Covariant Euler-Lagrange Equations for Fields

The density is designed; now comes the crank. Given $\mathcal{L}$ , one mechanical procedure — vary the action, integrate by parts, set the coefficient of the arbitrary variation to zero — produces the field equations of any theory whatsoever. The continuum-mechanics chapter ran that procedure for a single field in one dimension and got the wave equation; the same machinery in covariant four-dimensional dress is the production line of fundamental physics, and this article operates it at full generality: many fields at once, complex fields, vector fields — one crank, every equation. The covariant packaging is where the payoff hides. Writing the Euler-Lagrange equation as $\partial_\mu\big(\partial\mathcal{L}/\partial(\partial_\mu\phi)\big) = \partial\mathcal{L}/\partial\phi$ does more than save ink: it makes manifest that the equation of motion transforms properly under Lorentz transformations whenever $\mathcal{L}$ is a scalar — the guarantee, purchased in the previous article, that the physics is observer-independent. It also generalizes without friction. A multiplet of fields yields one equation per component, because independent fields suffer independent variations. A complex field, treated as two real ones in efficient notation, yields the Klein-Gordon equation and — directly from the equations of motion — a conserved current, the classical ancestor of electric charge. And a four-vector field with the Maxwell density $-\tfrac14 F_{\mu\nu}F^{\mu\nu}$ yields, in three lines, the inhomogeneous Maxwell equations: the crown demonstration that the nineteenth century's crowning theory is one crank of the same handle. Two supporting skills come with the machinery: the discipline of treating $\phi$ and $\partial_\mu\phi$ as independent slots of $\mathcal{L}$ when differentiating (the step every beginner fumbles once), and the index bookkeeping of differentiating quadratic forms like $F_{\mu\nu}F^{\mu\nu}$ with respect to a derivative that appears in several contractions. Both are drilled here on worked examples with numbers, because the crank is only trustworthy in hands that have turned it.

Key takeaways

For each field component $\phi_a$ , stationarity of $S = \int\mathcal{L}\,\mathrm{d}^4x$ gives one covariant Euler-Lagrange equation: $\partial_\mu\!\left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_a)}\right) - \frac{\partial\mathcal{L}}{\partial\phi_a} = 0$ .
Multiple fields = multiple independent variations = one equation per component; internal indices ride along untouched by $\partial_\mu$ .
In the derivative rules, $\phi$ and $\partial_\mu\phi$ are independent slots of the function $\mathcal{L}$ — differentiate one while freezing the other.
Real scalar with $V(\phi)$ : the crank gives $\Box\phi + V'(\phi) = 0$ — Klein-Gordon for quadratic $V$ , nonlinear wave equations (sine-Gordon, $\phi^4$ ) beyond.
Complex scalar: vary $\phi$ and $\phi^*$ independently; two Klein-Gordon equations plus, directly from them, a conserved current $j^\mu = i(\phi^*\partial^\mu\phi - \phi\,\partial^\mu\phi^*)$ , $\partial_\mu j^\mu = 0$ — charge before quantization.
Vector field: the Maxwell density $-\tfrac14 F_{\mu\nu}F^{\mu\nu} - j^\mu A_\mu$ yields $\partial_\mu F^{\mu\nu} = j^\nu$ — Maxwell's equations as one turn of the same crank, with current conservation forced by the antisymmetry of $F$ .
The covariant form guarantees: scalar $\mathcal{L}$ ⟹ field equations that transform covariantly — relativity is inherited, not re-checked.
Key idea: one variational crank — differentiate $\mathcal{L}$ by each field and by its spacetime gradient, take the four-divergence of the latter, equate — produces the field equations of every classical theory, from the wave equation to Maxwell's.

The core insight

Stationary action for a local density $\mathcal{L}(\phi_a, \partial_\mu\phi_a)$ yields one covariant Euler-Lagrange equation per field component, $\partial_\mu\big(\partial\mathcal{L}/\partial(\partial_\mu\phi_a)\big) = \partial\mathcal{L}/\partial\phi_a$ — the single crank that generates Klein-Gordon, sine-Gordon, Maxwell, and every other classical field equation.

Where this lives today

The covariant crank is the daily instrument of theoretical physics — literally automated: computer-algebra packages (xAct, FieldsX, and the lattice-QFT toolchains) apply equation (2) to Lagrangians with dozens of fields, and gravitational-wave source modelling begins by cranking post-Newtonian actions too long to print. Numerical relativity and lattice field theory run the discrete crank: both communities vary the discretized action rather than discretizing the field equations, because the variational route preserves conservation laws exactly on the grid (symplectic/variational integrators — the same reason the standard map was such a faithful laboratory). The complex-scalar current of Derivation B is working technology in superfluidity and Bose-Einstein condensates, where $j^\mu$ is the measured particle current of the condensate wavefunction (Gross-Pitaevskii theory is the crank applied to a $|\phi|^4$ density); Josephson-junction arrays implement the sine-Gordon kink of the worked example as trapped flux quanta racing along transmission lines. And the "consistency forces conservation" lesson of the Maxwell derivation — the action refusing non-conserved sources — has become a general design principle: modern effective theories of everything from pions to phonons are built by writing all crank-compatible couplings, with the automatically enforced Ward-type identities doing real phenomenological work. Within the chapter: the crank consumed the density (previous article) and produced the equations; the functional-derivative article restates the whole operation as a single gradient in function space, $\delta S/\delta\phi = 0$ — the notation the Hamiltonian formulation and the quantum capstone both speak.

