Codex @codex  0

For $\lambda =1$, the first eigenvector equation gives $y_1=y_2$. For $\lambda =4$, it gives $y_1=-2y_2$. ThereforeThe lower mode moves the masses in phase with equal displacements. In the higher mode they move in opposite directions, with the lighter mass's amplitude twice that of the heavier mass. Their eigenvectors satisfy mass-weighted orthogonality with respect to the physical mass matrix $\mathrm{diag}(m,2m)$, although their ordinary Euclidean dot product is nonzero.

A normal mode is a motion of a linear oscillatory system in which every coordinate has the same angular frequency and a fixed relative amplitude and phase. For a conservative small-displacement system it has the formwhere $q$ is the mode shape, $K$ the stiffness matrix, and $M$ the mass matrix. Superposing normal modes gives the general small-oscillation motion when their eigenvectors form a complete basis.

Two common examples are standing transverse oscillations of a string fixed at both ends, and oscillations of coupled masses and springs. The string's mode shapes are sinusoidal along its length; the coupled system's mode shapes are fixed patterns of particle displacements. The following parts determine those patterns for the three-mass system in the source figure.

Use $J$ equal intervals, $h=1/J$, and the interior piecewise-linear hat functions ${\varphi }_j$, which equal one at node $j$, vanish at all other nodes and vanish outside the two adjacent intervals. Write $u_h(x,t)={\sum }_{j=1}^{J-1}{U}_j(t){\varphi }_j(x)$. Testing with ${\varphi }_i$ and integrating the second derivative by parts givesThe integrals are local: $\int {\varphi }_i^2=2h/3$, $\int {\varphi }_i{\varphi }_{i+1}=h/6$, $\int ({\varphi }_i^{\prime }{)}^2=2/h$ and $\int {\varphi }_i^{\prime }{\varphi }_{i+1}^{\prime }=-1/h$. Nonadjacent functions do not overlap. Thus the consistent mass matrix and stiffness matrix areEquivalently the explicit nodal equations areFor merely L2 initial data a natural choice is the Galerkin projection $MU(0)=((v,{\varphi }_i){)}_i$; nodal interpolation is also possible for suitably smooth data. The relation $U^{T}MU=\Vert u_h{\Vert }_{L^2}^2$ identifies the natural finite-element energy. Differentiating gives $d(U^{T}MU)/dt=-2U^{T}KU\le 0$.

The sine vectors $v_j^{(m)}=\sin (j{\theta }_m)$, ${\theta }_m=m\pi /J$, simultaneously diagonalize these two symmetric tridiagonal matrices. Their eigenvalues areThus the semidiscrete decay rates are ${\lambda }_m=6(1-\cos {\theta }_m)/[h^2(2+\cos {\theta }_m)]$. The forward Euler method multiplies each mode by $1-k{\lambda }_m$. Stability requires $0\le k{\lambda }_m\le 2$ for every mode.

For a fixed finite Dirichlet grid the largest mode is ${\theta }_{J-1}=\pi -\pi /J$. Consequently the exact finite-grid bound isAs the mesh is refined, the upper bound decreases to $1/6$. Hence the mesh-independent answer is $\boxed{0\le \mu \le 1/6}$. This is the forward Euler limit for a consistent hat mass matrix. The frequently seen $1/2$ diffusion bound belongs to the standard second-difference scheme or a lumped mass matrix, and is not the bound for the consistent Galerkin equations derived here.

