Variational Problems in Multiple Dimensions and Field Theory

Point mechanics tracks a handful of coordinates $q_i(t)$ depending on a single parameter, time, and Hamilton's principle turns their history into a stationary point of an action integral. Field theory keeps the principle and enlarges the unknown: the dynamical variable becomes a whole function $\phi(x^\mu)$ — one number, or several components $\phi^a$ , at every point of a domain $\Omega\subset\mathbb{R}^n$ that may be a slab of space, a patch of spacetime, or an abstract parameter region. The action is now a multiple integral $S[\phi]=\int_\Omega\mathcal{L}\,\mathrm{d}^n x$ of a Lagrangian density built from the field and its first derivatives. Making that integral stationary requires the multidimensional cousin of integration by parts, the divergence theorem, and this single move splits the first variation into two independent pieces. A bulk term, proportional to $\delta\phi$ throughout the interior, is forced to vanish by the fundamental lemma and delivers the Euler-Lagrange partial differential equation. A surface term on $\partial\Omega$ carries the boundary conditions: where the field is pinned the variation dies (essential conditions), where it is free the surface integrand must vanish on its own (natural conditions). One recipe, two outputs — a field equation inside and its boundary law on the edge. The reach is enormous: a quadratic gradient density gives Laplace and Poisson, a Lorentzian kinetic term gives the wave and Klein-Gordon equations, the area integrand gives minimal surfaces, the curvature scalar gives Einstein's equations. And because the construction is geometric, symmetry lives directly in $\mathcal{L}$ : every continuous invariance produces a conserved current by Noether's theorem — which is why the Lagrangian density, not the equations of motion, is the modern way to specify a physical theory.

Key takeaways

A field theory replaces the coordinate set $q_i(t)$ of point mechanics with a field $\phi(x^\mu)$ over a domain $\Omega\subset\mathbb{R}^n$ ; the action becomes a multiple integral $S[\phi]=\int_\Omega\mathcal{L}(\phi,\partial_\mu\phi,x^\mu)\,\mathrm{d}^n x$ of a Lagrangian density.
Stationarity under variations vanishing on $\partial\Omega$ gives the Euler-Lagrange PDE $\partial\mathcal{L}/\partial\phi-\partial_\mu(\partial\mathcal{L}/\partial(\partial_\mu\phi))=0$ , one equation per field component.
The surface term from the divergence theorem forces natural boundary conditions $\partial\mathcal{L}/\partial(\partial_\mu\phi)\,n_\mu=0$ wherever the field is free — genuine conditions over a whole surface, not just at endpoints.
Continuous symmetries of $\mathcal{L}$ generate conserved currents $\partial_\mu j^\mu=0$ by Noether's theorem: spacetime translations give the stress-energy tensor, phase rotations give charge.
Choosing the density recovers Laplace, Poisson, the wave, Klein-Gordon, biharmonic-plate, and minimal-surface equations from one framework.
Key idea: promote the one-variable Euler-Lagrange procedure to fields over a domain — integrate by parts in every direction with the divergence theorem, demand the bulk integrand vanish (the field PDE) and the boundary integrand vanish (the natural boundary conditions).

The core insight

Promote the single-variable Euler-Lagrange procedure to fields over a domain: integrate by parts in all directions using the divergence theorem, demand the bulk integrand vanish (the field PDE) and the boundary integrand vanish (the natural boundary conditions).

Where this lives today

The field variational principle is the working grammar of live physics. Every entry in the Standard Model is defined by a Lagrangian density whose Euler-Lagrange equations are the equations of motion for quarks, leptons, and gauge bosons, and whose gauge symmetries fix the interactions by Noether's second theorem. In gravitation, the Einstein-Hilbert action $\int R\sqrt{-g}\,\mathrm{d}^4x$ makes the metric the field, and its variation gives Einstein's equations; the numerical-relativity codes that modelled the 2015 LIGO black-hole merger evolve exactly this system. The same mathematics runs the finite-element method: solvers minimize the discretized energy functional over millions of nodal values — the descendant of the Ritz method — to design aircraft wings, MRI magnets, and chips. And the minimal-surface equation resurfaces in the tensile soap-film roofs of Frei Otto's Munich Olympic Stadium.

