3.1.1 Weierstrass-Erdmann corner conditions

Recall from Section 2.2.1 that a piecewise
curve
on
is
everywhere except possibly at a finite number of points where it is
continuous but its derivative
is discontinuous. Such points
of discontinuity of
are known
as *corner points*. A corner point
is characterized by the property
that the left-hand derivative
and the
right-hand derivative
both exist but have
different values. For example, if a hanging chain (catenary) is suspended
too close to the ground, then it will not look as in Figure 2.3
on page but will instead touch the ground and have
two corner points; see Figure 3.1. Below is another example
in which a corner point arises.

Suppose that a piecewise curve is a strong extremum for the Basic Calculus of Variations Problem, under the same assumptions as in Section 2.3. Clearly, is then also a weak extremum, with respect to the generalized 1-norm

As we stated in Section 2.3.3 (see in particular footnote 2 there), such an extremum must satisfy the integral form (2.23) of the Euler-Lagrange equation almost everywhere, i.e., at all noncorner points. Extending our previous terminology to the present setting, we will refer to piecewise solutions of (2.23) as

For simplicity, we assume that has only one (unspecified) corner point . As a generalization of (2.10), we will let two separate perturbations and act on the two portions of (before and after the corner point). To make this construction precise, denote these two portions by and ; their perturbed versions will then be and . Clearly, we must have to preserve the endpoints. Furthermore, since the location of the corner point is not fixed, we should allow the corner point of the perturbed curve to deviate from . Let this new corner point be for some , with the same as before for convenience. Our family of perturbed curves (parameterized by ) is thus determined by the two curves , and one real number . We label these new curves as , with . There will be an additional condition on and to guarantee that is continuous for each ; see (3.4) below. We take both and to be , to ensure that is piecewise with a single corner point at . (Note that the difference is piecewise with two corner points, one at and the other at .) Figure 3.2 should help visualize this situation and the argument that follows.

The reader might have noticed a small problem which we need to fix before proceeding. The domain of is , whereas we want the domain of to be . For , such a perturbed curve is ill defined. To deal with this issue, let us agree to extend beyond via linear continuation: define for by . The linearity is actually not crucial, all we need is that the function be at , with

If the perturbation is also defined on an interval extending to the right of , then the earlier construction makes sense (at least for close enough to 0). Of course we need to make a similar modification to , extending it linearly to the left of .

Let us write the functional to be minimized as a sum of two components:

After the perturbation, the first functional becomes

(note that should also be an argument on the left-hand side, but we omit it for simplicity). We can now compute the corresponding first variation:

Applying integration by parts and recalling (3.2) and the constraint , we can bring the above expression to the form

Similarly, for the second functional we have

and the first variation of at is

For close to 0, the perturbed curve is close to the original curve in the sense of the 0-norm. Therefore, the function must attain a minimum at , implying that

Next, observe that each of the two portions , of the optimal curve must be an extremal of the corresponding functional . Indeed, this becomes clear if we consider the special case when the perturbation vanishes at and . Therefore, the integrals in the preceding expressions for and should both vanish, and we are left with the condition

Now we need to take into account the fact that the two perturbations and are not independent: they have to be such that the perturbed curve remains continuous at . This provides the additional relation

The quantity describes the first-order (in ) vertical displacement of the corner point, in much the same sense that describes the first-order horizontal displacement; and are independent of each other. Equating the first-order terms with respect to in (3.4) and using the second equality in (3.2) along with its counterpart , we obtain

Using (3.5), we can eliminate and from (3.3) and rewrite that formula in terms of and as follows:

Since and are independent and arbitrary, we conclude that the terms multiplying them must be 0. This means that and are in fact continuous at .

The above reasoning can be extended to multiple corner points,
yielding the necessary conditions for optimality known as the
**Weierstrass-Erdmann corner conditions**: *If a
curve
is a strong extremum, then
and
must be continuous at each corner point
of
.*
More precisely, their discontinuities (due to the fact that
does not exist at corner points) must be *removable*.
The quantities
and
are of course familiar to us from Chapter 2;
they are, respectively, the momentum and the Hamiltonian.
Weierstrass presented these conditions in 1865
during his lectures on calculus of
variations, but never formally published them.
They were independently derived and published by Erdmann in
1877.

For the case of a single corner point, the two Weierstrass-Erdmann corner conditions together with the two boundary conditions provide four relations, which is the correct number to uniquely specify two portions of the extremal (each satisfying the second-order Euler-Lagrange differential equation). In general, to uniquely specify an extremal consisting of portions (i.e., having corner points) we need conditions, and these are provided by corner conditions plus the two boundary conditions.