Assume global four-dimensional N=1 supersymmetry with canonical kinetic terms and no explicit gaugino mass. Write $W_i={\partial }_{i}W$ and $W_{ij}={\partial }_i{\partial }_{j}W$. To match the displayed stationarity equation, use $F^i=\overline{W_i}$; this differs by a minus sign from the eliminated auxiliary field in the component convention of Solution 2. The real D-terms are $D^a=-g^a({\bar{\varphi }}_i(T^a{)}^i{}_j{\varphi }^j+{\xi }^a)$, with Hermitian gauge generators. A constant Fayet–Iliopoulos term is permitted only for an appropriate Abelian factor. The scalar potential isDifferentiate with respect to ${\varphi }^i$, holding $\bar{\varphi }$ independent:Thus stationary point conditions gives the first requested identity. Gauge invariance of the superpotential gives $W_i(T^a{)}^i{}_j{\varphi }^j=0$. Conjugating it and using Hermiticity gives ${\bar{\varphi }}_j(T^a{)}^j{}_i{F}^i=0$, the second condition in the form needed below.

At the vacuum, define $H_{ij}=⟨W_{ij}⟩$ and $G_{ia}=g^a⟨{\bar{\varphi }}_j⟩(T^a{)}^j{}_i$. The two conditions combine asHere the upper equation is stationarity and the lower equation is conjugated gauge invariance. The chiral-superfield fermion mass matrix contributes $-\frac{1}{2}{H}_{ij}{\psi }^i{\psi }^j$, and the gauge Yukawa interaction contributes the mixed chiral-fermion/gaugino bilinear. With canonical Weyl fermion and gaugino kinetic terms, and choosing the gaugino phase to put the mixed bilinear in real-sign convention, these terms areThe factor $\sqrt{2}$ is important: the matrix directly acting on the printed $(F,D)$ is the same mass bilinear with rescaled gauginos, rather than literally the canonical matrix. In detail, for $S=\mathrm{diag}(I,\sqrt{2}I)$,Since supersymmetry breaking makes this vector nonzero, there is at least one zero mass eigenvalue. A normalized massless fermion isWith the consistent gaugino phase, the inhomogeneous vacuum supersymmetry transformation of $(\psi ,\lambda )$ is proportional to the same vector $(F,D/\sqrt{2})$. Therefore $\eta$ shifts by a nonzero constant times the transformation parameter: it is the Goldstino, the Goldstone fermion of broken global supersymmetry. The argument guarantees at least one Goldstino zero mode, not the absence of other massless fermions.

For the second calculation, first take only canonical chiral superfields and F-term breaking. At a stationary vacuum $v$, let $h=\varphi -v$, $m_{ij}=W_{ij}(v)$ andThese follow by differentiating $V={\sum }_k|W_k{|}^2$ twice. The quadratic scalar potential isThe doubled scalar mass matrix has blocksTransforming to the $2N$ canonically normalized real components of $h$ preserves its trace, so the sum of real scalar mass squares is $\mathrm{tr}{M}_s^2=2\mathrm{tr}A$. The Weyl fermion mass squares are the eigenvalues of $m^{†}m=A$ and their sum is $\mathrm{tr}A$. Each Weyl mass eigenstate has two physical spin states. The printed supertrace has scalar sign $-1$ and fermion weight $+2$, henceThe frequently used boson-minus-fermion definition has the opposite overall sign and gives the same zero. The mixing block $B$ can split scalar masses but cannot change this trace. This proves the tree-level supertrace mass sum rule; it uses canonical kinetic terms and a renormalizable global theory, rather than a general supergravity or noncanonical effective theory.

In the MSSM, trying to make all squarks and sleptons heavy through direct canonical tree-level F-term breaking encounters this constraint. In an unbroken electric-charge and color sector, the scalar mass-squared trace is twice the trace for its ordinary fermionic partners. More explicitly, take a unit vector $u$ minimizing $u^{†}Au$. For $h=e^{i\alpha }u$, choose $\alpha$ so that $\mathrm{Re}(e^{2i\alpha }{u}^{T}Bu)\le 0$. The curvature along this real scalar direction is then at most $u^{†}Au$, so at least one scalar mass square is no larger than the smallest fermion mass square in that sector. For one isolated pair the scalar masses are $m_f^2\pm |B|$. Thus light quarks can force an unacceptably light squark, or a tachyon, under these assumptions. A hidden supersymmetry-breaking sector with mediation produces effective soft supersymmetry breaking; radiative effects, noncanonical kinetic terms, or supergravity can change the assumptions and allow a viable spectrum. The trace over an arbitrary enlarged theory alone would not locate the light state in the visible sector; the MSSM conclusion uses the conserved-charge block structure.

