Natural boundary condition

Summary

A natural boundary condition is a boundary condition that a minimiser satisfies automatically at an end where no value is prescribed. For $J(y) = \int_a^b L(x, y, y') \, dx$ with $y(b)$ free, it reads $L_p = 0$ at $x = b$. It comes from the first variation, not from the statement of the problem. For the Dirichlet energy in several variables, it is the Neumann boundary condition.