Euler-Lagrange equations for two fields
= Euler-Lagrange equations for two fields
{c}
For
$$
\mathcal L[u,v]=\iint
f(x,y,u,v,u_x,u_y,v_x,v_y)\,dx\,dy,
$$
independent variations and integration by parts give
$$
f_u-\partial_xf_{u_x}-\partial_yf_{u_y}=0,
\qquad
f_v-\partial_xf_{v_x}-\partial_yf_{v_y}=0.
$$