Differential Geometry of Manifolds

2022 №53

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Finding symmetries for the problem of water waves with surface tension



T. Brooke Benjamin and P. J. Olver “Hamiltonian structure, symmet­ries and conservation laws for water waves” study the behavior of Hamil­to­nian systems with an infinite-dimensional phase space. The methods of va­riational problems and infinite-dimensional differential geometry are applicable to this problem. A special case of the problem is an abstract prob­lem of hydrodynamics for an ideal fluid. Its configuration space is the group of volume-preserving diffeomorphisms of some manifold in  or  filled with fluid. Even more special is the problem of waves on water. Its non-standard nature is due to the presence of boundary con­di­tions on the free surface. These boundary conditions can be interpreted in terms of the functional derivatives of the energy integral, which plays the role of the Hamiltonian. Here we consider in detail the case of this prob­lem in R2, taking into account surface tension, and find symmetries for it, which was not considered in detail in the article. Finding symmet­ries can be achieved without recourse to the Hamiltonian structure of the gi­ven problem.


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