Functional

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 Euler-Lagrange equation is $y^{\prime \prime }+ky=0$. For positive $k$ a nonzero stationary function exists exactly when $\sqrt{k}$ is a positive integer $n$, and all such functions are $B\sin (nx)$. These values are the Dirichlet eigenvalues of $-d^2/d{x}^2$ on the interval. At $k=0$ the solutions are affine and the endpoint conditions leave only zero; negative $k$ likewise leaves only zero. Stationarity alone does not assert minimization of the quadratic functional.

Impose zero Dirichlet boundary conditions on an interval. The isoperimetric constraint $I[y]=1$ is regular because $D{I}_y[y]=2I[y]=2\ne 0$. The Lagrange multiplier rule applied to $J-\lambda (I-1)$ gives the Sturm-Liouville equationMultiplying by $y$ and using integration by parts gives $J[y]=\lambda I[y]=\lambda$. Thus the multiplier is the stationary value of the original functional, or its Rayleigh quotient. This calculation does not require $w$ to have one sign; positive-weight spectral conclusions require additional hypotheses.

If two stationary functions have different multipliers, multiplying their equations crosswise and subtracting gives $({\lambda }_n-{\lambda }_m)\int w{y}_n{y}_m=0$. The endpoint terms vanish. This proves the weighted orthogonality of Sturm-Liouville eigenfunctions. Higher stationary values need not be local minima of the constrained functional.

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

The first variation of a differentiable action functional gives its change under a small field deformation. With $E=\delta S/\delta \varphi$, the change $\varphi \mapsto \varphi -\eta f$ generates $-\eta \int fE$ to first order. This is the basic cancellation behind the equivalence theorem for field redefinitions; the functional Jacobian and transformed operator insertions must also be treated consistently.

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

A field configuration is critical for an energy functional $E$ if its first variation vanishes for every admissible infinitesimal field variation. For $E[\varphi ]=\int (\frac{1}{2}|\nabla \varphi {|}^2+U(\varphi ))$, integration by parts against compactly supported variations gives $\delta E=\int (-\Delta \varphi +U^{\prime }(\varphi ))\delta \varphi$, so stationary field configurations satisfy the Euler-Lagrange equation $\Delta \varphi =U^{\prime }(\varphi )$. A constrained target instead allows only tangent variations, changing that field equation. Criticality is stationarity, not necessarily energetic stability or a local minimum.

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.

In Lagrangian, the action integrates such a density over spacetime.

A finite-dimensional optimization Lagrangian combines an objective function with Lagrange multiplier terms for its constraints.

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 $(a,b)$.