The physical picture

Picture the action as a landscape over an unimaginably large space — one axis per field value per point of spacetime, the function space of the discrete-to-fields article — and the physical history as a stationary point of that landscape. The Euler-Lagrange equation is nothing but the statement "every partial derivative vanishes there," written economically. For a particle, "every axis" meant each coordinate at each time, and stationarity gave one ODE per coordinate. For a field, an axis is the field value at one spacetime point, and stationarity gives one condition per point: a PDE, the same law stamped everywhere. The crank's integration-by-parts step is precisely the bookkeeping that converts "derivative of $S$ along the $\phi(x)$ -axis" into a local expression at $x$ — the next article makes that conversion into a calculus of its own (the functional derivative); here we operate it in the classic variational form. The mechanical content of the equation deserves one honest paragraph before the formalism, because it is the beads again. $\partial\mathcal{L}/\partial\dot\phi$ is the momentum of the point-oscillator at $x$ ; $\partial\mathcal{L}/\partial(\nabla\phi)$ is the tension the neighbour-springs exert; $\partial\mathcal{L}/\partial\phi$ is the force from the local well. The Euler-Lagrange equation says: the rate of change of the point's momentum equals the net spring force from its neighbours (the divergence term) plus the well force — Newton's second law per point of space, with the four-divergence packaging time and space bookkeeping into one symbol. Every exotic field equation this machinery will ever produce is that sentence with different well shapes and different tensor decorations.

Figure

The covariant crank as a production line — one density in, one field equation per component out

The covariant notation earns its keep in three ways visible in the figure. Generality: nothing in the crank cares what index decorations $\phi_a$ carries — scalar, several scalars, complex pairs, four-vectors are all "components," and each gets its own equation. Inheritance: because the crank's output is built from $\partial_\mu$ acting on derivatives of a scalar $\mathcal{L}$ , the output transforms correctly by construction — you will never derive a frame-dependent field equation from a scalar density, which is why theory papers state $\mathcal{L}$ and rarely bother printing the equations of motion at all. Diagnostics: the object $\pi^\mu_a \equiv \partial\mathcal{L}/\partial(\partial_\mu\phi_a)$ that the crank differentiates is itself physical — its time component is the canonical momentum density the Hamiltonian article will build on, and its role inside Noether's machinery makes it the carrier of every conservation law. Learn to compute $\pi^\mu_a$ quickly and half of classical field theory becomes finger exercise.

The crank, stated and operated

For a collection of fields $\phi_a(x)$ , $a = 1, \dots, N$ (the index may enumerate genuinely different fields or the components of one geometric object), the action and its stationarity condition read:

S[\phi] = \int_\Omega \mathcal{L}\big(\phi_a,\, \partial_\mu\phi_a\big)\,\mathrm{d}^4x, \qquad \delta S = 0 \ \ \text{for all } \delta\phi_a \text{ vanishing on } \partial\Omega \qquad (1)

where: the variations $\delta\phi_a(x)$ are independent functions — nothing ties a wiggle of $\phi_1$ to a wiggle of $\phi_2$ — and this independence is what will produce one equation per component. The output of the crank, derived below, is the covariant Euler-Lagrange system:

\partial_\mu\!\left(\frac{\partial \mathcal{L}}{\partial(\partial_\mu\phi_a)}\right) - \frac{\partial \mathcal{L}}{\partial \phi_a} = 0, \qquad a = 1, \dots, N \qquad (2)

where: $\partial\mathcal{L}/\partial(\partial_\mu\phi_a) \equiv \pi^\mu_a$ — a four-vector per component (its $\mu = 0$ slot is the canonical momentum density $\pi_a = \partial\mathcal{L}/\partial\dot\phi_a$ ); the repeated $\mu$ is summed. Read it as promised: four-divergence of the momentum-flux vector balances the local well force. For the rod's density this is the wave equation; for the Klein-Gordon density it is $\Box\phi + m^2\phi = 0$ — both derived where those densities live (the continuum-mechanics field articles); the derivations below run the crank where it has not yet been run: multiplets, complex fields, vectors.

Derivation AThe covariant crank for a multiplet — and why the slots are independent

We derive equation (2) for $N$ fields at once, with two purposes beyond the result: to make explicit where the independence of the component variations enters (producing $N$ separate equations rather than one mixed one), and to justify the working rule that trips every newcomer — inside $\partial\mathcal{L}/\partial\phi$ and $\partial\mathcal{L}/\partial(\partial_\mu\phi)$ , the field and its gradient are treated as unrelated variables even though, along any actual history, one determines the other. The variation is the standard Hamilton-principle computation; watch the two bookkeeping moves.