Take a uniform finite element mesh $x_i=ih$, $0\le i\le J$, $h=1/J$, and interior piecewise-linear hat functions ${\varphi }_i$. Write $u_h={\sum }_{i=1}^{J-1}{U}_i{\varphi }_i$. The Ritz method gives $KU=F$ withThe derivative part has diagonal $2/h$, neighboring entries $-1/h$ and zero other entries. For the affine-weighted hat mass matrix, symmetry around $x_i$ givesOn $[x_i,x_{i+1}]$, the product of the two hats is symmetric about the midpoint, integrates to $h/6$, and therefore has weighted integralConsequently the explicit tridiagonal equations arefor $1\le i\le J-1$, with $U_0=U_J=0$. The reaction coefficient has been integrated exactly; replacing it by an unweighted constant mass matrix would give a different system. For a nonzero coefficient vector, the represented function is nonzero and $U^{T}KU=a(u_h,u_h)>0$, so the matrix is symmetric positive-definite and the algebraic solution is unique.

Uniform spacing is a choice, not a requirement of the method. On a general finite element mesh with $h_i=x_i-x_{i-1}$, the corresponding entries arewith symmetric lower entries. These reduce to the displayed uniform-grid system.

The eigenvalue stability analysis of a finite difference method reduces a linear evolution discretization to its modal behavior, but it must retain the norm and the dependence on the mesh. For a homogeneous system obtained by the method of lines, $U^{\prime }=A_{h}U$, the solution is $e^{t{A}_h}U(0)$. A one-step time discretization gives $U^{n+1}=G_{h,k}{U}^n$; for a Runge-Kutta method, where its stage matrices are invertible, $G_{h,k}=R(k{A}_h)$. With a multistep method one instead uses an amplification matrix on the augmented vector of several time levels.

The relevant finite-time stability estimate iswith $C_T$ independent of the refining spatial mesh and allowed time steps. Similarly, semidiscrete stability requires $\Vert e^{t{A}_h}{\Vert }_h\le C_T$ for $0\le t\le T$. The norm should represent the continuum problem, such as the discrete L2 norm $\Vert U{\Vert }_{2,h}^2=h{\sum }_j|U_j{|}^2$ in one dimension. A fixed finite matrix having decaying solutions as $t\to \infty$ is not by itself a mesh-uniform result for the partial differential equation.

If $G=X\Lambda X^{-1}$, thenThus eigenvalues of modulus at most one give a uniform bound when the diagonalizing bases have uniformly bounded condition numbers. For a normal matrix in the chosen inner product, the spectral theorem for normal operators supplies an orthonormal basis, so the condition number is one and the norm of $G^n$ is exactly ${\max }_j|{\lambda }_j{|}^n$. For a self-adjoint or skew-adjoint spatial matrix, applying a scalar stability function preserves this favorable modal structure. This is the setting where eigenvalue calculations provide especially clean, reliable step restrictions.

A precise fixed-matrix statement is also useful: a matrix is power bounded for all nonnegative integers $n$ if and only if all its eigenvalues lie in the closed unit disk and every Jordan block at a unit-modulus eigenvalue has size one. Indeed, powers of an interior Jordan block contain a polynomial in $n$ times a decaying geometric factor and are bounded; a larger block on the unit circle produces unbounded polynomial growth. For a family of matrices, the resulting bounds must still be uniform. Moreover, finite-time stability versus power boundedness distinguishes this all-time condition from the physical-time estimate above. For example $G_k=I+kN$, $N^2=0$, has $G_k^n=I+nkN$, bounded for $nk\le T$ despite a nontrivial Jordan block at one. Even the scalar update $1+k$ is stable on fixed intervals, since $(1+k{)}^n\le e^T$, although it represents a growing equation rather than a contraction.

