Let be a function space over 1
and be a functional,
e.g. an Action functional.
Loosely speaking, the first variation tells you how the functional varies in response to an infinitesimal variation in its input, var
i.e.
This is made rigorous by considering to be a functional in its own right.
A variation of is a map such that .
Let denote the function space of all variations of .2
We define the functional so that
This generalizes easily to the th variation
Extrema of functionals
The main utility of th variation is for identifying extrema of functionals as a necessary (but in general insufficient) condition under certain hypotheses.
Suppose is topological and is continuous.
Further let be a local extremum of .
If exists, then it is zero.
If is a local minimum and exists, then it is strictly positive.
If is a local maximum and exists, then it is strictly negative.