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The Hamiltonian structure for dynamic free boundary problems. (English) Zbl 0638.58044

Authors’ summary: “Hamiltonian structures for 2- or 3-dimensional incompressible flows with a free boundary are determined which generalize a previous structure of Zakharov for irrotational flow. Our Poisson bracket is determined using the method of Arnold, namely reduction from canonical variables in the Lagrangian (material) description. Using this bracket, the Hamiltonian form for the equations of a liquid drop with a free boundary having surface tension is demonstrated. The structure of the bracket in terms of a reduced cotangent bundle of a principal bundle is explained. In the case of two-dimensional flows, the vorticity bracket is determined and the generalized enstrophy is shown to be a Casimir function. This investigation also clears up some confusion in the literature concerning the vorticity bracket, even for fixed boundary flows.”
Reviewer: N.Jacob

MSC:

58Z05 Applications of global analysis to the sciences
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
76D99 Incompressible viscous fluids
76A02 Foundations of fluid mechanics
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References:

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