2.3.1 Euler-Lagrange equation

We continue to follow the notational convention of Chapter 1 and denote by , , , , , etc. the partial derivatives of the Lagrangian . To keep things simple, we assume that all derivatives appearing in our calculations exist and are continuous. While we will not focus on spelling out the weakest possible regularity assumptions on , we will make some remarks to clarify this issue in Section 2.3.3.

Let be a given test curve in . For a perturbation in (2.10) to be admissible, the new curve (2.10) must again satisfy the boundary conditions (2.8). Clearly, this is true if and only if

In other words, we must only consider perturbations vanishing at the endpoints. Now, the first-order necessary condition (1.37) says that if is a local extremum of , then for every satisfying (2.11) we must have . (We denoted extrema by in (1.37) but here we drop the asterisks to avoid overly cluttered notation.) In the present case we want to go further and use the specific form of given by (2.9) to arrive at a more explicit condition in terms of the Lagrangian .

Recall that the first variation was defined via

The left-hand side of (2.12) is

We can write down its first-order Taylor expansion with respect to by expanding the expression inside the integral with the help of the chain rule:

Matching this with the right-hand side of (2.12), we deduce that the first variation is

Note that, proceeding slightly differently, we could arrive at the same result by remembering from (1.34)-(1.36) that

and using differentiation under the integral sign on the right-hand side of (2.13).

We see that the first variation depends not just on but also on . This is not surprising since has as one of its arguments. However, we can eliminate the dependence on if we apply integration by parts to the second term on the right-hand side of (2.14):

where the last term is 0 when satisfies the boundary conditions (2.11). Thus we conclude that if is an extremum, then we must have

for all curves vanishing at the endpoints and .

The condition (2.16) does not yet give us a practically useful test for optimality, because we would need to check it for all admissible perturbations . However, it is logical to suspect that the only way (2.16) can hold is if the term inside the parentheses--which does not depend on --equals 0 for all . The next lemma shows that this is indeed the case.

PROOF. Suppose that for some . By continuity, is then nonzero and maintains the same sign on some subinterval containing . Just for concreteness, let us say that is positive on .

Construct a function that is positive on and 0 everywhere else (see Figure 2.7). For example, we can set for and otherwise. This gives , and we reach a contradiction.

It follows from (2.16) and Lemma 2.1 that for to be an extremum, a necessary condition is

This is the celebrated

We must keep in mind, however, that the correct interpretation of the Euler-Lagrange equation is (2.17): and are treated as independent variables when computing the partial derivatives and , then one plugs in for these variables the position and velocity of the curve, and finally the differentiation with respect to is performed using the chain rule. Written out in detail, the right-hand side of (2.17) is

It might not be necessary to actually perform all of these operations, though; sometimes a shortcut is possible, as in the next example.

The functional to be minimized is given by the integral of the Lagrangian along a path, while the Euler-Lagrange equation involves derivatives of and must hold for every point on the optimal path; observe that the integral has disappeared. The underlying reason is that if a path is optimal, then every infinitesimally small portion of it is optimal as well (no ``shortcuts" are possible). The exact mechanism by which we pass from the statement that the integral is minimized to the pointwise condition is revealed by Lemma 2.1 and its proof.

Trajectories satisfying the Euler-Lagrange equation (2.18) are
called *extremals*. Since the Euler-Lagrange equation is only a
necessary condition for optimality, not every extremal is an extremum.
We see from (2.19) that the equation (2.18) is a
second-order differential equation; thus we expect that generically, the two
boundary conditions (2.8) should be enough to specify a unique
extremal. When this is true--as is the case in the above example--and when
an optimal curve is known to exist, we can actually conclude that the unique
extremal gives the optimal solution.
In general, the question of existence
of optimal solutions is not trivial, and the following example
should serve as a warning. (We will come back to this issue later in the context of optimal control.)