17. The multiplier rule

The reader has been told that the great twentieth-century quests in the calculus of variations have involved existence and multiplier rules. Progress in functional analysis, together with the direct method, has largely resolved the former issue; we turn now to that of multipliers. We consider the classical problem of Lagrange for this purpose. It consists of the basic problem (P) to which has been grafted an additional pointwise equality constraint φ(t,x,x ′) = 0. The additional constraint makes this problem much more complex than (P), or even the isoperimetric problem. In part, this is because we now have infinitely many constraints, one for each t. Given our experience in optimization, we expect the multiplier rule to assert, in a now familiar pattern, that if x _∗ solves this problem, then there exist multipliers η,λ, not both zero, with η= 0 or 1, such that x _∗ satisfies the necessary conditions for the Lagrangian ηΛ+λ(t)φ. Note that λ is a function of t here, which is to be expected, since there is a constraint φ(t,x,x ′) = 0 for each t.