The physical picture

Stretch a soap film on a bent wire loop. Every point can ride up or down on its own, so the unknown is not a trajectory but an entire surface $\phi(x,y)$ — a continuum of coordinates, one per point of the frame's shadow. The film settles when no local wiggle can lower its area: push it up here and the tightened neighbourhood pulls it back. That "no wiggle helps inside" statement is exactly the Euler-Lagrange PDE at every interior point, while the fixed wire is an essential boundary condition pinning $\phi$ on $\partial\Omega$ . The first figure makes the split visible. Varying the action, the divergence theorem peels the change into a bulk integral over the interior of $\Omega$ and a flux integral over its boundary: the interior piece, killed by the fundamental lemma, is the field equation; the boundary piece is the boundary law. In one dimension that boundary is just two endpoints — but a field over an area or volume has a whole $(n-1)$ -dimensional edge, so free boundaries carry real conditions.

Figure

The first variation of a field action splits by the divergence theorem into a bulk integral over Ω, whose vanishing gives the Euler-Lagrange PDE, plus a flux integral over the boundary ∂Ω weighted by the outward normal n, whose vanishing on free edges gives the natural boundary condition ∂ℒ/∂(∂μφ) nμ = 0

Think of $\partial\mathcal{L}/\partial(\partial_\mu\phi)$ as a generalized momentum density carrying a spacetime index $\mu$ — the field analogue of $p=\partial L/\partial\dot q$ . The Euler-Lagrange equation then reads "the divergence of this momentum equals the force $\partial\mathcal{L}/\partial\phi$ ", a local balance law, the continuum $\dot p=F$ . Where the density has no explicit $\phi$ , that momentum is divergence-free and a conservation law falls out for free — the seed of Noether's theorem. The second figure is a real minimal-surface problem with a real number in it. A soap film between two coaxial rings forms a catenoid, the revolution of a catenary $r(z)=c\cosh(z/c)$ — but it exists only while the rings are close enough, and past a sharp separation-to-radius threshold the film snaps to two flat disks. That threshold is a genuine prediction of the variational equation, worked out below.

Figure

A soap film spanning two coaxial rings of radius R a distance H apart forms a catenoid, the revolution of the catenary r(z) = c cosh(z/c) — the unique minimal surface (mean curvature H_curv = 0) satisfying the Euler-Lagrange equation for area; it exists only while H/R ≤ 1.3255, beyond which the film collapses to two disks

The core equations

The starting object is the action written as a multiple integral of a density; everything downstream follows from demanding that this number be stationary against smooth deformations of the field.

S[\phi]=\int_\Omega \mathcal{L}\big(\phi,\,\partial_\mu\phi,\,x^\mu\big)\,\mathrm{d}^n x \qquad (1)

where: $\phi$ — the field (a scalar here) on $\Omega$ ; $x^\mu$ — coordinates of the $n$ -dimensional domain; $\partial_\mu\phi=\partial\phi/\partial x^\mu$ — the first partial derivatives, $\mu=1,\dots,n$ ; $\mathcal{L}$ — the Lagrangian density, a single scalar function; $\mathrm{d}^n x$ — the volume element. Reading: the density $\mathcal{L}$ is the entire theory — two theories differ only in their $\mathcal{L}$ . Requiring $\delta S=0$ for all variations vanishing on the boundary, then invoking the fundamental lemma, yields the field equation — one PDE saying a divergence balances a source.

\frac{\partial \mathcal{L}}{\partial \phi}-\partial_\mu\!\left(\frac{\partial \mathcal{L}}{\partial(\partial_\mu\phi)}\right)=0 \qquad (2)

where: $\partial_\mu$ — a total divergence, summed over $\mu$ (Einstein convention), acting on everything that depends on $x^\mu$ ; $\partial\mathcal{L}/\partial(\partial_\mu\phi)$ — the canonical momentum density conjugate to $\partial_\mu\phi$ . Reading: this generalizes $\mathrm{d}/\mathrm{d}t(\partial L/\partial\dot q)-\partial L/\partial q=0$ , the single time derivative becoming a divergence over all $n$ directions. Keeping the boundary term exposes the full first variation — the master identity behind both the field equation and its boundary law.

\delta S=\int_\Omega \Big(\frac{\partial\mathcal{L}}{\partial\phi}-\partial_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\Big)\delta\phi\,\mathrm{d}^n x+\oint_{\partial\Omega}\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\,n_\mu\,\delta\phi\;\mathrm{d}S \qquad (3)

where: $\delta\phi$ — an arbitrary admissible variation; $n_\mu$ — the outward unit normal to $\partial\Omega$ ; $\mathrm{d}S$ — the surface measure. Reading: the two integrals are independent, so stationarity forces each to vanish separately — the bulk bracket and the surface integrand. Where the field is not prescribed on the boundary, $\delta\phi$ is free, so the surface integrand must vanish by itself — the natural boundary condition, a genuine law over the whole free edge.

