4 · classical field dynamics

the principle of least action and the euler-lagrange equations

Let $\Phi$ denote a collection of fields (scalar, tensor, or spinor) defined on a manifold $M$. An action is a functional $$ S[\Phi]=\int_M\mathcal{L}(\Phi,\nabla\Phi,g)\,d\mathrm{vol}_g, $$ where the Lagrangian density $\mathcal{L}$ is a scalar constructed from the fields, their covariant derivatives, and the metric. Stationarity under compactly supported variations, $\delta S=0$, produces the Euler-Lagrange equations $$ \frac{\delta\mathcal{L}}{\delta\Phi}=0. $$ When the fields take values in a vector bundle, the variational derivative is understood with respect to the appropriate fiber metric. Gauge symmetries of the Lagrangian imply, via Noether's theorem, the existence of conserved currents; diffeomorphism invariance likewise yields covariant conservation of the energy-momentum tensor.

definition (action and lagrangian density). given a fiber bundle $E\to M$ with configuration sections $\Phi\in\Gamma(E)$, a Lagrangian density is a smooth map assigning to the $k$-jet of $\Phi$ and to $g$ a scalar density $\mathcal{L}\,\mathrm{vol}_g$ (often written as a scalar $\mathcal{L}$ times the metric volume form). the action on a region $U\subset M$ is $S_U[\Phi]=\int_U\mathcal{L}\,d\mathrm{vol}_g$.

definition (euler-lagrange equations). $\Phi$ is critical for $S$ if for every compactly supported vertical variation $\Phi_\varepsilon$ with $\Phi_0=\Phi$, $$ \frac{d}{d\varepsilon}\Big|_{\varepsilon=0}S[\Phi_\varepsilon]=0. $$ the resulting differential equations are the Euler-Lagrange equations of $\mathcal{L}$.

theorem (euler-lagrange for first-order scalar fields). on Minkowski space, for $\mathcal{L}=\mathcal{L}(\phi,\partial_\mu\phi)$ of class $C^2$, a smooth field $\phi$ is critical for compactly supported variations if and only if $$ \partial_\mu\Bigl(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\Bigr) -\frac{\partial\mathcal{L}}{\partial\phi}=0. $$

proof. let $\phi_\varepsilon=\phi+\varepsilon\eta$ with $\eta$ smooth and supported in a compact set $K\subset\mathrm{int}\,U$. then $$ \frac{d}{d\varepsilon}\Big|_{0}S =\int_U\Bigl( \frac{\partial\mathcal{L}}{\partial\phi}\,\eta +\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\,\partial_\mu\eta \Bigr)\,d^4x. $$ integrate the second summand by parts: $$ \int_U\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\,\partial_\mu\eta\,d^4x =-\int_U\partial_\mu\Bigl(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\Bigr)\eta\,d^4x, $$ the boundary term vanishing because $\eta|_{\partial U}=0$. thus $$ \frac{d}{d\varepsilon}\Big|_{0}S =\int_U\eta\Biggl( \frac{\partial\mathcal{L}}{\partial\phi} -\partial_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)} \Biggr)\,d^4x. $$ this vanishes for all such $\eta$ if and only if the Euler-Lagrange expression is identically zero. $\square$

theorem (noether for internal symmetries). suppose $\mathcal{L}$ is invariant under a one-parameter group of field transformations $\delta_\varepsilon\phi=K(\phi)\varepsilon$ (to first order), up to a pure divergence $\delta\mathcal{L}=\partial_\mu K^\mu$. then the current $$ j^\mu =\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\,K(\phi)-K^\mu $$ is conserved on shell: $\partial_\mu j^\mu=0$ whenever the Euler-Lagrange equations hold.