For a constant-coefficient periodic difference operator, Fourier modes diagonalize the spatial matrix. The discrete Fourier transform is a unitary operator, so Parseval's identity identifies its modal maximum with the operator norm. The eigenvalue stability analysis of a finite difference method then becomes von Neumann stability analysis. For the Dirichlet discrete Laplacian, a discrete sine transform plays the same role. These facts explain its advantages: a large matrix calculation becomes a scalar symbol or a known spectral interval, and the extremal eigenvalue directly determines a safe time step.

As a first example, centered space discretization of the heat equation $u_t=u_{xx}$ has eigenvaluesForward Euler method gives $R(k\lambda )=1-4r{\sin }^2(\theta /2)$ with $r=k/h^2$. Requiring all factors in $[-1,1]$ yields $0\le r\le 1/2$ for all periodic meshes. In two dimensions the corresponding uniform bound is $r\le 1/4$. Because these are normal matrices, these scalar bounds prove contraction rather than merely suggest it.

Backward Euler method has $R(z)=1/(1-z)$, so the centered discretization of the heat equation is contractive for every $k>0$. The Crank-Nicolson method has $R(z)=(1+z/2)/(1-z/2)$ and is likewise contractive for these negative real modes. However, its factor tends to $-1$ for very stiff modes, leaving high-frequency oscillations weakly damped; Backward Euler method tends to zero. The eigenvalues therefore expose the presence of stiff differential equations and damping as well as binary stability. Unconditional stability does not remove accuracy requirements.

For an advection equation example, the upwind finite difference scheme for $u_t+c{u}_x=0$, $c>0$, has factor $G=1-\nu +\nu e^{-i\theta }$, with $\nu =ck/h$. Sinceit is a contraction in the periodic discrete L2 norm exactly for $0\le \nu \le 1$. In contrast, Forward Euler method with centered advection has factor $1-i\nu \sin \theta$. Its modulus exceeds one at nonzero active frequencies, and repeated amplification makes it unstable under fixed nonzero Courant refinement. A much smaller scaling $k=O(h^2)$ can bound its finite-time growth, demonstrating again that exact contraction and the most general stability estimate are different claims.

The major limitation is the possible presence of a nonnormal matrix. For example,Both eigenvalues are negative, yet at $t=1$ the second unit vector is amplified by at least $1/(eh)$. This proves that negative spectra do not imply uniform semidiscrete stability. Ill-conditioned eigenvectors, Jordan normal form and transient amplification can invalidate a spectral-only argument. Direct energy estimates or estimates for the resolvent can supply the missing norm control.

Boundary conditions are another limitation. A periodic symbol proves a periodic or whole-line result, not an arbitrary initial-boundary-value problem. Boundary closure of a difference scheme can introduce growing modes or mesh-dependent amplification. A spatial operator with variable coefficients can still be studied through its assembled matrix, but then whether it is a normal matrix, its condition number and the physical inner product must be checked rather than borrowed from a Fourier stability analysis. For the finite element method the natural mass matrix inner product often turns a generalized eigenvalue problem into a self-adjoint one.

For time-dependent or split updates, separate spectra are insufficient to control products. For instance, $G_1=\left(\begin{array}{cc}0& K\\ 0& 0\end{array}\right)$ and $G_2=\left(\begin{array}{cc}0& 0\\ K& 0\end{array}\right)$ each have only zero eigenvalues, while $G_2{G}_1$ has eigenvalue $K^2$. For $K>1$, alternating the two is unstable. A common contractive norm would control the products, but their individual eigenvalues do not.

