ABSTRACT
J b aGd(by) dx = _Jb~ (aG)by dx + aG by1b (12.8) a ay' dx a dx ay' ay' a
where the last term in Eq. (12.8) is zero since by = 0 at the boundaries for fixed end points. Substituting Eq. (12.8) into Eq. (12.7) and setting (jJ = 0 gives
J b[aG d (aG)](jJ = - - - - by dx = 0 a ay dx ay'
(12.9)
Equation (12.9) must be satisfied for arbitrary distributions of by, which requires that
aG d (aG) --- - -0 ay dx ay' - (12.10)
Equation (12.10) is known as the Euler equation of the calculus of variations. How does the calculus of variations relate to the solution of a boundary-value
ordinary differential equation? To answer this question, consider the following simple linear boundary-value problem with Dirichlet boundary conditions:
where Q = Q(x) and F = F(x). The problem is to determine a functional I[Y(x)] whose extremum (i.e., minimum or maximum) is precisely Eq. (12.11). If such a functional can be found, extremizing that functional yields the solution to Eq. (12.11).
