslash /2){\sigma }_y$. The Time-dependent Schrödinger equation is $i\hslash \dot{\psi }=H\psi$ with $\psi (0)=(1,0{)}^T$. The uniform-field spin precession follows from ${\sigma }_y^2=I$:

The sign of the precession is contained in the given $\alpha$. The normalized $x$-axis eigenstates are $(1,1{)}^T/\sqrt{2}$ and $(1,-1{)}^T/\sqrt{2}$.

The measurement direction is $\mathbf{n}=(0,\sin \theta ,\cos \theta )$, as confirmed by the original diagram. Write $c=\cos (\theta /2)$ and $s=\sin (\theta /2)$. The spinors ${\chi }_{+}=(c,is{)}^T$ and ${\chi }_{-}=(is,c{)}^T$ have unit norm and inner product $ics-isc=0$. Applying $\mathbf{n}\cdot \mathbf{\sigma }$ gives eigenvalues $+1$ and $-1$. At $\theta =0$ they are the usual $z$ basis; at $\pi /2$ they are the $y$ basis up to a phase; at $\pi$ the $z$ up/down roles are exchanged, with a factor $i$; and at $2\pi$ both return with a minus sign. A spinor changes sign under a full rotation, while its isolated-state measurement probabilities are unchanged.

For spin measurement in the yz plane, projection givesFor $\theta =0$ these are ${\cos }^2\varphi ,{\sin }^2\varphi$; for $\pi /2$ both are $1/2$ at all times; for $\pi$ they are exchanged; and for $2\pi$ they are the same as at zero angle. The constant $y$ probabilities also follow from conservation of $S_y$.

Define $s_i=\sin [2\pi (t_i-T)/P+\pi ]$ and use the independent quoted Gaussian errors $\sigma =10\mathrm{m}{\mathrm{s}}^{-1}$, with period and exoplanet transit epoch fixed. The model is linear in $K,V_0$ despite its sinusoidal time dependence. Minimize ${\chi }^2={\sum }_i(V_i-K{s}_i-V_0{)}^2/{\sigma }^2$. With $N=5$ the normal equations areUsing $\sum s_i=-0.741472$ and $\sum s_i^2=2.824633$, the solution isThe residuals in observation order are $(3.67,11.01,2.74,-7.90,-9.51)\mathrm{m}{\mathrm{s}}^{-1}$. Therefore ${\chi }_{\min }^2=2.950$ for $5-2=3$ degrees of freedom, giving reduced ${\chi }^2=0.983$. The upper-tail probability is about $0.40$: the fit is consistent with the stated errors, although five points do not establish a unique physical explanation.

For known errors the covariance matrix is ${\sigma }^2(X^{T}X{)}^{-1}$. Equivalently, profiling out the fitted offset givesThis is the $\Delta {\chi }^2=1$ one-parameter uncertainty, not the curvature with $V_0$ incorrectly held fixed. Propagating it through the supplied mass relation, and treating its coefficient and $P$ as fixed,The symbol $\pm nP$ in the exoplanet transit epoch denotes periodic copies of the same exoplanet transit phase, not an uncertainty distribution to propagate. No independent error for the empirical mass coefficient was supplied.

Additional independent observations retain positive weight even with twice the error: each has one quarter of the previous per-observation weight. Their information adds to the existing information matrix, so the formal mass uncertainty decreases for informative additional orbital phases and cannot increase under this fixed-error model. It can remain unchanged in a degenerate phase configuration that supplies no further amplitude information after fitting the offset. The conclusion assumes the same correct model and independent errors, not a new fitted error rescaling or unmodelled systematic effects.

A density operator describes both a quantum state prepared in one pure state and a statistical preparation whose members occupy different pure states. If normalized vectors $|{\psi }_i⟩$ occur with probabilities $p_i$, defineThe resulting operator is Hermitian, positive semidefinite and has trace one. Positivity follows because $⟨v|\rho |v⟩={\sum }_i{p}_i|⟨{\psi }_i|v⟩{|}^2\ge 0$. Conversely, the spectral theorem gives such a representation of any finite-dimensional operator with these properties, using its eigenvectors and eigenvalues. The matrix representing $\rho$ depends on the chosen basis; the operator and its predictions do not.

One operator gives every measurement probability. For a projective measurement with orthogonal projections ${\Pi }_a$ and outcomes $a$,For example, the first formula follows by averaging $⟨{\psi }_i|{\Pi }_a|{\psi }_i⟩$ over the preparation. The trace expression is basis independent. A general measurement replaces ${\Pi }_a$ by positive measurement effects that sum to the identity, without changing the probability formula. Thus $\rho$ specifies the statistics accessible to measurements on that system.

The distinction between pure and mixed states is especially transparent in the eigenvalues. A pure state has $\rho =|\psi ⟩⟨\psi |$, ${\rho }^2=\rho$, and $\mathrm{Tr}{\rho }^2=1$. A mixed state has at least two positive eigenvalues and $\mathrm{Tr}{\rho }^2<1$. For a qubit,pure states lie on the Bloch sphere; mixed states lie inside the Bloch ball. The maximally mixed qubit $\rho =I/2$ can be prepared by choosing equally between the two $z$ spin eigenstates, or equally between the two $x$ spin eigenstates. These ensembles are different preparation procedures but give the same density operator. No measurement on the resulting qubit alone can identify which ensemble was used.

quantum coherence must be distinguished from merely having two populated levels. In the $z$ basis, the state $|+x⟩$ hasBoth give equal probabilities for the two $z$ outcomes. A measurement along $x$ gives the positive result with probability one for the first state and one half for the second. The off-diagonal entries encode the relative phase information responsible for this difference. Whether a matrix entry is off diagonal is basis dependent; purity is not.

