The goal of this section is to discover a suitable definition of a ”functional derivative”, such that we can take the derivative of a functional and still have the same rules of differentiation as normal calculus. For example, we wish to δJ find a definition for δy , where J[y(x)] is a functional of y(x) such that things δ 2 δJ like δy J = 2J δy are still true.
Definitions
Functional
Stone’s definition of local functional where f is a multivariable function
Z x2 Z x2 J[y] = f(x, y(x), y0(x), y00(x), ··· , y(n)(x))dx = fdx (1) x1 x1
Notice the functional J ”eats” an entire function y, which is defined using its local values y(x), y0(x) etc, and spits out a number through integration. In short, a functional is just a number that depends on an input function.
Variation
A variation of the functional is the amount the functional changes when the input function is changed by a little bit. Suppose we let y(x) → y(x)+δy(x), d then since dx is linear
0 0 d 0 0 y (x) → y (x) + dx δy(x) = y (x) + δy (x) 2 y00(x) → y00(x) + d δy(x) = y00(x) + δy00(x) dx2 (2) . . n (n) (n) d (n) (n) y (x) → y (x) + dxn δy(x) = y (x) + δy (x) thus the integrant of the new output J[y + δy] can be expanded to first order using Taylor expansion of a multivariable function around the old input y
Z x2 J[y + δy] = f(x, y + δy, y0 + δ0y, ··· , y(n) + δ(n)y)dx x1 Z x2 (n) ∂f ∂f ∂f 0 ∂ f (n) = f + dx + δy + 0 δy + ··· + (n) δy dx (3) x1 ∂x ∂y ∂y ∂y 2
The variation of the functional is thus, by definition, the new output minus the old output taken to first order. δJ =J[y + δy] − J[y] Z x2 (n) ∂f ∂f ∂f 0 ∂ f (n) = dx + δy + 0 δy + ··· + (n) δy dx (4) x1 ∂x ∂y ∂y ∂y
d we can moved all the dx on δy to f using integration by parts
x2 x2 2 x2 d ∂f d ∂f 0 d ∂f δJ = ( )δy(x) + ( )δy (x) − ( )δy(x) dx ∂y0 dx ∂y00 dx2 ∂y00 x1 x1 x1 n x2 Z x2 n−1 d ∂f (n) ∂f + ··· + (−1) ( )δy (x) + dx+ dxn ∂y(n) ∂x x1 x1 Z x2 2 n ∂f d ∂f d ∂f n−1 d ∂f − 0 + 2 00 − · · · + (−1) n (n) δy(x)dx x1 ∂y dx ∂y dx ∂y dx ∂y (5) and Voila! (5) is the variation of the local functional defined in (1) with no additional assumption.
Functional Derivative
For a normal multi-variable function f(x1, x2, ··· , xn) we have a nice form for its variation n X ∂f df = dx (6) ∂x i i=1 i and we know how to calculate the derivatives ∂f . Here we wish to rewrite ∂xi (5) such that we have a similar form for the variation of a functional
Z x2 δJ δJ = dx (x)δy(x) (7) x1 δy Unfortunately, this is only possible under special circumstances. That is, we (n) (n) need the variation to have ”fixed-ends” (δy (x1) = δy (x2) = 0) and ∂f 1 that we require implicit f ( ∂x = 0) . Basically, we want everything before the last line of (5) to vanish. This way, comparing (5) and (7) we finally have
δJ ∂f d ∂f d2 ∂f dn ∂f = − + − · · · + (−1)n−1 (8) δy ∂y dx ∂y0 dx2 ∂y00 dxn ∂y(n)
1This is slightly over kill, since we just want R x2 ∂f dx = 0 x1 ∂x 3
As long as δJ can be written as (7), we will have our nice rules of differenti- ation. For example
δ(J 2) ≡(J + δJ)2 − J 2 = 2JδJ + O(δJ 2) Z x2 δJ = dx 2J (x)δy(x) ⇒ x1 δy δJ 2 δJ = 2J (9) δy δy A word of caution: This definition of functional derivative is nice, but as f involves higher derivatives of y, the fixed-end condition becomes harsher and the range of y this derivative applies to quickly diminishes. Therefore it is sometimes more useful to make variations by hand according to (5).
Lagrangian Mechanics When the integrant of the functional only has dependence on y and y0 (f(y, y0)), (8) reduces to the popular Fr´echetderivative
δJ ∂f d ∂f = − (10) δy ∂y dx ∂y0 this form should look familiar to all physicists, since it laid the foundation for basic Lagrangian mechanics. In a typical classical mechanics problem, we wish to minimize the action S, which is often a functional of a configuration function q, whose basic independent variable is time t. That is
Z t2 S[q] = L(q(t), q˙(t))dt (11) t1 where L is the Lagrangian. To find extrema, we set derivative to 0 δS = 0 ⇒ δq ∂L d ∂L = (12) ∂q dt ∂q˙ Lo and behold, the Lagrangian equation of motion.