ABSTRACT
Minimize J[ y ] b a
F ( x, y(x), y′(x)
) dx
Subject to y(a) = ya y(b) = yb
⎫ ⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
(14.1)
or subject to conditions on or including y′(a), y′(b). Also, the integral may involve higher order derivatives of y. Often the problem has a natural interpretation in terms of energy in a system. In Chapter 13 we studied minimization of a function that depends on several inde-
pendent variables, that is, unknowns to be solved for. In Chapter 14 we will minimize an integral which depends on a function which is the unknown to be solved for. That is inherently a more mind boggling problem, but results from Chapter 13 will still be relevant. A function is admissible if it is continuous and piecewise continuously differentiable
on the interval [ a, b ]. If higher order derivatives are in the integrand then the class of admissible functions may be further restricted to involve higher order differentiability. A functional is a mapping from a vector space to scalar values. For example, in (14.1)
the mapping
F ( x, y(x), y′(x)
) dx
is a functional.
