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, ... .