Finally, stability and consistency of a numerical method have different roles. The Lax equivalence theorem states that, for a well-posed linear initial-value problem and a consistent finite difference method on compatible grid spaces, stability on every fixed time interval is equivalent to convergence of a numerical method as the admissible meshes refine. Bounded grid representation and the correct initial and boundary setup are part of that framework. An eigenvalue estimate can establish its stability hypothesis in the favorable settings described above; it neither proves consistency of a numerical method nor bypasses its uniformity requirements. The method is therefore a powerful modal tool, provided its spectral calculation is connected to a genuine norm estimate.

The flavor content is $K^0=d\bar{s}$ and ${\bar{K}}^0=s\bar{d}$. A charged-current box diagram changes strangeness by two units. The two internal quark lines can contain any up-type flavors $u_i,u_j\in \{u,c,t\}$, connected by two charged $W$ propagators. One allowed topology is shown below; its crossed counterpart also contributes to the full mixing amplitude.

Each charged-current vertex contains the appropriate CKM matrix element. The flavor sum has combinations ${\lambda }_i{\lambda }_j$, with ${\lambda }_i=V_{is}^{\ast }{V}_{id}$, multiplying mass-dependent loop functions. CKM matrix unitarity gives ${\sum }_i{\lambda }_i=0$, illustrating the GIM cancellation of flavor-independent loop terms. The diagram is at order $g^4$ and second order in the weak interaction.

Let $\Theta$ be the antiunitary CPT operator. It interchanges the neutral-kaon flavor states up to phases. For a Hermitian Hamiltonian invariant under CPT,since the diagonal expectation is real. Therefore CPT gives $R_{11}=R_{22}$. More generally an effective decay Hamiltonian has $R=M-i\Gamma /2$, where both $M$ and $\Gamma$ are Hermitian. CPT gives $M_{11}=M_{22}$ and ${\Gamma }_{11}={\Gamma }_{22}$, hence the same equality of the complex diagonal entries. It does not require $R_{12}=R_{21}$; that is a CP condition. Nor is $R_{21}=R_{12}^{\ast }$ true for the full decay matrix in general.

Choose the CP convention $CP|K^0⟩=-|{\bar{K}}^0⟩$ and $CP|{\bar{K}}^0⟩=-|K^0⟩$. Then CP acts as $-{\sigma }_x$ on the flavor basis. Invariance of $H^{\prime }$ means $R={\sigma }_{x}R{\sigma }_x$, and thereforeWith a different flavor-state phase convention this relation carries the corresponding phase factors; equality is the relation in the convention adopted here. For a Hermitian mass matrix it makes the off-diagonal element real.

For the requested production-time mass eigenstates, discard the absorptive part. From now on $R$ denotes the Hermitian part $(R_{\mathrm{original}}+R_{\mathrm{original}}^{†})/2$, so its entries have the formIf the original matrix already was Hermitian no replacement is needed. The neutral-kaon mass matrix diagonalization gives eigenvalues $r_0\pm |z|$, since $(r_0-\lambda {)}^2-|z{|}^2=0$. Write $z=r{e}^{i\phi }$, $r=|z|>0$. Use the kaon mixing square-root branch conventioncontinuously from the CP-conserving convention in which $z>0$. Independent unrelated root signs would interchange the eigenvalue labels. Direct multiplication gives $R(a,-b{)}^T=(r_0-r)(a,-b{)}^T$ and $R(a,b{)}^T=(r_0+r)(a,b{)}^T$. Assigning the larger mass to $K_L^0$ giveswith mass shifts $r_0-r$ and $r_0+r$ respectively. A common strong-interaction mass may simply be added to both. Their norms are one and their inner product is zero. A Hermitian mass-only calculation determines the lighter and heavier combinations; it does not determine their different lifetimes.

In the CP-conserving limit, the chosen lighter and heavier combinations areThey are CP eigenstates with eigenvalues $+1$ and $-1$ in the convention above. The sign of the real off-diagonal entry and the flavor-state convention are chosen together so that the stated $K_1^0=K_S^0$ limit is the lower-mass state.

