Page 165 - Basic Structured Grid Generation
P. 165
154 Basic Structured Grid Generation
It is also straightforward to generalize the argument to cover the case
b
I = F(x 1 ,x 2 ,.. . , x n , ˙x 1, ˙x 2 ,..., ˙x n ,t) dt (6.2)
a
for n extremal functions x 1 (t), x 2 (t), . . . , x n (t), to be determined subject to the condi-
tions that they make I stationary and that they satisfy certain prescribed end conditions.
The Euler-Lagrange equations are
d ∂F ∂F
− = 0, i = 1, 2,... ,n, (6.3)
dt ∂ ˙x i ∂x i
n second-order differential equations to be satisfied subject to 2n boundary conditions.
Of course, this also holds for the case n = 1.
The differentiation operators appearing in eqn (6.3) have to be interpreted carefully.
The partial derivatives involve formal differentiation of F as a function of (2n + 1)
variables, while the ordinary (or total) derivative operator d/dt acting in the first term
regards everything as a function of the single variable t.
In the case where F does not depend explicitly on t, we can infer from the ‘total
derivative’ Chain Rule and eqn (6.3) that extremals satisfy the equation
n n
dF ∂F dx i ∂F d˙x i
= +
dt ∂x i dt ∂ ˙x i dt
i=1 i=1
n
n
d ∂F dx i ∂F d˙x i d ∂F
= + = ˙ x i .
dt ∂ ˙x i dt ∂ ˙x i dt dt ∂ ˙x i
i=1 i=1
Hence we obtain
n
d ∂F
F − ˙ x i = 0. (6.4)
dt ∂ ˙x i
i=1
A further generalization of (6.2) is the functional
b
I = F(x 1 ,x 2 ,...,x n , ˙x 1 , ˙x 2 ,.. . , ˙x n , ¨x 1 , ¨x 2 ,..., ¨x n ,t) dt, (6.5)
a
where the unknown functions x 1 (t), ...,x n (t) have their values and the values of their
derivatives prescribed at the end-points. Variations δx 1 ,...,δx n (all functions of t)in
these functions from some supposed extremal functions should produce a variation in
I which is zero to first order. Thus, using first-order increment formulas,
n n n
b
∂F ∂F ∂F
b
0 = δI = δF dt = δx i + δ ˙x i + δ ¨x i dt
a a ∂x i ∂ ˙x i ∂ ¨x i
i=1 i=1 i=1
∂F d ∂F d ∂F
n
b
= − δx i − δ ˙x i dt
a ∂x i dt ∂ ˙x i dt ∂ ¨x
i=1
after integration by parts, the integrated parts vanishing because all δx i and δ ˙x i , i =
1, 2,...,n, vanish at the ends because of the boundary conditions. Note that for the
integration by parts we assume that δ(˙x i ) = d(δx i )/dt and δ(¨x) = d(δ ˙x)/dt.