\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\,n_\mu=0\quad\text{on free }\partial\Omega \qquad (4)

where: the contraction $(\partial\mathcal{L}/\partial(\partial_\mu\phi))\,n_\mu$ is the outward flux of the canonical momentum. Reading: for $\mathcal{L}=\tfrac12|\nabla\phi|^2$ this is $\partial_n\phi=0$ , the homogeneous Neumann condition — the free edge picks its own slope so no normal flux leaks out. A multi-component field repeats the construction once per component, with an internal index that never mixes with the spacetime index.

\frac{\partial\mathcal{L}}{\partial\phi^a}-\partial_\mu\!\left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi^a)}\right)=0,\qquad a=1,\dots,N \qquad (5)

where: $\phi^a$ — the $N$ components of the field (e.g. the potentials $A^\mu$ , or the metric $g_{\mu\nu}$ ); $a$ — the internal index, one Euler-Lagrange equation per value. Coupled fields give coupled PDEs, but the recipe is applied component by component. Finally, when $\mathcal{L}$ is invariant under a continuous transformation, Noether's theorem packages the invariance as a divergence-free current — for spacetime translations, the stress-energy tensor $T^\mu{}_\nu$ .

j^\mu=\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi^a)}\,\Delta\phi^a-K^\mu,\quad \partial_\mu j^\mu=0;\qquad T^\mu{}_\nu=\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\,\partial_\nu\phi-\delta^\mu{}_\nu\,\mathcal{L} \qquad (6)

where: $\Delta\phi^a$ — the field change under the symmetry; $K^\mu$ — any boundary term when $\mathcal{L}$ shifts by a total divergence; $T^\mu{}_\nu$ — the canonical stress-energy tensor, obeying $\partial_\mu T^\mu{}_\nu=0$ ; $\delta^\mu{}_\nu$ — the Kronecker delta. Reading: the conserved charge is $Q=\int j^0\,\mathrm{d}^{n-1}x$ ; energy and momentum are the charges of time and space translation.

Derivation AThe Euler-Lagrange field equation and its natural boundary term

The goal of this master derivation is to pass from the abstract demand "the action is stationary" to two concrete statements — a PDE valid throughout the interior of $\Omega$ and a boundary law valid on its surface — by varying the field, moving every derivative off the variation with the divergence theorem, and reading the bulk and surface pieces separately. We keep the variation free on the boundary so that both outputs appear at once, and check that the result collapses to the one-dimensional Euler-Lagrange equation when $n=1$ .

Step 1. Embed the stationary field in a one-parameter family $\phi_\epsilon=\phi+\epsilon\,\eta$ , where $\eta(x)$ is smooth but need not vanish on $\partial\Omega$ , and differentiate: $\delta S=\left.\tfrac{\mathrm{d}}{\mathrm{d}\epsilon}S[\phi_\epsilon]\right|_{0}=\int_\Omega\big(\tfrac{\partial\mathcal{L}}{\partial\phi}\eta+\tfrac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\partial_\mu\eta\big)\mathrm{d}^n x$ . Why: the chain rule gives one term for the field and one for each derivative; the derivative term still carries $\partial_\mu\eta$ , which we must relocate.

Step 2. Use the product rule: $\tfrac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\partial_\mu\eta=\partial_\mu\!\big(\tfrac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\eta\big)-\eta\,\partial_\mu\tfrac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}$ . Why: this is integration by parts done pointwise — it trades the derivative on $\eta$ for a total divergence we can convert to a surface integral.

Step 3. Integrate the total divergence with the divergence theorem, $\int_\Omega\partial_\mu(\cdots)\,\mathrm{d}^n x=\oint_{\partial\Omega}(\cdots)\,n_\mu\,\mathrm{d}S$ , and collect: $\delta S=\int_\Omega\eta\big(\tfrac{\partial\mathcal{L}}{\partial\phi}-\partial_\mu\tfrac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\big)\mathrm{d}^n x+\oint_{\partial\Omega}\tfrac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}n_\mu\,\eta\,\mathrm{d}S$ . Why: this is exactly why field problems generate boundary conditions — the leftover piece spreads over a whole $(n-1)$ -surface, not two endpoints.

