Functionals and first variations

A functional maps a function to a number, often by integrating an energy density depending on its position and derivative. For a classical calculation take continuously differentiable real functions on a compact interval with fixed endpoint values. Their differences vanish at the endpoints. Alternatively the Sobolev space $H^1(a,b)$ consists of square-integrable functions with square-integrable weak derivative. In one dimension these functions have continuous representatives and endpoint traces, so prescribed endpoint values remain meaningful even when a classical derivative does not exist everywhere.

The admissible set is usually an affine set rather than a vector space: adding two functions with the same nonzero boundary value doubles that value. Perturbations instead lie in its associated homogeneous space. A fixed-endpoint perturbation $h$ must vanish at both ends. Additional inequalities can permit motion in only one direction, changing a zero-derivative condition into an inequality. A norm also matters: being close in maximum displacement allows sharp changes of slope, whereas being close in the continuously differentiable norm controls both shape and slope.

Fix a candidate and an admissible direction, then consider the scalar function of the perturbation parameter obtained by substituting $u+\varepsilon h$ into the objective. Its derivative at zero is the first variation. In a classical setting, a continuously differentiable integrand on a neighborhood of the compact set traced by the curves supplies bounded derivatives and justifies differentiation under the integral. In a Sobolev setting, appropriate growth and integrability hypotheses are needed instead; formal differentiation alone does not establish them.

For smooth data the chain rule yields an integral linear in the direction and its derivative. Directional derivatives considered one direction at a time are weaker than Fréchet differentiability, which requires an error small relative to the chosen norm uniformly over nearby directions. The quadratic energy on $H^1$ has the stronger property directly from its exact expansion. For a general functional, existence of directional derivatives does not automatically supply a continuous linear derivative or a uniform approximation suitable for a numerical method.

At a local minimum with two-sided admissible line perturbations, the scalar function has a minimum at zero for each direction, so every first variation must vanish. Call this stationarity. A local maximum satisfies the same first-order condition, as can a saddle. The topology specifies which nearby functions are compared, and the phrase local minimum is incomplete without that context. For convex differentiable functionals on convex sets, first-order inequalities can become sufficient, but that requires a separate structural argument.

Existence is a different question again. On an open admissible interval even an ordinary smooth function can decrease toward an excluded boundary without attaining its infimum. In function spaces, minimizing sequences can oscillate or concentrate, leaving their limiting behavior outside a classical admissible class or defeating passage of the energy to the limit. Later lessons address these failures using weak compactness and lower semicontinuity. For now, keep a clear record of the space, constraints, derivative calculation, and conclusion actually proved.