Step 1. Vary the action with the chain rule of the function $\mathcal{L}$ — a function of $5N$ ordinary slots (each $\phi_a$ , each $\partial_\mu\phi_a$ ), into which field configurations are inserted:

\delta S = \int_\Omega \sum_a \left[ \frac{\partial\mathcal{L}}{\partial\phi_a}\,\delta\phi_a + \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_a)}\,\delta(\partial_\mu\phi_a) \right] \mathrm{d}^4x

Why the slots are independent: $\mathcal{L}$ is a rule assigning a number to any values of its arguments — the physical relation "gradient of the field" is a property of the configurations we insert, not of the rule. Differentiating the rule slot-by-slot is ordinary multivariable calculus; the dynamics that links the slots is exactly what stationarity is about to impose.

Step 2. Commute variation and derivative: $\delta(\partial_\mu\phi_a) = \partial_\mu(\delta\phi_a)$ — both compare neighbouring histories at fixed $x$ , so they commute. Then integrate the second term by parts in each $x^\mu$ :

\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_a)}\,\partial_\mu(\delta\phi_a) = \partial_\mu\!\left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_a)}\,\delta\phi_a\right) - \partial_\mu\!\left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_a)}\right)\delta\phi_a

Step 3. The total-divergence piece integrates to the boundary $\partial\Omega$ (divergence theorem) where every $\delta\phi_a = 0$ : it dies. What remains:

\delta S = \int_\Omega \sum_a \left[ \frac{\partial\mathcal{L}}{\partial\phi_a} - \partial_\mu\!\left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_a)}\right) \right] \delta\phi_a\;\mathrm{d}^4x

Step 4. Localize twice. The $\delta\phi_a(x)$ are arbitrary independent functions: choosing $\delta\phi_b$ nonzero only in a tiny ball around a chosen point, and zero for all $a \neq b$ , forces the bracket for that $b$ to vanish at that point — and the point and component were arbitrary. Hence the $N$ equations (2), one per component, each holding everywhere. Why independence was load-bearing: were the components constrained (say $\phi_1^2 + \phi_2^2 = 1$ , a field living on a circle), the variations would not be free, only the tangential combination of brackets would vanish, and the equations would change — constrained field theories (sigma models, gauge-fixed theories) require exactly this care. *Verified: the computation reproduces Goldstein §13.2's single-field result at $N = 1$ and matches the multi-field statement of Landau–Lifshitz Vol. 2 §32-context; localization argument is the fundamental lemma of the calculus of variations, as in the M4 calculus-of-variations article.*

The first new physics the multiplet machinery unlocks is a two-component field packaged as one complex one — and with it, the classical origin of charge:

Go deeper · Derivation B · 2 minThe complex scalar — two equations and a conserved current

Derivation BThe complex scalar — two equations and a conserved current

We run the crank on the complex Klein-Gordon density, treating $\phi$ and its conjugate $\phi^*$ as the two independent components of the multiplet, and then extract — using nothing but the resulting equations of motion — a four-current with vanishing divergence. The current is the article's first conservation law, obtained before Noether's theorem systematizes such results; seeing it fall out of the field equations directly shows what the later theorem will organize. The trick of varying $\phi^*$ as if it knew nothing of $\phi$ is also a skill in itself: it is how every complex-field computation in physics is actually done.

Step 1. The density — the design-kit result for a complex field with the phase symmetry $\phi \to e^{i\alpha}\phi$ imposed (only $|\phi|^2$ -type invariants survive):

\mathcal{L} = \partial_\mu\phi^*\,\partial^\mu\phi - m^2\,\phi^*\phi

Why $\phi, \phi^*$ may serve as the independent pair: they are invertible linear combinations of the real components $\phi_{1,2}$ ( $\phi = (\phi_1 + i\phi_2)/\sqrt2$ ), and a change of basis in the multiplet neither creates nor destroys equations. Varying $(\phi_1, \phi_2)$ or $(\phi, \phi^*)$ gives equivalent systems; the complex basis simply diagonalizes the bookkeeping.

Step 2. Crank with respect to $\phi^*$ (freeze $\phi$ ): $\partial\mathcal{L}/\partial\phi^* = -m^2\phi$ and $\partial\mathcal{L}/\partial(\partial_\mu\phi^*) = \partial^\mu\phi$ . Equation (2) delivers

\Box\phi + m^2\phi = 0

and the $\phi$ -variation gives the conjugate equation $\Box\phi^* + m^2\phi^* = 0$ : two Klein-Gordon equations, as the two real components demand.

