Gravity Waves and Finite Turbulent Flow Fields 



where tf is an unspecified ternninal time of integration between 

 points A and B. It is required to find ck'(t) and ts such that the 

 above constraints are satisfied and that the performance index J(Qr) 

 of elapsed time t- is a minimum expressed mathematically. 



[a) = r 



3{a) = \ dt (11) 







Is a minimum. 



From the methods of Ccdculus of variations , the Hamlltonlan 

 of the system is: 



H(x,y,a,\^,\) = 1 + \Ji- V^- C cos or) - \ C sin a (12) 



where \y^ and \y are Lagrange multipliers. The Euler-Lagrange 

 equations are: 



The terminal conditions are: 



^y(tp = 



H(tp = 



(14) 



Since the Hamiltonian is not an explicit function of time, H = and 

 H is a constant. Further, since H = at the terminal condition, 

 then it follows that H = for all < t S t^. 



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