Functional Derivative

Author

Eric W. Weisstein
April 15, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/CalculusofVariations/FunctionalDerivative.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FunctionalDerivative.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Additional Authors

Michael Trott
Wolfram Research, Inc.
April 5, 2002

Definition

The functional derivative is a generalization of the derivative. Instead of differentiating a function wrt a variable, one differentiates a functional wrt a function. The definition for the univariate case is

((δ F[f(x)] )/(δ f(y)))=(lim_(ϵ->0))((F[f(x)+ϵ δ(x-y)]-F[f(x)])/ϵ).

FunctionalDerivative[functional_, f_[y_]] := 
Module[{x}, 
Limit[Normal[Series[#, {ϵ, 0, 1}]]& //@ 
  ((((functional /. f :> Function[x, f[x] + 
    ϵ DiracDelta[x - y]]) - functional)/ϵ) //. 
       Integrate[a_, b_] - Integrate[c_, b_] :>
       Integrate[a - c, b]), ϵ -> 0]]

Example

The Lagrange equations are the result of functional differentiating the Hamilton action (functional).

FunctionalDerivative[Integrate[1/2 x'[t]^2-V[x[t]],{t,-Infinity,Infinity}],x[τ]]