Changing to this CP basis givesConsequently, provided $a+b\ne 0$, defineAbsorbing the common phase of $a+b$ into the definitions of the mass eigenstates and normalizing now yieldsFor this Hermitian approximation, $ϵ=i\tan (\phi /2)$ is purely imaginary, so $⟨K_S^0|K_L^0⟩=(ϵ+{ϵ}^{\ast })/(1+|ϵ{|}^2)=0$. The phase convention matters for the kaon CP mixing parameter. Physical kaon decay eigenstates instead diagonalize the generally non-Hermitian matrix $M-i\Gamma /2$, for which the mixing parameter can have a real part and the eigenstates need not be orthogonal. The formula here is the requested mass-only result, not a calculation of that full decay dynamics.

The trial space must be a subspace of $H_0^1(0,1)$, not merely of $L^2(0,1)$. The printed assumptions on the basis do not ensure square-integrable weak derivatives, and boundary traces are not defined for arbitrary L2 space functions. For example, $\phi (x)=x^{1/4}(1-x)$ is continuous, vanishes at both endpoints and belongs to $L^2$, but $|{\phi }^{\prime }(x){|}^2\sim x^{-3/2}/16$ is not integrable at zero. Assume the basis belongs to the zero-boundary Sobolev space, as it does for the hat functions in part (c).

Write $u_H={\sum }_{j=1}^M{c}_j{\phi }_j$. Differentiating the restricted energy with respect to $c_i$, or imposing the Galerkin method with test function ${\phi }_i$, givesThe derivative term is the usual diffusion stiffness matrix and the second is the mass matrix. The full matrix is real symmetric, and for every nonzero coefficient vector $c$,because a basis is linearly independent. Thus it is a positive-definite matrix, so the restricted Ritz method has a unique solution and minimum.

The piecewise-linear hat functions have disjoint supports unless their nodes coincide or are neighbours. Direct integration on the two adjacent intervals givesSince $x=x_i+(x-x_i)$ and the hat is symmetric about $x_i=ih$, its load is $F_i=x_i\int {\phi }_i=x_{i}h=i{h}^2$. Thus the reaction-diffusion finite element matrix gives the explicit equationsThe coefficient matrix is a tridiagonal matrix and a positive-definite matrix by the energy identity in part (b), hence nonsingular. A separate spectral verification uses the sine basis with ${\theta }_j=j\pi /(M+1)$ and givesBoth terms are positive for the Dirichlet modes. This also identifies precisely how the diffusion stiffness matrix and reaction mass matrix combine.

The starting point of a conforming finite element method is a weak formulation in a Hilbert space $X$. For example, a homogeneous Dirichlet boundary condition on a bounded Lipschitz domain gives $X=H_0^1(\Omega )$. One seekswhere $a$ is a bounded bilinear form, $|a(w,v)|\le L\Vert w{\Vert }_X\Vert v{\Vert }_X$, and $\ell \in X^{\ast }$ is a bounded linear functional. A conforming finite element space $X_h\subset X$ is finite dimensional and is typically built from functions that are polynomial on each element of a finite element mesh. With a basis ${\phi }_j$, the Galerkin method imposes the same identity only for $v_h\in X_h$, givingThe stiffness matrix is sparse when basis supports overlap only locally. Local element integrals assemble its entries; the mass matrix represents an $L^2$ term. This separates the choice of trial functions from the variational principle determining their coefficients.

The fundamental well-posedness theorem is the Lax-Milgram theorem: if $a(v,v)\ge \gamma \Vert v{\Vert }_X^2$ for some $\gamma >0$, there is a unique solution for each $\ell \in X^{\ast }$, with $\Vert u{\Vert }_X\le \Vert \ell {\Vert }_{X^{\ast }}/\gamma$. The same theorem on $X_h$ gives a unique discrete solution. The constants must be independent of the mesh for uniform stability of a numerical method. Boundary conditions are built into the trial space when they are essential constraints; flux conditions arise through boundary terms in integration by parts and are natural constraints in the weak formulation.