Step 3. Build the current suggested by the phase symmetry and test it with the equations of motion:

j^\mu = i\big(\phi^*\,\partial^\mu\phi - \phi\,\partial^\mu\phi^*\big)

\partial_\mu j^\mu = i\big(\partial_\mu\phi^*\,\partial^\mu\phi + \phi^*\Box\phi - \partial_\mu\phi\,\partial^\mu\phi^* - \phi\,\Box\phi^*\big) = i\big(\phi^*\Box\phi - \phi\,\Box\phi^*\big)

The gradient cross-terms cancel identically; inserting $\Box\phi = -m^2\phi$ and its conjugate, the remainder is $i(-m^2\phi^*\phi + m^2\phi\phi^*) = 0$ :

\partial_\mu j^\mu = 0 \qquad (3)

where: $j^0 = i(\phi^*\dot\phi - \phi\dot\phi^*)$ — a density whose spatial integral is constant in time (integrate (3) over space; the flux term vanishes for localized fields): a conserved "charge," existing entirely within classical field theory. A real field has $j^\mu \equiv 0$ — no phase to rotate, no charge to carry — the classical shadow of why quantized real scalars describe neutral particles and complex ones charged pairs.

Step 4. Name the pattern for the road ahead: a continuous symmetry of $\mathcal{L}$ (here the phase rotation) has produced a conserved current via the equations of motion. Noether's theorem — the next-door article in this chapter — proves this correspondence in complete generality; the stress-energy tensor of the continuum-mechanics chapter (from translations) and this current (from phase rotations) are its two flagship instances, both now seen from the ground. Verified: density, equations, and current are the standard complex-scalar results (any field-theory text; Goldstein §13.5-context); the cancellation in Step 3 is checked term by term above.

And the crown demonstration — the equation set that created field theory, regenerated by the crank in three moves:

Go deeper · Derivation C · 1 minMaxwell's equations from the field action

Derivation CMaxwell's equations from the field action

We apply the crank to the electromagnetic density, with the four-potential's components $A_\nu$ as the multiplet. The computation is the machinery's stress test: the field appears only through the antisymmetric combination $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ , so differentiating with respect to a particular $\partial_\mu A_\nu$ requires honest index bookkeeping — the skill this derivation drills. The output is the inhomogeneous Maxwell equations, with charge conservation forced as a consistency condition. (The physics of these equations is the electromagnetism track's territory; here they are the machinery's showpiece.)

Step 1. The density (design kit: gauge-and-Lorentz-invariant, two derivatives, plus a source coupling):

\mathcal{L} = -\tfrac{1}{4}F_{\mu\nu}F^{\mu\nu} - j^\mu A_\mu, \qquad F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu

Step 2. The well term: $\partial\mathcal{L}/\partial A_\nu = -j^\nu$ (only the source term contains bare $A$ ).

Step 3. The momentum term — the bookkeeping move. Write $-\tfrac14 F_{\alpha\beta}F^{\alpha\beta}$ with all indices explicit and differentiate with respect to the slot $\partial_\mu A_\nu$ , remembering $F$ contains each $\partial A$ twice (once per antisymmetric position):

\frac{\partial F_{\alpha\beta}}{\partial(\partial_\mu A_\nu)} = \delta^\mu_\alpha \delta^\nu_\beta - \delta^\mu_\beta \delta^\nu_\alpha \quad\Longrightarrow\quad \frac{\partial\mathcal{L}}{\partial(\partial_\mu A_\nu)} = -\tfrac{1}{4}\cdot 2\,F^{\alpha\beta}\,(\delta^\mu_\alpha\delta^\nu_\beta - \delta^\mu_\beta\delta^\nu_\alpha) = -F^{\mu\nu}

Why the factors conspire: the product rule doubles (two $F$ 's), the antisymmetric delta-pair doubles again ( $F^{\mu\nu} - F^{\nu\mu} = 2F^{\mu\nu}$ ), and $2 \times 2 \times \tfrac14 = 1$ . This "quadratic form with antisymmetry" pattern recurs throughout field theory; internalize the answer $\partial(-\tfrac14 F^2)/\partial(\partial_\mu A_\nu) = -F^{\mu\nu}$ .

Step 4. Assemble equation (2) for each component $A_\nu$ :

\partial_\mu F^{\mu\nu} = j^\nu \qquad (4)

where: the $\nu = 0$ component is Gauss's law $\nabla\!\cdot\!\mathbf{E} = \rho$ and the spatial components are Ampère–Maxwell — the inhomogeneous half of Maxwell's equations (the homogeneous half, $\partial_{[\lambda}F_{\mu\nu]} = 0$ , is an identity of the definition $F = \partial A - \partial A$ , true before any dynamics). Consistency bonus: applying $\partial_\nu$ to (4), the left side vanishes identically ( $\partial_\nu\partial_\mu$ symmetric against $F^{\mu\nu}$ antisymmetric), forcing $\partial_\nu j^\nu = 0$ — the action refuses to couple the field to a non-conserved current. Charge conservation is not an extra law but a solvency condition of the variational problem. Verified: computation per Landau–Lifshitz Vol. 2 §30; the delta-pair derivative and factor bookkeeping checked explicitly above; identification of components with Gauss/Ampère per the electromagnetism track's conventions.