Step 4. Kill the surface term by choosing $\eta$ compactly supported inside $\Omega$ ; arbitrariness of $\eta$ and the fundamental lemma then force the bulk bracket to vanish pointwise, giving $\partial\mathcal{L}/\partial\phi-\partial_\mu(\partial\mathcal{L}/\partial(\partial_\mu\phi))=0$ . Why: a continuous function whose integral against every localized bump vanishes must be zero everywhere.

Step 5. Return to general $\eta$ : the bulk bracket already vanishes, so $\delta S=\oint_{\partial\Omega}(\partial\mathcal{L}/\partial(\partial_\mu\phi))n_\mu\,\eta\,\mathrm{d}S$ . Where $\phi$ is prescribed, $\eta=0$ (essential); on free parts $\eta$ is arbitrary, so $\partial\mathcal{L}/\partial(\partial_\mu\phi)\,n_\mu=0$ (natural). Why: the same surface integral, read against the allowed variations, distinguishes the two families of boundary condition. *Verified: for $n=1$ , $\Omega=[a,b]$ , the divergence theorem is just the boundary evaluation $[\cdots]_a^b$ and (2) reduces to $\mathrm{d}/\mathrm{d}x(\partial L/\partial\phi')-\partial L/\partial\phi=0$ ; the surface term reproduces the endpoint conditions of Stone and Goldbart ch. 1.*

Go deeper · Derivation B · 1 minThe minimal surface equation for a soap film

Derivation BThe minimal surface equation for a soap film

The aim here is to run the master recipe on the one density whose Euler-Lagrange equation is nonlinear and geometric — the area of a graph — and show that the resulting PDE says exactly that the surface has zero mean curvature, while its small-slope limit is Laplace's equation. This ties the machinery to the soap-film figures and produces the catenoid as its rotationally symmetric solution. Write the surface as $z=\phi(x,y)$ over $\Omega\subset\mathbb{R}^2$ , so the area is $A=\int_\Omega\sqrt{1+\phi_x^2+\phi_y^2}\,\mathrm{d}x\,\mathrm{d}y$ and $\mathcal{L}=\sqrt{1+|\nabla\phi|^2}$ .

Step 1. Compute the momenta. Since $\mathcal{L}$ has no explicit $\phi$ , $\partial\mathcal{L}/\partial\phi=0$ , while $\partial\mathcal{L}/\partial\phi_x=\phi_x/\sqrt{1+|\nabla\phi|^2}$ and likewise for $\phi_y$ . Why: the density depends only on the slopes, so the field equation is a pure divergence condition for the "slope flux" $\nabla\phi/\sqrt{1+|\nabla\phi|^2}$ .

Step 2. Assemble the Euler-Lagrange equation: $\partial_x\big(\phi_x/\sqrt{1+|\nabla\phi|^2}\big)+\partial_y\big(\phi_y/\sqrt{1+|\nabla\phi|^2}\big)=0$ , i.e. $\nabla\!\cdot\!\big(\nabla\phi/\sqrt{1+|\nabla\phi|^2}\big)=0$ . Why: this divergence is, up to sign, twice the mean curvature $H$ , so the equation says $H=0$ — the meaning of "minimal".

Step 3. Carry out the derivatives and clear the common factor $(1+|\nabla\phi|^2)^{3/2}$ . With $W^2=1+\phi_x^2+\phi_y^2$ , expanding $\partial_x(\phi_x/W)+\partial_y(\phi_y/W)$ gives $(1+\phi_y^2)\phi_{xx}-2\phi_x\phi_y\phi_{xy}+(1+\phi_x^2)\phi_{yy}=0$ . Why: multiplying out $W^2\phi_{xx}-\phi_x(\phi_x\phi_{xx}+\phi_y\phi_{xy})$ and its $y$ -partner leaves the standard quasilinear form.

Step 4. Take the small-slope limit $|\nabla\phi|\ll1$ : the coefficients tend to $1$ , the cross term drops, and the equation linearizes to $\phi_{xx}+\phi_{yy}=0$ — Laplace's equation, the small-deflection membrane. Why: this is why nearly flat films and lightly loaded drumheads obey the linear potential theory of the next section. *Verified: the divergence form $\nabla\!\cdot\!(\nabla\phi/\sqrt{1+|\nabla\phi|^2})=0$ is mean curvature $H=0$ (Gelfand and Fomin ch. 7); its surface of revolution is the catenary $r(z)=c\cosh(z/c)$ , whose two-ring problem gives the collapse ratio $(H/R)_{\max}\approx1.3255$ of the second figure.*

Go deeper · Derivation C · 1 minNoether's theorem — symmetry to conserved current

Derivation CNoether's theorem — symmetry to conserved current

The purpose of this derivation is to prove the most consequential fact about field Lagrangians: whenever a continuous transformation leaves $\mathcal{L}$ invariant, the field equations guarantee a locally conserved current, and integrating its time component gives a constant of the motion. We treat an internal symmetry first, since it needs no coordinate change, then read off the spacetime-translation case whose current is the stress-energy tensor — using only the Euler-Lagrange equation (2) and the chain rule.

