Calculus of variations

Calculus of variations finds stationary points of functionals, often among functions or regular curves.

A sequence of functionals Gamma-converges in a specified topology when every convergent $u_n\to u$ satisfies $F(u)\le {\mathrm{liminf}}_n{F}_n(u_n)$, and each $u$ has a recovery sequence $u_n\to u$ with $F_n(u_n)\to F(u)$. With appropriate compactness of energy sublevels, limits of approximate global minimizers minimize $F$, and the minimum values converge. This is suited to variational approximation such as the Ambrosio–Tortorelli approximation; it does not describe arbitrary local minima.

A variational problem seeks a stationary point or minimizer of a functional on a specified function space. For a symmetric coercive bilinear form $a$ and bounded linear functional $\ell$ on a real Hilbert space, minimizing $J(v)=a(v,v)/2-\ell (v)$ is equivalent to the weak formulation $a(u,v)=\ell (v)$ for every test vector. The Lax-Milgram theorem gives a unique solution, and $J(u+w)-J(u)=a(w,w)/2$ proves unique minimization. Specifying the admissible space and its boundary constraints is part of the problem.

The direct method takes a minimizing sequence, extracts a subsequence using compactness, and uses lower semicontinuity to show that its limit attains the infimum.

For real $f\in L^2({\mathbb{R}}^3)$, this energy functional on $H^1({\mathbb{R}}^3)$ is coercive and attains its minimum by the direct method in the calculus of variations. The nonnegative potential term is weakly lower semicontinuous by local Sobolev compactness gives lower semicontinuity of a nonnegative integral. Its Euler-Lagrange equation is $-\Delta u+u+\sin u=f$ in the weak solution sense. The extra linear restoring term screens the Sine-Gordon equation nonlinearity. The scalar potential $V(s)=s^2/2+1-\cos s$ is convex because $V^{\prime \prime }(s)=1+\cos s\ge 0$; hence any weak critical point is a minimizer. The elliptic regularity estimate for $-\Delta +1$, together with $|\sin u|\le |u|$, gives $u\in H^2$. The Sobolev inequality then puts $u$ in $W^{1,6}$, so Morrey's inequality and uniformly continuous integrable functions vanish at infinity give a continuous representative tending to zero.

Take $a=3\pi /2$, $L=10$ and $u(x)=a\exp (-|x{|}^2/L^2)$ on ${\mathbb{R}}^3$. Set $f=-\Delta u+u+\sin u$. Both functions are smooth and belong to L2 space, and $u\in H^1$. The strictly increasing reaction $h(s)=s+\sin s$ has $0\le h(u)\le h(a)=a-1$, whileConsequently $\Vert f{\Vert }_{\infty }\le a-1+6a/L^2<a=\Vert u{\Vert }_{\infty }$. This $u$ is a minimizer of the screened sine-Gordon energy: it solves the Euler-Lagrange equation and that energy is convex. Thus even the minimizing and smooth hypotheses do not justify the stronger source-size bound. The appropriate general estimate is maximum bound for a monotone reaction term.

A minimizing sequence satisfies $E(u_n)\to \inf E$. Coercivity bounds an appropriate sublevel set, Compactness supplies a convergent subsequence, and sequential lower semicontinuity turns its limit into a global minimizer.

Boundary data $g$ satisfy the bounded slope condition with constant $K$ when every boundary point admits affine lower and upper supporting functions of Lipschitz constant at most $K$. Comparison with these barriers gives a global Lipschitz bound for convex variational problems.

For suitable convex integral functionals, ordered boundary values of two minimizers imply the same ordering in the domain. Affine minimizers can therefore serve as upper and lower barriers.

A functional is a function whose inputs are themselves functions, curves, or other elements of a function space.

A quadratic functional on a real vector space combines a quadratic form $B(v,v)$, a linear functional $\ell (v)$ and a constant. If $B$ is a symmetric bounded bilinear form, its first variation is $DI(v)[w]=2B(v,w)-2\ell (w)$. For a strictly positive $B$, an existing stationary point is the unique minimizer by the quadratic variational principle for a symmetric positive operator.