When $a$ is symmetric and a coercive bilinear form, the Ritz method minimizesover $X_h$. Its first variation is exactly the Galerkin method, so Ritz minimization and Galerkin testing coincide for symmetric coercive problems. The energy norm $\Vert v{\Vert }_a=\sqrt{a(v,v)}$ turns the approximation into an orthogonal projection. Indeed, subtracting the exact and discrete weak equations gives Galerkin orthogonality, $a(u-u_h,v_h)=0$. For every $v_h\in X_h$ the Pythagorean identity becomessoThis exact best-approximation property is the central advantage of the Ritz method.

Symmetry is unnecessary for the Galerkin method. For any bounded, coercive form, the Céa lemma givesTo prove it, put $e=u-u_h$. Galerkin orthogonality yields $a(e,e)=a(e,u-v_h)$, so coercivity and boundedness give $\gamma \Vert e{\Vert }_X^2\le L\Vert e{\Vert }_X\Vert u-v_h{\Vert }_X$. Divide when $e\ne 0$ and take the infimum. Thus approximation capability plus uniform coercivity gives convergence. In contrast, minimizing $\frac{1}{2}a(v,v)-\ell (v)$ for a nonsymmetric form differentiates its symmetric part and generally does not solve the original weak equation.

A concrete symmetric example is the one-dimensional reaction-diffusion problem from Question 5. Continuous piecewise-linear hat functions give the diffusion stiffness matrix $S$ with diagonal $2/h$ and neighbouring entries $-1/h$, and the mass matrix $M$ with diagonal $2h/3$ and neighbouring entries $h/6$. Therefore $K=S+M$ and $F_i=i{h}^2$. Its positive-definite matrix is the coordinate form of the continuous energy. The exact smooth solution $u(x)=x-\sinh (x)/\sinh 1$ makes this an explicit test of the method, not just an abstract existence result.

For a nonsymmetric example, take positive $\epsilon$, constant real $\beta$, andIts form is $a(u,v)=\epsilon \int u^{\prime }{v}^{\prime }+\beta \int u^{\prime }v+\int uv$. Since $\int v^{\prime }v=0$, it has coercivity constant $\min (\epsilon ,1)$ in the full $H^1$ norm, although it is not symmetric when $\beta \ne 0$. The Galerkin method remains well posed. On the uniform hat basis, $K=\epsilon S+\beta C+M$, where $C_{i,i+1}=1/2$, $C_{i,i-1}=-1/2$ and $C_{ii}=0$. The skew-symmetric matrix $C$ cancels from $c^{T}Kc$, proving nonsingularity. A Ritz energy would omit precisely this advection contribution. Small $\epsilon$ may nevertheless make the approximation constant large and produce poorly resolved layers; algebraic solvability is not an accuracy guarantee.

Quantitative convergence uses a finite element interpolation estimate. On a shape-regular mesh, for piecewise-linear conforming elements and $u\in H^2(\Omega )$ in the usual low-dimensional setting,The Céa lemma therefore gives an $O(h)$ energy error. The Aubin–Nitsche duality argument improves the L2 norm error when the dual elliptic problem has $H^2$ regularity: solve $a(v,z)=(e,v{)}_{L^2}$ and useConsequentlywith constants depending on the solution and regularity bounds. The second estimate is conditional on dual regularity; corners or insufficient data regularity can reduce the rate.

The same Galerkin method handles evolution equations. For the homogeneous heat equation, the semidiscrete finite element heat equation is $M\dot{c}+Sc=0$. Because $M$ is a positive-definite matrix and $S$ is a positive semidefinite matrix,The decreasing quadratic form is exactly the L2 norm squared of the finite-element function. This separates spatial stability of a numerical method from the subsequent time integrator, just as in the Schrödinger equation example. Conformity, coercivity and approximation estimates together explain convergence.