proof. compute $\delta\mathcal{L}$ by the chain rule: $$ \delta\mathcal{L} =\frac{\partial\mathcal{L}}{\partial\phi}\delta\phi +\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\partial_\mu(\delta\phi). $$ rewrite the second term as $$ \partial_\mu\Bigl(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\delta\phi\Bigr) -\Biggl(\partial_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\Biggr)\delta\phi. $$ the coefficient of $\delta\phi$ is the Euler-Lagrange expression. on shell it vanishes, so $\delta\mathcal{L}$ equals a pure divergence built from $j^\mu$. but by hypothesis $\delta\mathcal{L}=\partial_\mu K^\mu$ identically (off shell for a strict symmetry of the Lagrangian density up to $K^\mu$). equating the two expressions for $\delta\mathcal{L}$ on shell yields $\partial_\mu j^\mu=0$. $\square$

remark (bundle-valued fields). when $\Phi$ is a section of a vector bundle, the same argument applies in local frames after replacing $\partial_\mu$ by a connection $\nabla_\mu$; the variational derivative becomes the fiberwise dual of the Euler-Lagrange one-form. gauge symmetries are then vertical automorphisms of the principal bundle, and Noether currents take values in the dual of the gauge Lie algebra (chapter 9).

special-relativistic mechanics and electrodynamics

In Minkowski space the principle of relativity fixes the causal structure and the invariance of the interval. The four-momentum $p^\mu=mu^\mu$ of a massive particle satisfies $p\cdot p=-m^2$, while the kinematics of collisions and decays follow from four-momentum conservation.

definition (relativistic particle). a massive particle of rest mass $m>0$ has a future-directed unit (or normalized) four-velocity $u^\mu$ with $u\cdot u=-1$ and four-momentum $p^\mu=mu^\mu$. free motion extremizes proper time, equivalently $p$ is parallel-transported along the worldline.

theorem (geodesic equation from proper time). among future-directed timelike curves from $p$ to $q$ in Minkowski space, the straight line maximizes proper time $\tau=\int\sqrt{-g(\dot\gamma,\dot\gamma)}\,d\lambda$. its Euler-Lagrange equation is $\ddot x^\mu=0$ in inertial coordinates.

proof. in inertial coordinates $L=\sqrt{-\eta_{\mu\nu}\dot x^\mu\dot x^\nu}$. the Euler-Lagrange equation for a reparametrization-invariant Lagrangian reduces on the constant-speed gauge $\eta(\dot x,\dot x)=-1$ to $\ddot x^\mu=0$. uniqueness of solutions through given endpoints yields the straight-line maximizer. $\square$

Electromagnetism is introduced through a $\mathrm{U}(1)$ connection: the four-potential $A_\mu$. The field strength $$ F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu $$ is the curvature of this connection. Gauge invariance $A_\mu\mapsto A_\mu+\partial_\mu\lambda$ leaves $F$ unchanged and guarantees that physical observables depend only on the curvature. The homogeneous Maxwell equations $$ dF=0\qquad\bigl(\partial_{[\lambda}F_{\mu\nu]}=0\bigr) $$ are the Bianchi identity for an abelian connection and are therefore geometric. The inhomogeneous equations $$ d{*F}=4\pi{*J}\qquad\bigl(\partial_\mu F^{\mu\nu}=4\pi J^\nu\bigr) $$ arise as the Euler-Lagrange equations of the Maxwell action $$ S_{\mathrm{EM}} =-\frac14\int F_{\mu\nu}F^{\mu\nu}\,d^4x +\int A_\mu J^\mu\,d^4x. $$

definition (maxwell field strength). for a $\mathrm{U}(1)$ connection one-form $A$, set $F=dA\in\Omega^2(M)$. in components $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$.

theorem (gauge invariance of the maxwell action). if $\partial_\mu J^\mu=0$ (and boundary terms vanish), then $S_{\mathrm{EM}}$ is invariant under $A\mapsto A+d\lambda$. moreover $F$ itself is gauge invariant.

proof. $d(A+d\lambda)=dA$, so the $F_{\mu\nu}F^{\mu\nu}$ term is unchanged. the coupling shifts by $\int(\partial_\mu\lambda)J^\mu=-\int\lambda\,\partial_\mu J^\mu=0$ after integration by parts and using current conservation. $\square$

theorem (maxwell equations from the action). critical points of $S_{\mathrm{EM}}$ with respect to compactly supported variations of $A$ satisfy $\partial_\mu F^{\mu\nu}=4\pi J^\nu$. the identity $dF=0$ holds for any $A$ of class $C^2$.

