first variation

In ordinary calculus you find a maximum or minimum by nudging x a tiny bit and seeing whether f changes to first order; at a peak or valley that first-order change is zero. The first variation is exactly this idea lifted to functionals. Instead of nudging a number, you nudge a whole curve: replace the candidate curve y(x) by a wiggled version y(x) + epsilon eta(x), where eta is an arbitrary small bump that vanishes at the fixed endpoints, and epsilon is a tiny dial. The first variation measures how the functional responds to first order in epsilon.

Concretely, J becomes a plain function of the single number epsilon once you fix the bump shape eta, namely g(epsilon) = J[y + epsilon eta]. The first variation, written delta J, is the derivative g'(0) — the rate of change of the functional as you start to deform the curve. Working it out for J[y] = integral of L(x, y, y') dx, you differentiate under the integral sign, then integrate by parts to peel the derivative off eta'. The result is delta J = integral of (dL/dy minus d/dx of dL/dy') eta dx, plus a boundary term that the fixed endpoints kill. For y to make J stationary, delta J must vanish for every allowed bump eta.

The first variation is the engine of the entire subject. Setting delta J = 0 is the analogue of setting a derivative to zero; combined with the fundamental lemma it forces the Euler-Lagrange equation. Just as in ordinary calculus, delta J = 0 is a necessary condition for a minimum or maximum but does not by itself say which — that distinction needs the second variation, the analogue of the second-derivative test. A vanishing first variation only marks a stationary path, also called an extremal.