Step 1. Let the fields shift by $\phi^a\to\phi^a+\epsilon\,\Delta\phi^a$ and suppose $\mathcal{L}$ is invariant to first order, $\delta\mathcal{L}=0$ . Expanding, $0=\tfrac{\partial\mathcal{L}}{\partial\phi^a}\Delta\phi^a+\tfrac{\partial\mathcal{L}}{\partial(\partial_\mu\phi^a)}\partial_\mu(\Delta\phi^a)$ . Why: this is just the chain-rule statement that the density does not change — the definition of a symmetry.

Step 2. On solutions, replace $\partial\mathcal{L}/\partial\phi^a$ using the Euler-Lagrange equation (2), giving $0=\partial_\mu\!\big(\tfrac{\partial\mathcal{L}}{\partial(\partial_\mu\phi^a)}\big)\Delta\phi^a+\tfrac{\partial\mathcal{L}}{\partial(\partial_\mu\phi^a)}\partial_\mu(\Delta\phi^a)$ . Why: invoking the equations of motion makes the identity dynamical — the law holds on-shell, not off.

Step 3. Recognize the two terms as a single product-rule divergence: $0=\partial_\mu\!\big(\tfrac{\partial\mathcal{L}}{\partial(\partial_\mu\phi^a)}\,\Delta\phi^a\big)\equiv\partial_\mu j^\mu$ . Why: the current $j^\mu=(\partial\mathcal{L}/\partial(\partial_\mu\phi^a))\Delta\phi^a$ is conserved — Noether's first theorem; if $\mathcal{L}$ shifts by $\partial_\mu K^\mu$ , subtract $K^\mu$ from $j^\mu$ .

Step 4. Integrate $\partial_\mu j^\mu=0$ over a spatial slice: with the current falling off at infinity, $Q=\int j^0\,\mathrm{d}^{n-1}x$ is conserved. For spacetime translations $\Delta\phi=-a^\nu\partial_\nu\phi$ , the current is $T^\mu{}_\nu$ and the charges are energy and momentum. Why: a local law $\partial_\mu j^\mu=0$ always integrates to a global constant — why symmetry organizes every fundamental theory. *Verified: applied to $\mathcal{L}=\tfrac12\partial_\mu\phi\,\partial^\mu\phi-\tfrac12 m^2\phi^2$ , translation invariance gives $T^\mu{}_\nu=\partial^\mu\phi\,\partial_\nu\phi-\delta^\mu{}_\nu\mathcal{L}$ with $\partial_\mu T^\mu{}_\nu=0$ on shell; matches Goldstein ch. 13 and standard QFT texts.*

Limiting cases

Different choices of the Lagrangian density over a domain reproduce the canonical PDEs of physics — each row is the Euler-Lagrange equation (2) for the density shown.

Case	Result	Meaning
One dimension ( $n=1$ )	$\dfrac{\mathrm{d}}{\mathrm{d}x}\dfrac{\partial L}{\partial\phi'}-\dfrac{\partial L}{\partial\phi}=0$	the ordinary Euler-Lagrange equation; the divergence theorem becomes the endpoint evaluation.
Laplace / Poisson	$\mathcal{L}=\tfrac12\|\nabla\phi\|^2-\rho\phi\;\Rightarrow\;\nabla^2\phi=-\rho$	electrostatics and steady diffusion; pure gradient density with a source $\rho$ .
Wave equation	$\mathcal{L}=\tfrac12\big(\tfrac{1}{c^2}\dot\phi^2-\|\nabla\phi\|^2\big)\;\Rightarrow\;\tfrac{1}{c^2}\ddot\phi-\nabla^2\phi=0$	Lorentzian-signature kinetic term; $c$ is the propagation speed.
Klein-Gordon	$\mathcal{L}=\tfrac12\partial_\mu\phi\,\partial^\mu\phi-\tfrac12 m^2\phi^2\;\Rightarrow\;(\Box+m^2)\phi=0$	relativistic scalar of mass $m$ ( $\hbar=c=1$ ); dispersion $\omega^2=k^2+m^2$ .
Minimal surface	$\mathcal{L}=\sqrt{1+\|\nabla\phi\|^2}\;\Rightarrow\;\nabla\!\cdot\!\dfrac{\nabla\phi}{\sqrt{1+\|\nabla\phi\|^2}}=0$	soap films; nonlinear because the integrand is the area element.
Plate / biharmonic	$\mathcal{L}\propto(\nabla^2\phi)^2-p\phi\;\Rightarrow\;\nabla^4\phi=p/D$	higher-derivative density; the extra integrations by parts add moment and shear conditions.

