Euler-Lagrange equations for two fields

For

L [u, v] = \iint f (x, y, u, v, u_{x}, u_{y}, v_{x}, v_{y}) d x d y,

independent variations and integration by parts give

f_{u} - \partial_{x} f_{u_{x}} - \partial_{y} f_{u_{y}} = 0, f_{v} - \partial_{x} f_{v_{x}} - \partial_{y} f_{v_{y}} = 0.

 Ancestors (5)