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Itisnaturalto ask whetheror not thesebrackets areinsome senseequivalent. Isitpossible by a coordinate changeto map one intothe other? 96) with the inverse u = sgn (_)v/_. 98) into the KdV bracket yields {F,G} = - 6F 6u Ox @ 6G 6u dx = -13 6F 0 6G _ _. ax 65 _G 0 6_ ax _u dx. 99) and from above, where we changed fluid variables form (v,p,s) to (M,p,a), that sometimes brackets can be mapped into the Lie-Poisson form by an invertible _razsformation. The study of when this can be done is an interesting area that we will not address here.

J. Morrisonand J. M. Greene,Phys. Rev. Lett. 45, 790 (1980); ibid. 48, 509 (1982). tSee, for three-dimensionalvortex dynamics, E. A. Kuznetsov and A. Mikhailov,Phys. Lett. 77A, 37 (1980). 49 .......... 89) This transformation requires the use of the chain rule for functional derivatives, which gives formulas like the following: ,,. 92) Now consider the conditon for the Casimir invariants, {F, C} = 0 for all F. 86) it is seen that this implies v. vs . 6c= 0 6v v6C + (Vxv) p p - x 6v 6C V86C _v p 68 = o.

44) O¢_(a,b) 0b_ = L_C¢C_,b))LT'_(b). 45) is a system of partial differential equations of Mayer-Lie type. Here ¢(a, b) is the unknown and a is a fixed parameter. A similar equation holds where the roles of a and b are reversed. 46) which implies Ob" [L,x¢',a'b//L_l_xb/J =_L ,-- - -- ,- --J 57 ...... 49) Now the left hand side can be made a function of c alone and the right hand side can be made a function of b alone, by multiplying by "L(b)L(b)L-_(c) " with the appropriate indices. 50) must equal the same constant, which is determined by setting c = 0.