proof. $dF=d^2A=0$ is immediate. for the inhomogeneous equation, vary $A\mapsto A+\varepsilon a$ with $a$ compactly supported. using $\delta(F_{\mu\nu}F^{\mu\nu})=4F^{\mu\nu}\partial_\mu a_\nu$ and integrating by parts, $$ \delta S_{\mathrm{EM}} =\int a_\nu\bigl(\partial_\mu F^{\mu\nu}+4\pi J^\nu\bigr)\,d^4x $$ up to the conventional overall normalization matching the $-1/4$ prefactor (the same computation with exterior calculus reads $\delta\int F\wedge{*F}\propto\int a\wedge d{*F}$). vanishing for all $a$ yields $\partial_\mu F^{\mu\nu}=-4\pi J^\nu$ or $\partial_\mu F^{\mu\nu}=4\pi J^\nu$ according to the metric signature and ${*}$ convention; the text adopts the form $\partial_\mu F^{\mu\nu}=4\pi J^\nu$ as stated for the action above. $\square$

The energy-momentum tensor $$ T_{\mu\nu} =F_{\mu\lambda}F^\lambda{}_\nu-\frac14 g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta} $$ is symmetric, traceless (in four dimensions), and divergenceless on shell; it supplies the energy density, Poynting flux, and Maxwell stresses.

theorem (properties of the maxwell stress-energy). for $F=dA$ on Minkowski space in four dimensions, $T_{\mu\nu}$ is symmetric, $T^\mu{}_\mu=0$, and $\partial^\mu T_{\mu\nu}=J^\lambda F_{\lambda\nu}$ (hence $\partial^\mu T_{\mu\nu}=0$ in source-free regions where Maxwell's equation holds with $J=0$).

proof. symmetry is manifest. the trace identity $F_{\mu\lambda}F^{\lambda\mu}-\frac14\cdot 4\,F_{\alpha\beta}F^{\alpha\beta}=0$ is four-dimensional. for the divergence, compute $$ \partial^\mu T_{\mu\nu} =(\partial^\mu F_{\mu\lambda})F^\lambda{}_\nu +F_{\mu\lambda}\partial^\mu F^\lambda{}_\nu -\frac12 F_{\alpha\beta}\partial_\nu F^{\alpha\beta}. $$ the homogeneous equation $\partial_{[\nu}F_{\alpha\beta]}=0$ rearranges the last two structures so that they cancel against $F_{\mu\lambda}\partial^\mu F^\lambda{}_\nu$ up to $F^\lambda{}_\nu\partial^\mu F_{\mu\lambda}$, and Maxwell $\partial^\mu F_{\mu\lambda}=4\pi J_\lambda$ produces $4\pi J^\lambda F_{\lambda\nu}$ (sign fixed with the same conventions as the field equation). in free space $J=0$, so $\partial^\mu T_{\mu\nu}=0$. $\square$

The field of an arbitrarily moving point charge is given by the Lienard-Wiechert potentials. At large distances the radiation field falls as $1/r$ and carries energy whose angular distribution is controlled by the multipole moments of the source. Dipole radiation dominates for nonrelativistic accelerations; synchrotron radiation appears for ultrarelativistic charges in magnetic fields. The back-reaction of the emitted radiation on the particle is described by the Lorentz-Dirac equation, whose radiation-damping term follows from energy-momentum balance.

definition (lienard-wiechert potentials). for a point charge $e$ with worldline $z^\mu(\tau)$ and four-velocity $u^\mu$, the retarded distance $R^\mu=x^\mu-z^\mu(\tau_{\mathrm{ret}})$ is null and future-directed from the emission event. the potentials are $$ A^\mu(x)=\frac{e u^\mu}{u\cdot R}\Big|_{\tau_{\mathrm{ret}}}, $$ up to overall $4\pi\epsilon_0$-type constants fixed by unit conventions.

remark (radiation and radiation reaction). the $1/r$ piece of $F_{\mu\nu}$ constructed from the Lienard-Wiechert field carries a null Poynting flux; integrating $T_{\mu\nu}$ over large spheres yields the Larmor (or relativistic Larmor) power. the Lorentz-Dirac self-force balances that loss against the mechanical four-momentum, at the price of familiar runaway and preacceleration issues in the pure point-particle idealization.

