Page 164 - Basic Structured Grid Generation

Variational methods and adaptive grid generation 153

of the variational approach [see Liseikin (1999) for detailed references] has been that
it can facilitate intuitive control over these various factors, while at the same time
providing firm theoretical foundations.
In the following section we review some basic ideas of variational calculus.

6.2 Euler-Lagrange equations

We have already considered one application of variational methods in Chapter 3, where
an outline of the derivation of the general equations for geodesics on surfaces in E 3
was presented. These eqns (3.67) are necessary conditions that geodesics must satisfy
in order to make the integral in eqn (3.66) stationary. Such integrals, which depend
1
2
for their value on the particular functions, u (t) and u (t) in that case, being substi-
tuted into the integrand, are called functionals in the classical Calculus of Variations.
Sufficient conditions for maxima or minima are, however, usually more difficult to
establish than in the corresponding calculus of functions. Thus we may be able to
solve the Euler-Lagrange equations (the necessary conditions) to obtain solution func-
tions, which are called extremals, but it is not usually easy to establish their nature. It
may be pointed out, in addition, that variational problems may have no solution. For
example, the functional may be bounded below while at the same time a function that
actually minimizes it may not exist.
Here we give a brief account of the main types of variational problems that we shall
encounter, with a discussion of the corresponding Euler-Lagrange equations.
1
2
In Chapter 3 we met the functional of the form, writing x 1 , x 2 in place of u , u ,
b
I = F(x 1 ,x 2 , ˙x 1 , ˙x 2 ) dt,
a
where F is regarded formally as a function of four independent variables, and the
functions x 1 and x 2 are required to take prescribed values at the ends t = a and
b of the interval of integration. The Euler-Lagrange equations are the two ordinary
differential equations
d ∂F ∂F
− = 0,
dt ∂ ˙x 1 ∂x 1
d ∂F ∂F
− = 0. (6.1)
dt ∂ ˙x 2 ∂x 2
Since in general the terms ∂F/∂ ˙x 1 and ∂F/∂ ˙x 2 contain terms in ˙x 1 and ˙x 2 ,these
are two second-order differential equations, and there are four prescribed boundary
conditions to be satisfied.
The argument sketched in Chapter 3 also applies to the case

b
I = F(x 1 ,x 2 , ˙x 1 , ˙x 2 ,t) dt,
a
and the conclusions, eqns (6.1), are the same.