The epigraph is $\{(u,t):E(u)\le t\}$. It is a convex set exactly when $E$ is a convex function, and is a sequentially closed set in the product topology exactly when $E$ is sequentially lower semicontinuous.

The functional derivative is defined by the first variation

Along a differentiable path of fields $\varphi (x,t)$, a differentiable functional obeysIntegrating in time expresses the functional change as the line integral of its functional derivative along the path.

An energy functional assigns an integral energy to a function. Minimization of a suitable energy is the basis of the Dirichlet principle. For the modified Helmholtz equation, a natural example is

The p-energy of a Sobolev function isIts stationary points under fixed boundary data are weak solutions of the p-Laplacian equation.

A Lagrangian is a function or density whose stationary integral encodes an extremization problem. In Lagrangian mechanics its action gives the equations of motion; in geometry it describes geodesics and minimal surfaces.

For an objective $f(x)$ with equality constraints $g_i(x)=0$, the Lagrangian function is

A variation of a function $y$ is a one-parameter family $y+\epsilon \eta$, where $\eta$ satisfies the required boundary conditions.

The first variation of a functional $L$ at $y$ in the direction $\eta$ is

If a continuous function $f$ satisfiesfor every smooth compactly supported test function $\eta$, then $f=0$ on $(a,b)$.

A point is stationary for a differentiable function or functional when its first derivative or first variation vanishes in every admissible direction.

Among functions with fixed boundary values, the solution of $\nabla \cdot (\kappa \nabla \varphi )=0$ with $\kappa >0$ uniquely minimizes the weighted energy

Fermat-type optical paths in cylindrical coordinates extremize refractive index times Euclidean arclength.

A helical extremal has constant radius and a polar angle linear in axial position.

Noether’s theorem associates a conserved quantity to each continuous variational symmetry.

Suppose an action is invariant under a local transformation $\delta {\Phi }_a=R_a\alpha +R_a^{\mu }{\partial }_{\mu }\alpha$ for every compactly supported function $\alpha$. Variation and integration by parts give the off-shell identityThus arbitrary local symmetry parameters imply dependencies among the field equations. In gauge theory, these identities accompany constraints rather than independent physical charges for each function. This complements the conserved-current statement of Noether theorem.

For $\delta A_{\mu }={\partial }_{\mu }\alpha$, $\delta \varphi =-ie\alpha \varphi$ and $\delta {\varphi }^{\ast }=ie\alpha {\varphi }^{\ast }$, Noether second theorem givesThis holds without imposing the field equations. On the matter equations it becomes a divergence identity for the electromagnetic equation, expressing compatibility with charge conservation. The Hamiltonian description carries the corresponding Gauss law constraint in gauge theory.

If a time-independent Lagrangian $L(q,\dot{q})$ is invariant under a one-parameter point transformation with infinitesimal generator ${\xi }_i={\partial }_s{Q}_i{|}_{s=0}$, then every Euler-Lagrange trajectory conserves

The second variation is the quadratic term in a functional's expansion along $u+\epsilon \eta$; positivity on admissible variations is a local-minimum test.

The Jacobi equation is the linearization of an Euler-Lagrange equation about a stationary path. Its solutions describe infinitesimal one-parameter families of stationary paths.

A conjugate point marks a nonzero endpoint-vanishing Jacobi variation and the loss of positive definiteness of the second variation.

For a Jacobi field along a geodesic with nonpositive sectional curvature, the curvature convention with positive round-sphere curvature givesThus $|J{|}^2$ is nonnegative and convex. If it vanishes at both ends of a segment, convexity forces it to vanish everywhere. No nonzero Jacobi field can vanish at both ends, so the segment has no conjugate points. Only curvature of planes containing the geodesic tangent is needed, and completeness is unnecessary.

For $f(0)=f(T)=0$,with equality for a multiple of $\sin (\pi t/T)$.

If a continuously differentiable $L$-periodic function $f$ has mean zero, thenEquality holds exactly for $f(s)=A\cos (2\pi s/L)+B\sin (2\pi s/L)$.