general relativity: the metric as a dynamical field

In general relativity the metric itself becomes a dynamical variable. The Einstein-Hilbert action $$ S_{\mathrm{EH}} =\frac{1}{16\pi G}\int_M R\,d\mathrm{vol}_g +S_{\mathrm{matter}} $$ is varied with respect to $g_{\mu\nu}$. The resulting Euler-Lagrange equation is the Einstein equation $$ G_{\mu\nu}=R_{\mu\nu}-\frac12 R g_{\mu\nu}=8\pi G\,T_{\mu\nu}, $$ where $G_{\mu\nu}$ is the Einstein tensor and $T_{\mu\nu}$ is the energy-momentum tensor of all non-gravitational fields. The twice-contracted Bianchi identity $\nabla^\mu G_{\mu\nu}=0$ automatically enforces $\nabla^\mu T_{\mu\nu}=0$, so local energy-momentum conservation is a geometric necessity rather than an extra postulate.

definition (einstein-hilbert action). for a Lorentzian metric $g$ on an oriented manifold $M$, $$ S_{\mathrm{EH}}[g] =\frac{1}{16\pi G}\int_M R_g\,d\mathrm{vol}_g +S_{\mathrm{matter}}[g,\Phi], $$ with $R_g$ the scalar curvature of the Levi-Civita connection of $g$. boundary Gibbons-Hawking-York terms may be added when $M$ has boundary so that the variational principle is well-posed for fixed induced metric on $\partial M$.

definition (matter stress-energy). $$ T_{\mu\nu} :=-\frac{2}{\sqrt{|g|}}\frac{\delta S_{\mathrm{matter}}}{\delta g^{\mu\nu}} $$ (sign conventions vary; the main point is that $T_{\mu\nu}$ is the metric variational derivative of the matter action).

theorem (einstein equation from the einstein-hilbert action). if $g$ is critical for $S_{\mathrm{EH}}$ under compactly supported variations of the metric (with suitable boundary terms arranged to cancel), then $$ R_{\mu\nu}-\frac12 R g_{\mu\nu}=8\pi G\,T_{\mu\nu}. $$

proof. the Palatini identity states that the linearization of the Ricci tensor is a total covariant divergence of metric derivatives of $\delta g_{\mu\nu}$: $$ \delta R_{\mu\nu} =\nabla_\lambda(\delta\Gamma^\lambda_{\mu\nu}) -\nabla_\nu(\delta\Gamma^\lambda_{\mu\lambda}). $$ contracting and integrating against $\sqrt{|g|}$ converts $\int\sqrt{|g|}\,g^{\mu\nu}\delta R_{\mu\nu}$ into a pure boundary integral when $\delta g$ has compact support. the variation of the volume density contributes $-\frac12 g_{\mu\nu}\delta g^{\mu\nu}$ times $R$, while $\delta(g^{\mu\nu}R_{\mu\nu})=R_{\mu\nu}\delta g^{\mu\nu}+g^{\mu\nu}\delta R_{\mu\nu}$. collecting coefficients of $\delta g^{\mu\nu}$ yields $$ \delta S_{\mathrm{EH}} =\frac{1}{16\pi G}\int\bigl(R_{\mu\nu}-\tfrac12 R g_{\mu\nu}\bigr)\delta g^{\mu\nu}\,d\mathrm{vol} +\delta S_{\mathrm{matter}}. $$ stationarity for all compactly supported $\delta g^{\mu\nu}$ is Einstein's equation with the stated $T_{\mu\nu}$. $\square$

theorem (contracted bianchi implies conservation). for the Levi-Civita connection, $\nabla^\mu G_{\mu\nu}=0$ identically. consequently any metric satisfying $G_{\mu\nu}=8\pi G\, T_{\mu\nu}$ obeys $\nabla^\mu T_{\mu\nu}=0$.