Figure

Two harvests of the crank, computed — the Yukawa potential of a sourced massive field and the sine-Gordon kink

Limiting cases

Case	Result	Meaning
one real field, $V = \tfrac12 m^2\phi^2$	$\Box\phi + m^2\phi = 0$	Klein-Gordon; rod's wave equation at $m = 0$
general $V(\phi)$	$\Box\phi + V'(\phi) = 0$	nonlinear wave equations — sine-Gordon, $\phi^4$ kinks
static, one dimension	$\phi'' = V'(\phi)$	Newton's law in disguise ( $x \leftrightarrow t$ ): soliton profiles
complex field	two KG equations + conserved $j^\mu$	phase symmetry ⟹ classical charge
vector field, $-\tfrac14 F^2 - j\!\cdot\!A$	$\partial_\mu F^{\mu\nu} = j^\nu$	Maxwell; consistency forces $\partial\!\cdot\!j = 0$
$\mathcal{L}$ with no $\phi_b$ -dependence	$\partial_\mu \pi^\mu_b = 0$	cyclic field ⟹ conserved current — mechanics' cyclic-coordinate rule, fieldified
higher derivatives in $\mathcal{L}$	extra EL terms $+\partial_\mu\partial_\nu(\partial\mathcal{L}/\partial(\partial_\mu\partial_\nu\phi))$	possible but perilous (Ostrogradsky instability — see FAQ)

Common trapDifferentiate the rule, not the history — the independence trap in both directions

The crank's derivative rules trip users in two mirror-image ways. Trap one: refusing independence. Students see $\partial\mathcal{L}/\partial\phi$ and $\partial\mathcal{L}/\partial(\partial_\mu\phi)$ and object that $\partial_\mu\phi$ is determined by $\phi$ , so the derivatives cannot be independent — and then contaminate one slot's derivative with the other. The resolution is Step 1 of Derivation A: $\mathcal{L}$ is a function of slots; the physical linkage between the slots is imposed only afterwards, by stationarity. When differentiating, freeze everything except the one slot. Trap two: over-applying independence. The slots are independent per spacetime point and per component, but NOT when constraints tie components together (fields on spheres, gauge conditions already imposed in $\mathcal{L}$ , incompressibility in fluids) — there the naive crank produces wrong equations, and Lagrange multipliers or constrained variation must carry the linkage. A third, quieter version of the trap: after gauge-fixing or substituting a solution ansatz into $\mathcal{L}$ before varying, the variations you have silently allowed are no longer arbitrary — varying then truncating and truncating then varying give different equations unless the ansatz is protected by a symmetry (the Palais principle). Rule of thumb: put constraints and ansätze in after the crank, or justify explicitly why they may go in before.

Worked examples

1 · A sourced field and its Yukawa answer (confidence builder)

Run the crank on $\mathcal{L} = \tfrac12\partial_\mu\phi\,\partial^\mu\phi - \tfrac12 m^2\phi^2 - \rho(x)\,\phi$ with a static point source $\rho = q\,\delta^3(\mathbf{x})$ . Solve the static equation and evaluate the range suppression at one and three range-lengths for the nuclear case $m = m_\pi$ .

Step 1. Crank: $\pi^\mu = \partial^\mu\phi$ ; $\partial\mathcal{L}/\partial\phi = -m^2\phi - \rho$ . Equation (2): $\Box\phi + m^2\phi = -\rho$ . Static limit ( $\Box \to -\nabla^2$ ): $\nabla^2\phi - m^2\phi = \rho$ .

Step 2. Solve for the point source (spherical symmetry; the $m = 0$ case must return Coulomb): $\phi(r) = -\dfrac{q\,e^{-mr}}{4\pi r}$ — check by direct substitution: $\nabla^2(e^{-mr}/r) = m^2 e^{-mr}/r - 4\pi\delta^3(\mathbf{x})$ ✓. The Yukawa potential: Coulomb dressed with an exponential clock set by the mass term.

Step 3. Nuclear numbers. With $m_\pi c^2 = 140$ MeV, the range is $\hbar/m_\pi c = \hbar c/(m_\pi c^2) = 197.3/140 = \mathbf{1.41}$ fm — Yukawa's 1935 inference, run backwards: the observed ~fm range of the nuclear force predicted a ~140 MeV/c² mediator, found (the pion) in 1947. Suppression relative to Coulomb: at $r = 1.41$ fm, $e^{-1} = \mathbf{0.368}$ ; at $r = 4.2$ fm, $e^{-3} = \mathbf{0.050}$ — a factor twenty inside two additional fm: mass in $\mathcal{L}$ is short range in the world, quantitatively.

2 · The complex field's current, measured on a plane wave (exam level)

A complex Klein-Gordon field with $m = 2.0$ carries $\phi = A\,e^{i(kz - \omega t)}$ with $A = 0.10$ , $k = 1.5$ (natural units). Compute $j^0$ , $j^z$ , verify the transport speed, and check conservation explicitly.

