First Variation

The first variation of a functional at a function gives its linear first-order change under an admissible perturbation . Here must remain admissible for all sufficiently small positive and negative . When this linear functional exists, it is given by

For a fixed , this is an ordinary derivative with respect to the real parameter (Liberzon 2012).

For , differentiation under the integral gives

provided the integrand is sufficiently smooth. Here subscripts denote partial derivatives, evaluated along . For fixed endpoint values, admissible perturbations satisfy . An extremal makes the first variation vanish for every such perturbation. This is a necessary condition for a local extremum, but does not by itself imply a minimum or maximum (Liberzon 2012).