proof. the second Bianchi identity of chapter 2, after one contraction and use of the algebraic symmetries of the Riemann tensor, is $\nabla^\mu R_{\mu\nu}=\frac12\nabla_\nu R$, which rearranges to $\nabla^\mu G_{\mu\nu}=0$. Einstein's equation then transfers the vanishing divergence to $T_{\mu\nu}$. $\square$

remark (diffeomorphism Noether identity). the same conservation law is the Noether identity associated with diffeomorphism invariance of $S_{\mathrm{EH}}+S_{\mathrm{matter}}$: an infinitesimal diffeomorphism generated by a compactly supported vector field $X$ produces a variation $\delta_X g=\mathcal{L}_X g$ that must leave the action stationary whenever the matter Euler-Lagrange equations hold, forcing $\nabla^\mu T_{\mu\nu}=0$.

exact solutions, waves, and cosmology

The simplest nontrivial vacuum solutions are the centrally symmetric Schwarzschild and Kerr metrics, which describe the exterior fields of spherical and rotating bodies and govern planetary motion, light deflection, and gravitational collapse. Linearized theory about Minkowski space yields gravitational waves: transverse, traceless metric perturbations that propagate at the speed of light and carry energy quantified by the Landau-Lifshitz pseudotensor (or any of its equivalent superpotential constructions).

definition (schwarzschild metric). in standard coordinates, for mass parameter $M>0$, $$ ds^2 =-\Bigl(1-\frac{2GM}{r}\Bigr)dt^2 +\Bigl(1-\frac{2GM}{r}\Bigr)^{-1}dr^2 +r^2 d\Omega^2, $$ on $r>2GM$ (the exterior region). it is the unique spherically symmetric vacuum solution (Birkhoff).

theorem (birkhoff). every $C^2$ spherically symmetric solution of the vacuum Einstein equation $R_{\mu\nu}=0$ is locally isometric to a region of the Schwarzschild (or Minkowski) spacetime.

proof (outline). spherical symmetry implies a two-dimensional orbit space with metric of the form $A(t,r)dt^2+2B\,dt\,dr+C\,dr^2$ times the area radius function $r$. the vacuum Einstein equations force $B=0$ after a coordinate choice that diagonalizes the orbit-space metric, reduce $A$ and $C$ to functions of a single variable, and yield the Schwarzschild mass function constant. uniqueness is local in the region where the area radius is a regular coordinate. $\square$

definition (linearized gravitational wave). write $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with $|h|\ll 1$. in the Lorenz gauge $\partial^\mu\bar h_{\mu\nu}=0$ with $\bar h_{\mu\nu}=h_{\mu\nu}-\frac12\eta_{\mu\nu}h$, the linearized vacuum Einstein equation reduces to $\square\bar h_{\mu\nu}=0$. transverse-traceless (TT) solutions satisfy $\bar h_{0\mu}=0$, $\partial^i\bar h_{ij}=0$, and $\bar h^i{}_i=0$.

theorem (propagation of TT waves). plane-wave solutions $\bar h_{\mu\nu}=\mathrm{Re}\bigl(H_{\mu\nu}e^{ik_\alpha x^\alpha}\bigr)$ of the linearized vacuum equation in Lorenz gauge obey $k^\mu k_\mu=0$ and $k^\mu H_{\mu\nu}=0$. residual gauge freedom may be used to place $H_{\mu\nu}$ in TT form with two independent polarization amplitudes (plus and cross).

proof. $\square\bar h=0$ implies $k^2=0$. the Lorenz condition is $k^\mu H_{\mu\nu}=0$. residual gauges $\delta\bar h_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu-\eta_{\mu\nu} \partial\cdot\xi$ with $\square\xi_\mu=0$ remove the non-TT components without spoiling the gauge, leaving a two-dimensional space of spatial TT tensors orthogonal to $\vec k$. $\square$

Relativistic cosmology models the large-scale universe as a homogeneous and isotropic Robertson-Walker spacetime. The Friedmann equations derived from the Einstein equation govern the expansion; the spatial sections may be closed, flat, or open according to the sign of the curvature parameter. Observational redshift, nucleosynthesis, and the cosmic microwave background emerge as direct consequences. Homogeneous but anisotropic Bianchi models exhibit richer dynamics, including oscillatory regimes near the initial singularity. The Hawking-Penrose singularity theorems demonstrate that, under generic energy conditions and the existence of trapped surfaces, geodesic incompleteness is inevitable; the classical description of spacetime therefore breaks down at finite proper time in the past (or future) of realistic cosmologies.