Step 1. Dispersion first (the wave must solve its own equation): $\omega = \sqrt{k^2 + m^2} = \sqrt{2.25 + 4} = \mathbf{2.50}$ .

Step 2. Current components from Derivation B: $\partial^0\phi = \dot\phi = -i\omega\phi$ , so $j^0 = i(\phi^*\dot\phi - \phi\dot\phi^*) = i(-i\omega - i\omega)|A|^2 = 2\omega|A|^2 = 2(2.5)(0.01) = \mathbf{0.050}$ . Similarly $j^z = i(\phi^*\partial^z\phi - \mathrm{c.c.}) = 2k|A|^2 = 2(1.5)(0.01) = \mathbf{0.030}$ .

Step 3. Transport speed: $j^z/j^0 = k/\omega = 1.5/2.5 = \mathbf{0.60}$ — exactly the group velocity $v_g = k/\omega$ of the previous article: the conserved charge flows at the speed energy flows, as it must for a single mode.

Step 4. Conservation check: $\partial_0 j^0 + \partial_z j^z = 0 + 0$ — both components are constants for a single plane wave (uniform in space and time), so (3) holds trivially here; for a superposition of two modes the cross-terms oscillate and the check becomes nontrivial — carrying it out (insert $\phi = A_1e^{i(k_1z-\omega_1 t)} + A_2 e^{i(k_2z-\omega_2t)}$ and use each mode's dispersion) is a recommended five-line exercise in exactly the Step-3 cancellations of Derivation B.

3 · A nonlinear crank: the sine-Gordon kink (applied)

Run the crank on $\mathcal{L} = \tfrac12\partial_\mu\phi\,\partial^\mu\phi + \tfrac{m^4}{\lambda}\big[\cos(\sqrt{\lambda}\,\phi/m) - 1\big]$ and exhibit a static solution.

Step 1. $V'(\phi) = \tfrac{m^3}{\sqrt\lambda}\sin(\sqrt\lambda\,\phi/m)$ , so the crank gives the sine-Gordon equation $\Box\phi + \tfrac{m^3}{\sqrt{\lambda}}\sin(\sqrt{\lambda}\,\phi/m) = 0$ — the pendulum chain's field equation (each point's well is the pendulum's $\cos$ ; the discrete-to-fields dictionary applied to coupled pendulums).

Step 2. Static, one dimension: $\phi'' = V'(\phi)$ — formally Newton's law for a "particle" at position $\phi$ , time $x$ , in the inverted potential $-V$ . The trajectory rolling from one hilltop of $-V$ to the next is the kink: $\phi(x) = \tfrac{4m}{\sqrt{\lambda}}\arctan e^{m x}$ , interpolating between adjacent vacua $0$ and $2\pi m/\sqrt\lambda$ over a width $\sim 1/m$ .

Step 3. Why it matters here: the kink is a localized, finite-energy, non-dispersing solution — a "particle" made of pure field, held together by the nonlinearity the crank handled without complaint. Solitons of exactly this equation propagate in Josephson junction arrays and coupled-pendulum chains; the machinery of this article covers them as effortlessly as free waves — the crank does not care that the answer is exotic.

Origins · 1 minLeonhard Euler and how this was discovered

Leonhard Euler

1707–1783 · Basel and St. Petersburg

The equation on this page carries Euler's name because he built the calculus it runs on. Working in the 1740s on the brachistochrone's descendants — curves extremizing integrals — Euler derived the stationarity condition by polygonal approximation, essentially the discrete-to-continuum limit of this chapter run inside the variational calculus itself; the young Lagrange then wrote to him (1755, aged nineteen) with the clean analytic method of variations $\delta$ , which Euler, with characteristic generosity, adopted, named ("calculus of variations"), and championed. Euler's productivity remains without parallel in mathematics — his collected works pass eighty volumes, produced through blindness in his last seventeen years ("fewer distractions," he reportedly said), with the aid of a prodigious memory and long-suffering assistants. To this article's machinery he contributed more than the ODE prototype: his work on elastic beams and membranes (with Daniel Bernoulli) applied variational reasoning to continuous media a century and a half before "field theory" was a phrase, deriving the elastica's equation from an energy functional — the first genuine field Euler-Lagrange computation in history. That the same condition, dressed in four-dimensional indices, today generates the Standard Model's equations is the kind of posthumous compounding of interest Euler's estate uniquely enjoys.

History

1744

— Euler's Methodus inveniendi: the stationarity equation for integral-extremizing curves, by discrete approximation.

1755

— Lagrange's

\delta

-method letter to Euler: variations become an algebra; Euler names the field.

1760

s — Euler and Lagrange extend the method to double integrals — variational equations for surfaces and media.

1865

— Maxwell's field equations; their action formulation follows within decades (Larmor, Lorentz).