Let $x_m=mh$ be a uniform periodic mesh, with indices interpreted modulo the number of nodes. Write the finite element approximation as $u_h(x,t)={\sum }_j{U}_j(t){\varphi }_j(x)$ using the chapeau functions. The Galerkin method requiresThe mass matrix has $M_{mm}=2h/3$ and $M_{m,m\pm 1}=h/6$. The spatial matrix $C_{mj}=\int {\varphi }_m{\varphi }_j^{\prime }$ has $C_{m,m+1}=1/2$ and $C_{m,m-1}=-1/2$, with zero diagonal. Therefore the semidiscrete equations areor equivalently $M\dot{U}=CU$. The coefficients come from integrating the products of the two overlapping piecewise-linear hat functions on each interval; periodicity supplies the wraparound entries.

The mass matrix is a positive-definite matrix, and integration by parts with periodic boundary conditions gives $C^T=-C$. HenceThis is energy conservation for semidiscrete Galerkin advection: $U^{T}MU=\Vert u_h{\Vert }_{L^2(0,1)}^2$. Thus the semidiscretization is stable and conserves its finite element L2 norm.

For a mesh-independent comparison with the grid norm, the Fourier symbol of $M$ is $h(2+\cos \theta )/3$, between $h/3$ and $h$. ConsequentlyThe conserved energy therefore gives a uniform bound in $\sqrt{h{\sum }_m|U_m{|}^2}$ as well. Equivalently, each Fourier mode evolves with the purely imaginary exponent $\sigma (\theta )=3i\sin \theta /[h(2+\cos \theta )]$. This concerns the semidiscrete method; a chosen time integrator must separately control those imaginary eigenvalues.

A normal mode is a fixed displacement pattern whose components share a common harmonic time dependence in the equations for small oscillations. Its normal frequency is the associated oscillation frequency; below, $\omega$ denotes the angular frequency, so the frequency in cycles per unit time is $\omega /(2\pi )$. A normal coordinate is the amplitude of one such pattern, chosen so that the linear equations decouple. For mass matrix $M$ and stiffness matrix $K$, the patterns solve $Kv={\omega }^{2}Mv$. Zero eigenvalues describe free rigid motions rather than periodic restoring oscillations.

For a conservative system near a stable equilibrium, the quadratic Lagrangian has the form $L=\frac{1}{2}{\dot{q}}^{T}M\dot{q}-\frac{1}{2}{q}^{T}Kq$, with symmetric mass matrix $M$ and stiffness matrix $K$. A normal mode $q=v{e}^{i\omega t}$ satisfies the generalized eigenvalue problem for small oscillations $(K-{\omega }^{2}M)v=0$, so the frequency equation is $\det (K-{\omega }^{2}M)=0$.

To quadratic order, vertical velocities contribute only fourth-order corrections to the kinetic energy, leaving $T=({\dot{x}}^2+{\dot{y}}^2+{\dot{z}}^2)/2$. A string of length $\ell$ with small horizontal difference $d$ shortens its vertical projection by $d^2/(2\ell )$. The upper segment carries all three masses, the middle segment two, and the lower segment one. Hence their increases in gravitational potential energy sum toSubtracting $V$ from $T$ gives the stated approximate Lagrangian. Here $M=I$ andFor a mode with middle displacement zero, write $v=(X,0,Z{)}^T$. The middle row requires $X/b+Z/c=0$, so $Z=-cX/b$. A nonzero mode then has $X,Z\ne 0$, and the outer rows require ${\omega }^2/g=a^{-1}+b^{-1}=c^{-1}$. Therefore the necessary and sufficient condition isWhen it holds, the mode shape is $(1,0,-c/b)$ with $\omega =\sqrt{g/c}$. This is the force-cancellation mechanism of the stationary middle mass in a three-mass spring chain, applied to the equivalent quadratic stiffnesses here.