definition (robertson-walker metric). $$ ds^2=-dt^2+a(t)^2\Bigl( \frac{dr^2}{1-kr^2}+r^2 d\Omega^2 \Bigr), $$ with scale factor $a(t)$ and curvature parameter $k\in\{-1,0,1\}$.

theorem (friedmann equations). for a perfect fluid $T_{\mu\nu}=(\rho+p)u_\mu u_\nu +pg_{\mu\nu}$ comoving with the RW coordinates, Einstein's equation reduces to $$ \Bigl(\frac{\dot a}{a}\Bigr)^2=\frac{8\pi G}{3}\rho-\frac{k}{a^2},\qquad \frac{\ddot a}{a}=-\frac{4\pi G}{3}(\rho+3p). $$

proof. compute the Einstein tensor of the RW metric in the coordinate orthonormal frame; the only independent components are $G_{00}$ and the spatial diagonal. equate them to $8\pi G\,T_{\mu\nu}$ for the perfect fluid with $u=\partial_t$. the first equation is the Hamiltonian constraint; differentiating it and using $\nabla^\mu T_{\mu\nu}=0$, which becomes $\dot\rho+3(\dot a/a)(\rho+p)=0$, yields the acceleration equation (or derive it directly from $G_{rr}$). $\square$

theorem (hawking-penrose, schematic statement). let $(M,g)$ be a globally hyperbolic spacetime satisfying the null (or strong) energy condition, with a noncompact Cauchy surface, and suppose there exists a trapped surface. then $(M,g)$ is null-geodesically incomplete.

proof (strategy). the Raychaudhuri equation for a null congruence, $$ \frac{d\theta}{d\lambda}=-\frac12\theta^2-\sigma_{\mu\nu}\sigma^{\mu\nu}-R_{\mu\nu}k^\mu k^\nu, $$ together with the null energy condition $R_{\mu\nu}k^\mu k^\nu\ge 0$, forces the expansion $\theta$ of the outgoing null congruence from a trapped surface to reach $-\infty$ in finite affine time. conjugate points then form; standard causality arguments in globally hyperbolic spacetimes convert the existence of conjugate points on every generator into the incompleteness of the congruence. $\square$

Thus the variational principle, applied first to matter fields on a fixed background and then to the metric itself, generates the complete dynamical content of classical electromagnetism and gravitation. The geometric identities of the preceding chapters reappear as conservation laws, while the field equations determine the evolution of both the fields and the spacetime geometry in which they propagate.

remark (to later chapters). global topological constraints on field configurations (chapters 5 and 6), analytic index theory (chapter 8), and nonabelian gauge dynamics with instantons (chapter 9) refine the local variational picture constructed here.

exercises

exercise 1 (klein-gordon). for $\mathcal{L}=\frac12\partial_\mu\phi\partial^\mu\phi-\frac12 m^2\phi^2$ on Minkowski space, derive the Euler-Lagrange equation and obtain $(\square+m^2)\phi=0$.

exercise 2 (maxwell energy). from $T_{\mu\nu}$ as above in free space, compute $T_{00}$ and $T_{0i}$ in terms of $\mathbf{E}$ and $\mathbf{B}$ and identify the energy density and Poynting vector.

exercise 3 (dust conservation). for pressureless dust $T_{\mu\nu}=\rho u_\mu u_\nu$ with $u^\mu u_\mu=-1$, show that $\nabla^\mu T_{\mu\nu}=0$ implies both $u^\mu\nabla_\mu u^\nu=0$ (geodesic flow) and $\nabla_\mu(\rho u^\mu)=0$ (continuity).

exercise 4 (noether current for a complex scalar). let $\phi$ be complex with $\mathcal{L}=\partial_\mu\phi\,\partial^\mu\bar\phi-V(|\phi|^2)$. for the global $\mathrm{U}(1)$ symmetry $\phi\mapsto e^{i\alpha}\phi$, compute the Noether current $J^\mu$ and verify $\partial_\mu J^\mu=0$ on-shell.