VariationalD[f,u[x],x]
returns the variational derivative of the integral 
with respect to u[x], where the integrand f is a function of u, its derivatives, and x.
VariationalD[f,u[x,y,…],{x,y,…}]
returns the variational derivative of the multiple integral 
with respect to u[x,y,…], where f is a function of u, its derivatives, and the coordinates x, y, ... .
VariationalD
VariationalD[f,u[x],x]
returns the variational derivative of the integral 
with respect to u[x], where the integrand f is a function of u, its derivatives, and x.
VariationalD[f,u[x,y,…],{x,y,…}]
returns the variational derivative of the multiple integral 
with respect to u[x,y,…], where f is a function of u, its derivatives, and the coordinates x, y, ... .
VariationalD[f,{u[x,y,…],v[x,y,…],…},{x,y,…}]
gives a list of variational derivatives with respect to u, v, ... .
Details and Options
- To use VariationalD, you first need to load the Variational Methods Package using Needs["VariationalMethods`"].
Examples
Basic Examples (2)
Needs["VariationalMethods`"]Variational derivative for the arc length 
of a plane curve:
VariationalD[Sqrt[1 + y'[x] ^ 2], y[x], x]Needs["VariationalMethods`"]Variational derivative for the Dirichlet integral in two variables:
VariationalD[D[u[x, y], x] ^ 2 + D[u[x, y], y] ^ 2, u[x, y], {x, y}]