L .H ;- -- to ,E 11September 1995 PHYSICS LETTERS A FLSEVIER Physics Letters A 205 (1995) 149-157 The Poisson bracket for 2D hydrodynamics reduces to the Gardner bracket L. Piterbarg ’ Center for Applied Mathematical Sciences, Universiiy of Southern California, Los Angeles. CA 90089-I 113. USA Received 20 March 1995; accepted for publication 19 July 1995 Communicated by AI? Fordy lbstract The Hamiltonian structure related to the 2D hydrodynamics of the ideal incompressible fluid is studied. Solutions whose vorticities have a unique extremum are considered. For this class of solutions the corresponding Poisson bracket is reduced to he Gardner-Zakharov-Faddeev bracket by changing the functional variable. The new coordinates have a clear geometrical neaning: Cartesian coordinates of the extremum and the set of distances from it to the vorticity lines. The motion of an ideal incompressible fluid in the plane is governed by the following equation, -where g(r) is the stream function, r = (x, y); 0 = 021c, is the vorticity and J(f,g) = jXgY - f\,gX is the Jacobian. We consider functions O(r) decaying rapidly enough if jr/ goes to infinity. Under this condition Eq. (1) can be written in Hamiltonian form as follows [ 11, where the Hamiltonian and the Poisson bracket are given by and ’ This work was supported by ONR Grant No. NO0014 -91-J-1526 Elsevier Science B.V. SSDI 0375-9601(95)00542-O