Optimum Shapes of Bodtes in Free Surface Flows 
The original problem is equivalent to the minimization of a new func- 
tional 
I [Y, es), teas ae = i Ie = Ay ) , (13) 
where dg 's are undetermined Lagrange multipliers. 
We next seek the necessary conditions of optimality. Let 
[(&) denote the required optimal function which, together with its 
conjugate function B6(£) given by (9), minimizes I [T, B |. We 
further let 6 I'(&) denote an admissible variation of [ (£), 
which is Holder continuous, satisfies the isoperimetric constraints 
(12) and the end conditions (6). The corresponding variation in 
B(&) is found from (9) as 
6p(&) = -H,[6P] eet oes (14) 
The variation of the functional I due to the variations 6T 
and 6B is 
AW. = pleat sta dle poset 8.5 Seat OC. | - bl RoR ce 
(15) 
where 6c,,'s are variations of parameters c,. For sufficiently 
small [6 r| ; |6 B | and [aca] , expansion of the above integrand in 
Taylors'series yields 
1 z 
ARE Ve OYE ea 6 I + at 6°I FURS (16) 
2 1 
: eal Rebs =a 2 
where the first variation 61 andthe second variation 6 I are 
1 
l 
hee [ ot a 66 Jatt sc [tars separ. (17) 
-l 
Lit? 