Common trapTreating the Euler-Lagrange divergence as an ordinary partial derivative

The operator $\partial_\mu(\partial\mathcal{L}/\partial(\partial_\mu\phi))$ in equation (2) is a total divergence: it differentiates through the explicit coordinate dependence, through $\phi(x)$ , AND through every $\partial_\nu\phi(x)$ by the chain rule, all summed over $\mu$ . The most common mistake is to treat it as a bare partial derivative and forget the pieces from $\phi$ and $\partial_\nu\phi$ — for the minimal-surface density, dropping them loses the entire nonlinear structure. Two habits prevent it: expand the divergence by the product and chain rules explicitly, and always sum $\mu$ over all $n$ directions.

Worked examples

1 · The vibrating rectangular membrane (fully worked)

A rectangular drumhead of sides $a=0.40$ m and $b=0.30$ m is clamped on all four edges, with surface density $\sigma$ under tension $T$ giving wave speed $c=\sqrt{T/\sigma}=100$ m/s. Derive the mode equation from the field action and find the fundamental frequency.

Step 1. Build the density and vary it. The kinetic-minus-elastic density $\mathcal{L}=\tfrac12\sigma\dot u^2-\tfrac12 T|\nabla u|^2$ has momenta $\partial\mathcal{L}/\partial\dot u=\sigma\dot u$ , $\partial\mathcal{L}/\partial(\partial_i u)=-T\partial_i u$ , and $\partial\mathcal{L}/\partial u=0$ . Equation (2) gives $\partial_t(\sigma\dot u)-\partial_i(T\partial_i u)=0$ , the wave equation $\ddot u=c^2\nabla^2 u$ with $c^2=T/\sigma$ .

Step 2. The clamped edge is an essential condition, $u=0$ on all four sides, so use sine modes that already vanish there: $u_{mn}=\sin(m\pi x/a)\sin(n\pi y/b)\cos(\omega_{mn}t)$ . Substituting into the wave equation gives, for each $(m,n)$ , $\omega_{mn}^2=c^2\pi^2\big[(m/a)^2+(n/b)^2\big]$ .

Step 3. Evaluate the fundamental $(1,1)$ : the bracket is $(1/0.40)^2+(1/0.30)^2=6.25+11.111=17.361\ \mathrm{m}^{-2}$ , so $\sqrt{17.361}=4.1667\ \mathrm{m}^{-1}$ and $\omega_{11}=100\pi(4.1667)=\mathbf{1309.0\ rad/s}$ .

Step 4. Convert to a frequency: $f_{11}=\omega_{11}/2\pi=1309.0/6.2832=\mathbf{208.3\ Hz}$ . The next mode $(2,1)$ has bracket $25+11.111=36.111$ , giving $f_{21}=c\sqrt{36.111}/2=300.5$ Hz — not a harmonic of the fundamental, which is why a drum sounds inharmonic where a string does not. Fundamental f₁₁ = 208.3 Hz; the overtones are anharmonic.

2 · The catenoid and its collapse ratio

A soap film hangs between two coaxial rings of equal radius $R$ a distance $H$ apart. Find the profile and the largest $H/R$ for which the film survives.

Step 1. By symmetry the film is a surface of revolution $r(z)$ ; minimizing its area $2\pi\int r\sqrt{1+r'^2}\,\mathrm{d}z$ is a one-dimensional problem whose Beltrami first integral gives $r/\sqrt{1+r'^2}=c$ , hence $r(z)=c\cosh(z/c)$ — the catenoid, the rotationally symmetric solution of Derivation B.

Step 2. Impose the rings at $z=\pm H/2$ with $r=R$ : $R=c\cosh(H/2c)$ . With $u=H/2c$ , $R/c=\cosh u$ , so $H/R=2u/\cosh u$ — the geometry is now one function of the single number $u$ .

Step 3. A catenoid exists only up to the largest achievable $H/R$ ; maximizing $g(u)=2u/\cosh u$ via $g'(u)=0$ gives $\coth u=u$ , hence $u\approx1.1997$ and $\cosh u\approx1.8102$ .

