We propose dynamic non-linear equations for moving surfaces in an electromagnetic field. The field is induced by a material body with a boundary of the surface. Correspondingly the potential energy, set by the field at the boundary can be written as an addition of four-potential times four-current to a contraction of the electromagnetic tensor. Proper application of the minimal action principle to the system Lagrangian yields dynamic non-linear equations for moving three dimensional manifolds in electromagnetic fields. The equations in different conditions simplify to Maxwell equations for massless three surfaces, to Euler equations for a dynamic fluid, to magneto-hydrodynamic equations and to the Poisson-Boltzmann equation.