For an isolated system, each component evolves with the same unitary time-evolution operator. Therefore $\rho (t)=U(t)\rho (0)U^{†}(t)$ andUnitary evolution preserves the eigenvalues and purity of $\rho$. A density operator commuting with a time-independent Hamiltonian operator is stationary. A thermal equilibrium example is the canonical ensemble ${\rho }_{\beta }=e^{-\beta H}/Z$, where $Z=\mathrm{Tr}(e^{-\beta H})$ and $\beta =1/(k_{\mathrm{B}}T)$. expectation values then reproduce quantum statistical mechanics without assigning the system a definite energy eigenstate on every run.

A mixed state can also arise without classical randomness in the joint preparation. For a bipartite quantum system, the reduced density operator ${\rho }_A={\mathrm{Tr}}_B{\rho }_{AB}$ gives every measurement probability accessible to subsystem $A$. For the Bell state $(|00⟩+|11⟩)/\sqrt{2}$, the joint state is pure, whereas ${\rho }_A=I/2$. Its local mixedness reflects entanglement with the inaccessible system; the partial trace discards correlations that matter for joint measurements. More generally, interaction with an environment can suppress off-diagonal entries in a selected basis, producing decoherence while the combined system still evolves unitarily.

Finally, state evolution conditional on a measurement outcome differs from isolated evolution. An ideal projective measurement yielding $a$ updates the state by the Lüders rule $\rho \mapsto {\Pi }_a\rho {\Pi }_a/P(a)$. Discarding the outcome gives ${\sum }_a{\Pi }_a\rho {\Pi }_a$, which generally removes coherence between different eigenspaces. The Von Neumann entropy $-\mathrm{Tr}(\rho \ln \rho )$ quantifies mixedness: it vanishes for pure states and reaches $\ln d$ for $I/d$. Multiplying by $k_{\mathrm{B}}$ gives thermodynamic entropy units. These uses make the density operator a common language for preparation uncertainty, measurement, thermal states and entangled subsystems, while retaining their distinct physical origins.

The specified energy requires $n_x+n_y=2$, so the relevant energy eigenspace has the orthonormal basisApplying the creation and annihilation operators givesThus its matrix in this ordered basis isWriting $q=\hslash \lambda$, the characteristic polynomial gives $\lambda ({\lambda }^2-4)=0$. The possible angular momentum eigenvalues are thereforeFor completeness, normalized simultaneous energy and angular momentum eigenstates areEach value occurs once in this energy eigenspace. Equivalently, circular ladder operators of a two-dimensional oscillator give $\hat{H}=\hslash \omega ({\hat{N}}_{+}+{\hat{N}}_{-}+1)$ and ${\hat{L}}_z=\hslash ({\hat{N}}_{+}-{\hat{N}}_{-})$. With $N_{+}+N_{-}=2$, the difference can only be $2,0,-2$.

$Z$ is the terminating load impedance, $Z_0$ the line's characteristic impedance, and $k$ the propagation wave number; $l$ is its length. In the lossless convention printed, an open circuit gives $Z_{\mathrm{in}}=-i{Z}_0\cot kl$, a short circuit gives $i{Z}_0\tan kl$, and a matched load $Z=Z_0$ gives $Z_{\mathrm{in}}=Z_0$ for every length. The divergent or zero limits at isolated electrical lengths are understood continuously.

Put $\theta =kl$, $c_{\theta }=\cos \theta$, $s_{\theta }=\sin \theta$. The voltage/current matrix of one section isFor a pair beginning with $Z_1$, write $M_1{M}_2=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)$. Its entries are $A=c_{\theta }^2-(Z_1/Z_2)s_{\theta }^2$, $D=c_{\theta }^2-(Z_2/Z_1)s_{\theta }^2$, $B=i(Z_1+Z_2)s_{\theta }{c}_{\theta }$, and $C=i(1/Z_1+1/Z_2)s_{\theta }{c}_{\theta }$. The alternating-impedance transmission line has unchanged input $X$, its Bloch impedance, obeying $X=(AX+B)/(CX+D)$. Hence, away from the removable dividing zeros,For real positive $Z_1,Z_2$, select the branch carrying energy into the line or decaying into it. A positive real part selects the pass-band root; infinitesimal passive loss resolves the stop-band branch. At $\theta =m\pi$ the cell matrix is the identity, so the exact fixed-point condition alone does not determine $X$; at odd quarter-wave lengths the appropriate limiting impedance can be zero or infinite.

In a transmission-line stop band, a lossless semi-infinite ladder totally reflects because its physical input is reactive. Equivalently its cell hasThus, for nonzero contrast, the stop-band interiors satisfyWith ${\theta }_c=\arcsin [2\sqrt{Z_1{Z}_2}/(Z_1+Z_2)]$, these are $m\pi +{\theta }_c<kl<(m+1)\pi -{\theta }_c$; band-edge reflection follows by a limit. If $k=\omega /u$, multiply the endpoints by $u/l$ for angular frequencies. Identical impedances have no stop band. A finite ladder can transmit by tunnelling, unlike the semi-infinite idealization.

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 isThe 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: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 oscillationsA 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 $M^{-1/2}K{M}^{-1/2}$. 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 impliesThus 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 ${\ddot{Q}}_i+{\omega }_i^2{Q}_i=0$, with total energy ${\sum }_i({\dot{Q}}_i^2+{\omega }_i^2{Q}_i^2)/2$. 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.

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 $k_c$,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.

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 giveOrthogonality 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 $q_j\propto e^{i(kja-\omega t)}$ withTheir 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.

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. Normal modes remain useful when linearization, the boundary conditions and the treatment of damping accurately describe the motion being measured.