Lewis, D. and Marsden, J. and Montgomery, R. and Ratiu, T. (1986) The Hamiltonian structure for dynamic free boundary problems. Physica D, 18 (1-3). pp. 391-404. ISSN 0167-2789 http://resolver.caltech.edu/CaltechAUTHORS:20100823-111237772
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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.
|Additional Information:||© 1986, Elsevier Science Publishers. Available online 16 September 2002. We thank Henry Abarbanel, Brooke Benjamin, Darryl Holm, Alan Newell, Peter Olver, Harvey Segur, and Alan Weinstein for their comments, interest, and encouragement.|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Ruth Sustaita|
|Deposited On:||23 Aug 2010 19:56|
|Last Modified:||26 Dec 2012 12:21|
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