Shape-dependent natural boundary condition of Lagrangian field

In the framework of the Lagrangian field theory, we derive the equations characterizing shape-dependent natural boundary conditions from the Hamilton’s principle. Of these equations, one exhibits mathematical pattern similar to general relativity. In this equation, one side of the sign of equality is the energy–momentum tensor of field and another side is the combination of mean curvature and Gaussian curvature of boundary surface. Meanwhile, we verify that the shape-dependent natural boundary condition can be simplified into the shape equation of lipid vesicle or the generalized Young–Laplace’s equation under different condition.