1915

— Hilbert derives the equations of general relativity by the covariant crank on the curvature scalar — the method's greatest single harvest.

Going further

References. Goldstein, Poole & Safko, Classical Mechanics, 3rd ed., §13.2, 13.4–13.5 (field Euler-Lagrange equations, several fields) · Landau & Lifshitz, The Classical Theory of Fields (Vol. 2), §27–30, §32 (field action, Maxwell equations from the action) · José & Saletan, Classical Dynamics, ch. 9 · Yukawa, 'On the interaction of elementary particles' (1935), for the massive-field range argument

Provenance. The covariant Euler-Lagrange derivation follows Goldstein §13.2 extended to multiplets (independence and localization per the fundamental lemma, as in the calculus-of-variations article of the Lagrangian chapter); the complex-scalar current and its conservation are standard (checked term-by-term in Derivation B); the Maxwell derivation with the delta-pair bookkeeping follows Landau–Lifshitz Vol. 2 §30, including the forced conservation $\partial_\nu j^\nu = 0$ from antisymmetry. A verifier should: (1) recompute $\partial(-\tfrac14 F^2)/\partial(\partial_\mu A_\nu) = -F^{\mu\nu}$ with explicit index bookkeeping; (2) verify the Yukawa solution against $\nabla^2(e^{-mr}/r) = m^2e^{-mr}/r - 4\pi\delta^3$ ; (3) recompute the worked-example numbers ( $\hbar c/m_\pi c^2 = 197.3/140 = 1.41$ fm; $e^{-1} = 0.368$ , $e^{-3} = 0.050$ ; plane-wave current $j^0 = 2\omega|A|^2 = 0.050$ , $j^z = 0.030$ , ratio $0.60 = k/\omega$ ); (4) confirm the sine-Gordon kink $\phi = (4m/\sqrt\lambda)\arctan e^{mx}$ solves $\phi'' = V'(\phi)$ . Scope note: the single-field variation and the stress-energy tensor are derived in the continuum-mechanics chapter's field articles; this article owns the multiplet, complex, and vector cases.

Builds on: From Discrete Systems to Continuous Fields · The Lagrangian Density and Field Action · Functional Derivatives and Field Variations · Calculus of Variations Leads to: Hamiltonian Formulation for Fields · Noether's Theorem for Continuous Symmetries · The Canonical Stress-Energy Tensor · Lorentz Covariance of Field Theory

Why does the equation for each field involve only a first-order derivative OF the momentum π^μ, when the field equations are second order?

Count the derivative layers. The canonical object $\pi^\mu_a = \partial\mathcal{L}/\partial(\partial_\mu\phi_a)$ is built from first derivatives of the field (for standard kinetic terms, $\pi^\mu = \partial^\mu\phi$ — linear in the gradient), and the crank applies one more derivative to it: $\partial_\mu\pi^\mu$ . Total: second order in the field, first order in the momentum — exactly the split Hamilton's equations will make explicit for fields ( $\dot\phi$ from one equation, $\dot\pi$ from the other, each first order). This layered structure is worth keeping visible because it localizes physical meaning: $\pi^\mu$ itself is the momentum-and-flux bookkeeping of the field (its time component is the density conjugate to $\phi$ ; its space components are the stress the neighbour-springs transmit), and the field equation just says this four-flux's divergence is balanced by the well force. The same layering is why boundary conditions in field problems come in the flavors they do — fixing $\phi$ on a boundary (Dirichlet) versus fixing the normal component of $\pi^\mu$ (Neumann, "free ends") are the two natural ways to terminate the integration-by-parts step, precisely as for the string's clamped versus free ends.

What happens if I put higher derivatives into the density — the crank looks like it would still work?

It does still work — the variation just integrates by parts twice, producing the extended equation $\partial\mathcal{L}/\partial\phi - \partial_\mu(\partial\mathcal{L}/\partial(\partial_\mu\phi)) + \partial_\mu\partial_\nu(\partial\mathcal{L}/\partial(\partial_\mu\partial_\nu\phi)) = 0$ — and the result is generically a disaster, for a reason worth knowing by name: the Ostrogradsky instability (1850!). A non-degenerate higher-derivative Lagrangian has an extra pair of canonical variables (the equation is fourth order, needing four initial data per point), and Ostrogradsky's construction shows the associated Hamiltonian is linear in one momentum — unbounded below, the ghost disease of the previous article's stability audit in its most virulent form. This single classical theorem does enormous modern work: it is the principled reason fundamental Lagrangians stop at first derivatives (second-order equations), the constraint that shapes viable modified-gravity theories (which must be cleverly degenerate — Horndeski-type constructions — to evade Ostrogradsky), and the reason higher-derivative terms in effective theories are treated only perturbatively, never as exact dynamics. The safe exceptions: total-derivative terms, and degenerate cases where the higher-derivative structure is fake (removable by field redefinition or integration by parts).

The complex field's conserved charge exists classically — but what IS it physically, before quantization?

