What do people mean, when they speak of natural boundary conditions in the calculus of variations?

How do natural boundary conditions relate to the Euler-Lagrange equations?

An example would be fantastic!


2 Answers 2


Basically two types of boundary conditions are used: Essential or geometric boundary conditions which are imposed on the primary variable like displacements, and Natural or force boundary conditions which are imposed on the secondary variable like forces and tractions. Essential boundary conditions are imposed explicitly on the solution but natural boundary conditions are automatically satisfied after solution of the problem.

Natural Boundary Conditions of the Simplest Kind:

Let $~J : C^2[x_0, x_1] → \mathbb R~$ be a functional of the form of $$J(y) = \int^{x_1}_{x_0}f(x, y, y') dx$$ and assume that no boundary conditions have been imposed on $~y~$, then $~J~$ have an extremum $~y~$ if the following necessary conditions are satisfied:

$(i)~~$ The ordinate of the extremal satisfies the Euler-Lagrange Equation $$f_y - \frac{d}{dx} f_y' = 0$$

$(ii)~~$ At $~x = x_0~$ $$\left|\frac{\partial f}{\partial y'}\right|_{~x_0}=0$$

$(iii)~~$ At $~x = x_1~$ $$\left|\frac{\partial f}{\partial y'}\right|_{~x_1}=0$$

  • $\begingroup$ Thank you for your answer. Do you know how one can derive (i) and (ii) ? Do these conditions somehow follow from evaluation of the functional derivative at 0 ? $\endgroup$
    – Joker123
    Aug 6, 2019 at 15:15
  • $\begingroup$ Follow this link staff.www.ltu.se/~larserik/applmath/chap3en/part9.htm, we will get your answer. $\endgroup$
    – nmasanta
    Aug 7, 2019 at 3:54
  • $\begingroup$ hey, ok suppose now f(x,y,y',y'') and y(0) = 0, y(1) = 1, but there are no restrictions on y' at x=0 and x=1. What will the natural boundary conditions be in this case? the same as you pointed out above or something different now? $\endgroup$ Nov 6, 2019 at 18:20
  • $\begingroup$ @nmasanta What will be the boundary conditions when f is a function of higher order derivatives? Can you please explain? $\endgroup$
    – Kashif
    Mar 24, 2021 at 6:45
  • $\begingroup$ Please ask your query as a separate question. It will be more helpful. @Kashif $\endgroup$
    – nmasanta
    Mar 24, 2021 at 6:48

From the perspective of computational math: Solving a differential equation using e.g. finite elements you usually build a linear equation $Ax = b$ where $x$ is a discrete approximation of the solution. You set up $A$ and $b$ depending on different aspect of your problem, like how your domain looks like and also what kind of boundary conditions there are.

We speak of natural boundary conditions, if during assembly of the problem, this step of adding the information of boundary conditions is effectivly skipped. It's those boundary conditions that "naturally" arise when we don't pay attention to them.

See also here: Neumann = 0 boundary condition drops the term for boundary conditions completely, so we don't have to worry about them.


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