Functional derivative

Volterra derivative

One of the first concepts of a derivative in an infinite-dimensional space. Let $ I ( y) $ be some functional of a continuous function of one variable $ y ( x) $; let $ x _ {0} $ be some interior point of the segment $ [ x _ {1} , x _ {2} ] $; let $ y _ {1} ( x) = y _ {0} ( x) + \delta y ( x) $, where the variation $ \delta y ( x) $ is different from zero in a small neighbourhood $ [ a, b] $ of $ x _ {0} $; and let $ \sigma = \int _ {a} ^ {b} \delta y ( x) dx $. The limit

$$ \lim\limits _ {\begin{array}{c} \sigma \rightarrow 0, \\ a , b \rightarrow x _ {0} \end{array} } \ \frac{I ( y _ {1} ) - I ( y _ {0} ) } \sigma , $$

assuming that it exists, is called the functional derivative of $ I $ and is denoted by $ ( \delta I ( y _ {0} )/ \delta y) \mid _ {x = x _ {0} } $. For example, for the simplest functional of the classical calculus of variations,

$$ I ( y) = \ \int\limits _ { x _ {1} } ^ { {x _ 2 } } F ( x, y, \dot{y} ) dx, $$

the functional derivative has the form

$$ \left . \frac{\delta I ( y _ {0} ) }{\delta y } \right | _ {x = x _ {0} } = $$

$$ = \ \frac{\partial F ( x _ {0} , y _ {0} ( x _ {0} ), \dot{y} _ {0} ( x _ {0} )) }{\partial y } - { \frac{d}{dx} } \frac{\partial F ( x _ {0} , y ( x _ {0} ), \dot{y} ( x _ {0} )) }{\partial \dot{y} } , $$

that is, it is the left-hand side of the Euler equation, which is a necessary condition for a minimum of $ I ( y) $.

In theoretical questions the concept of a functional derivative has only historical interest, and in practice has been supplanted by the concepts of the Gâteaux derivative and the Fréchet derivative. But the concept of a functional derivative has been applied with success in numerical methods of the classical calculus of variations (see Variational calculus, numerical methods of).

Comments

The existence of the functional derivative of $ I $ at $ y = y _ {0} $ and $ x = x _ {0} $ apparently means that the Fréchet derivative $ dI $ of $ I $ at $ y = y _ {0} $, which is a continuous linear form on the space of admissible infinitesimal variations $ z $, is of the form $ \int u ( x) \cdot z ( x) dx $ for some continuous function $ u $, so that it can be continuously extended to $ z = $ the $ \delta $- function at $ x = x _ {0} $. In the example this happens only if $ y _ {0} $ is twice continuously differentiable.