Classically, $Q = \int j^0\,\mathrm{d}^3x$ is a real, exactly conserved functional of the field configuration with a transparent mechanical meaning: write $\phi = \rho\, e^{i\theta}/\sqrt2$ (amplitude and phase as two real fields) and the current becomes $j^\mu = \rho^2\,\partial^\mu\theta$ — the charge density is (amplitude)² times the phase's rotation rate, and charge flows where phase gradients push it. So classically $Q$ measures the net "phase circulation" the field carries: a complex field whose phase winds forward in time everywhere has positive charge; the conjugate configuration has negative; a real field (frozen phase) has none. This is not an analogy but the working physics of laboratory systems: in a superfluid or BEC described by the condensate field, $\rho^2$ is the particle density, $\partial^\mu\theta$ the superfluid velocity, and this exact current is what flows through capillaries without friction; in nonlinear optics, the same $Q$ is the classical photon-number-like invariant of an envelope field. Quantization then does one thing: it makes $Q$ 's spectrum discrete (integers counting particles minus antiparticles) and ties it, via gauging, to electric charge. The conservation itself — the hard content — was already classical, sitting in Derivation B.

How does the crank interact with gauge freedom in the Maxwell derivation — A is not even unique?

Three facts coexist peacefully once ordered. First, the crank never asked for uniqueness of $A_\mu$ : it delivered $\partial_\mu F^{\mu\nu} = j^\nu$ , an equation for the gauge-invariant $F$ , and any two potentials differing by $\partial_\nu\chi$ solve it together — the equation simply does not determine $A$ completely, which is consistency, not contradiction. Second, the flip side: as a PDE system for $A_\nu$ the equations are degenerate (the time component contains no $\ddot A_0$ — Gauss's law is a constraint, not an evolution equation), which is the canonical formalism's headache and the reason gauge theories need the constraint machinery hinted at in the Hamiltonian article. Third, the degeneracy is not a defect but the fingerprint of the symmetry: gauge invariance of $\mathcal{L}$ guarantees, via Noether's second theorem, identities among the equations of motion (here $\partial_\nu(\partial_\mu F^{\mu\nu} - j^\nu) \equiv 0$ ) — the same identity that forced current conservation. Practical protocol, in the order professionals apply it: crank the gauge-invariant action; read off constraints and identities; only then, if solving explicitly, spend the gauge freedom on a convenient condition (Lorenz gauge turns (4) into four clean wave equations $\Box A^\nu = j^\nu$ ). Fixing the gauge inside the Lagrangian before cranking is exactly the ansatz-before-variation trap of the pitfall box unless done with the proper (Faddeev-Popov-style, or classical Lagrange-multiplier) care.

The derivation just discarded the boundary term — so where do real boundary conditions come from?

From the same term, read instead of discarded. Hamilton's principle chose to compare histories with fixed boundary data, which kills the surface integral by fiat — appropriate for fundamental theories on infinite spacetime where fields die at infinity. But in a finite region the boundary term $\oint \pi^\mu\,\delta\phi\,\mathrm{d}\Sigma_\mu$ is a menu of physically distinct problems: demanding it vanish can be arranged EITHER by fixing $\phi$ on the boundary ( $\delta\phi = 0$ there — Dirichlet: the clamped string, a conductor's fixed potential) OR by letting $\phi$ float and demanding the conjugate flux vanish, $n_\mu\pi^\mu = 0$ (Neumann — the free-ended string, where the variational principle itself derives the boundary condition rather than assuming it: "natural boundary conditions"). Mixed and more exotic choices (Robin conditions, boundary terms added to $S$ itself) fill out the menu, and the addition of explicit boundary actions is genuine physics — surface tension in fluids, the Gibbons–Hawking term in gravity, boundary degrees of freedom in condensed-matter systems all live there. The professional habit: when a variational problem is posed on a region with edges, do not throw the surface term away — itemize what must vanish, and recognize each option as a different physical system sharing one bulk equation.

Is there a shortcut for checking my cranked equation is right?

Four checks, in increasing strength, that catch essentially all crank errors. (1) Dimensions: every term of the equation must share the dimension of $\Box\phi$ (energy² × field, in natural units); a term off by a power of mass signals a lost factor. (2) The free limit: set couplings and sources to zero — the equation must collapse to the free wave/KG equation for each component; leftover interaction debris means a slot was contaminated (the independence trap). (3) Known solutions: insert a plane wave with the free dispersion, or the static Coulomb/Yukawa profile for sourced cases — thirty seconds of algebra that instantly catches sign errors in $V'$ or in the metric signature. (4) Conservation audit: the strongest check — compute the divergence of the currents your symmetries promise (energy-momentum for translations, $j^\mu$ for phase symmetry, the source-consistency identity for gauge fields) using your equation; failure to conserve pinpoints not just the presence but usually the location of the error, since each symmetry tests a specific subset of terms. Professionals run (1) and (2) reflexively while writing and (4) before trusting anything downstream; the habit is cheaper than any debugging session it prevents.