Variational Approaches to Steady Ship Wave Problems 



which becomes 



E=iyy (cf)- ct>o)(cf>^- (j>J dS= Jo - J, (2.14) 



by Green's theorem. Here, 



Jo=iyy Mou^S (2.15) 



is the correct value. We see clearly that 



6E = - 6J. (2.16) 



Since E is non-negative, we have the inequality [ 10] 



Jo^ J. (2.17) 



It is well-known that among all functions ^ having a finite 

 energy integral, 



T =1^^^ [Vcj>]^ dr, (2.18) 



and a given normal derivative on S, the one which minimizes T is 

 a harmonic function [ 1,4] . Accordingly, if we solve this minimiza- 

 tion problem, say by the relaxation method, we have the inequality 



T > J^. [1,4] (2.19) 



This is the dual of (2.17) and we now have the variational problem 

 (2.7) as an involutory transformation of the latter minimization prob- 

 lem. (See, for example, the textbook on variational calculus [ 11] .) 



III. A VARIATIONAL PROBLEM FOR THE LAGRANGIAN 



The preceding principle can be easily extended to flow in a 

 gravitational field. Let us consider the functional 



L = T - V, (3.1) 



where 



551 