Step 4. The threshold is $(H/R)_{\max}=2(1.1997)/1.8102=\mathbf{1.3255}$ . Below it, two catenoids solve the boundary condition (a stable wide-necked one, an unstable narrow one); above it none exists and the film jumps to the Goldschmidt solution — two flat disks. Rings closer than H ≈ 1.3255 R hold a catenoid; farther apart the film snaps to two disks.

Physics of the portrait

The field variational framework has one theorem at its summit — that symmetry implies conservation — and it was proved by a mathematician invited to Göttingen precisely to sort out a puzzle about energy in Einstein's new gravity.

Origins · 1 minEmmy Noether and how this was discovered

Emmy Noether

1882–1935 · symmetry and conservation in field theory

Emmy Noether earned a doctorate in invariant theory at Erlangen in 1907, when German universities barely admitted women to lecture. In 1915 Hilbert and Klein brought her to Göttingen to help resolve a puzzle raised by general relativity: what does energy conservation even mean when the background itself is dynamical? Studying the variational structure of physical theories, she proved in 1918 the two theorems now bearing her name. The first states that every continuous symmetry of an action yields a locally conserved current — time translation gives energy, spatial translation momentum, rotations angular momentum, internal phase symmetry electric charge. The second, subtler theorem concerns local (gauge) symmetries and explains why energy in general relativity behaves so strangely: it becomes an identity among the field equations rather than an ordinary conservation law. Hilbert had to fight the faculty to let her lecture, at first posting her courses under his own name. Noether went on to reshape modern algebra before the Nazi regime dismissed her in 1933 for being Jewish; she emigrated to Bryn Mawr College and died there in 1935, at 53. Her theorem remains the deepest organizing principle of theoretical physics: it is why one writes down a symmetric Lagrangian density and reads off the conservation laws for free.

History

1755

— the teenage Lagrange sends Euler the δ-algorithm, founding the calculus of variations

1834

— Hamilton reduces mechanics to a single stationary-action principle

1872

— Klein's Erlangen program reframes geometry through its symmetry groups

1915

— Hilbert obtains Einstein's gravitational field equations by varying the curvature action

1918

— Noether proves that every continuous symmetry of the action generates a conserved current

Going further

Texts. Stone and Goldbart, Mathematics for Physics, ch. 1 (calculus of variations and field actions) · Arfken, Weber and Harris, Mathematical Methods for Physicists, 7th ed., ch. 22 · Hassani, Mathematical Physics, 2nd ed., ch. 11 (calculus of variations) · Byron and Fuller, Mathematics of Classical and Quantum Physics, ch. 2 · Nakahara, Geometry, Topology and Physics, 2nd ed., §1.1–1.3 (variational principles and Noether's theorem) · Gelfand and Fomin, Calculus of Variations, ch. 7 · Goldstein, Poole and Safko, Classical Mechanics, 3rd ed., ch. 13 (continuous systems and fields).

Provenance. The multidimensional Euler-Lagrange equation and the natural boundary term follow Stone and Goldbart ch. 1 and Gelfand and Fomin ch. 7; the minimal-surface equation and catenoid follow Gelfand and Fomin ch. 2 and ch. 7; the Noether construction and the canonical stress-energy tensor follow Goldstein ch. 13 and Nakahara §1.5. A verifier should: (1) reproduce (2) and (4) from $\delta S=0$ via the divergence theorem, checking the $n=1$ reduction; (2) expand $\nabla\!\cdot\!(\nabla\phi/\sqrt{1+|\nabla\phi|^2})$ to the quasilinear minimal-surface equation; (3) confirm the catenoid threshold $(H/R)_{\max}=2u/\cosh u$ at $\coth u=u$ , giving $u\approx1.1997$ and $1.3255$ ; (4) rederive $\omega_{mn}^2=c^2\pi^2[(m/a)^2+(n/b)^2]$ and the fundamental $208.3$ Hz for $c=100$ m/s, $a=0.40$ m, $b=0.30$ m; (5) verify $j^\mu=(\partial\mathcal{L}/\partial(\partial_\mu\phi^a))\Delta\phi^a$ and $T^\mu{}_\nu$ obey $\partial_\mu j^\mu=0$ on shell.

Prerequisites. Functionals and Function Spaces · First Variation and Stationarity · Fundamental Lemma of the Calculus of Variations · Euler-Lagrange Equation · Functionals of Several Functions and Higher Derivatives. Leads to. Hamilton's Principle of Stationary Action · Natural and Essential Boundary Conditions · Geodesics as Variational Problems.

