Page 165 - Basic Structured Grid Generation
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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.
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