Second Variation

The second variation of a functional at a function measures its second-order change under an admissible perturbation . Here must remain admissible for all sufficiently small positive and negative . In the second-derivative convention, it is

Some authors instead define the second variation as the coefficient of in the expansion, which is half this quantity (Liberzon 2012).

For and sufficiently smooth ,

The subscripts denote partial derivatives, evaluated along . For fixed endpoint values, .

At an extremal, a second variation taking both positive and negative values on admissible perturbations proves that the functional has a saddle point. Nonnegativity is necessary for a local minimum, and nonpositivity is necessary for a local maximum, but either condition alone can be inconclusive. At an extremal, a sufficient condition for a strict local minimum is uniform positivity for some , together with an expansion whose remainder is in the same norm. The condition , with the same remainder assumption, gives a strict local maximum. Pointwise positivity in each perturbation direction need not be uniform in an infinite-dimensional function space (Liberzon 2012).