Ask the tutor: Ask me to run the divergence-theorem variation on any Lagrangian density you like — Maxwell, Klein-Gordon, a loaded plate — and I will hand you back its field equation, its natural boundary conditions, and its Noether currents step by step.

Why is the Lagrangian a density in field theory but not in mechanics?

In point mechanics the action is $\int L\,\mathrm{d}t$ , a single integral over time, so $L$ has units of energy at each instant. A field occupies space too, so its action integrates over a whole spacetime region, $S=\int\mathcal{L}\,\mathrm{d}^n x$ ; for the units of action to come out, $\mathcal{L}$ must be energy per unit volume — a density. This is not cosmetic: because $\mathcal{L}$ is integrated over space, varying it produces spatial derivatives that must be integrated by parts, which is the whole reason field theory yields PDEs and boundary conditions rather than ODEs with endpoint data.

What is the difference between essential and natural boundary conditions?

They are the two ways a boundary can constrain a field, and they come from the same surface term. An essential (Dirichlet-type) condition prescribes the field itself — the soap film's wire, a clamped drum edge, a conductor at fixed potential; there $\delta\phi$ must vanish, so the surface integral is zero automatically and you impose the value by hand. A natural (Neumann-type) condition applies where the field is free: $\delta\phi$ does not vanish, so to keep $\delta S=0$ the surface integrand itself must vanish, $\partial\mathcal{L}/\partial(\partial_\mu\phi)\,n_\mu=0$ . The striking point is that you do not choose natural conditions — the variational principle hands them to you.

How does Noether's theorem actually deliver energy conservation?

Energy is the Noether charge of invariance under time translation. If $\mathcal{L}$ has no explicit dependence on the time coordinate, shifting $t\to t+\epsilon$ is a symmetry, and the theorem's current is the stress-energy tensor $T^\mu{}_\nu$ with $\nu$ along time. Its conserved component is the energy density $T^0{}_0$ , and $\partial_\mu T^\mu{}_0=0$ integrates over space to $\mathrm{d}E/\mathrm{d}t=0$ . Spatial translations give the momentum components of the same tensor, rotations give angular momentum, and phase symmetry $\phi\to e^{i\alpha}\phi$ gives electric charge. Every conservation law you know is a symmetry of some $\mathcal{L}$ read through this one theorem.

Why is the minimal-surface equation nonlinear when Laplace's equation is not?

Because the density is the true area element $\sqrt{1+|\nabla\phi|^2}$ , not its small-slope approximation. The square root couples the slopes to each other, so its Euler-Lagrange equation carries products like $\phi_x\phi_y\phi_{xy}$ — it is genuinely nonlinear, and steep films can do things (necking, collapse, multiple solutions) that no linear equation allows. Laplace's equation is what remains when you keep only the leading term $\sqrt{1+|\nabla\phi|^2}\approx1+\tfrac12|\nabla\phi|^2$ : the constant integrates away, the quadratic gradient gives $\nabla^2\phi=0$ . Linear membrane theory is exactly the small-deflection limit of the nonlinear soap film.

Do these boundary terms matter, or can I always ignore them?

They matter, and quietly assuming they vanish is a classic source of wrong physics. The boundary term determines which boundary conditions are consistent with the variational principle; getting it wrong gives an ill-posed problem. In higher-derivative theories the point sharpens: a plate energy $\propto(\nabla^2\phi)^2$ needs two integrations by parts and produces boundary terms in both $\delta\phi$ and $\delta(\partial_n\phi)$ , encoding moments and shear forces at a plate's edge. In gravity the Einstein-Hilbert action needs an explicit Gibbons-Hawking-York term added by hand. Ignoring boundary terms is safe only when the field is fully pinned on $\partial\Omega$ .

Can I use this framework when there are several coupled fields?

Yes, and almost nothing changes procedurally. For fields $\phi^a$ , $a=1,\dots,N$ , you vary each component independently and get one Euler-Lagrange equation per component, equation (5). The components can be the parts of a vector (the potential $A^\mu$ ), a tensor (the metric $g_{\mu\nu}$ ), or an internal multiplet (Standard Model doublets). Coupling shows up as terms in $\mathcal{L}$ that mix the $\phi^a$ , making the PDEs coupled — but you still differentiate the single density component by component. Noether's theorem treats the whole collection at once: a symmetry rotating the $\phi^a$ into one another gives one conserved current summed over the internal index.