Text 8. Hamiltonian Bifurcations
The energy-momentum method has also been used in the context of Hamiltonian bifurcation problems. We shall give some simple examples of this in §1.8. One such context is that of free boundary problems building on the work of Lewis, Marsden, Montgomery, and Ratiu , which gives a Hamiltonian structure for dynamic free boundary problems (surface waves, liquid drops, etc.), generalizing Hamiltonian structures found by Zakharov. Along with the Arnold method itself, this is used for a study of the bifurcations of such problems in Lewis, Marsden, and Ratiu , Lewis [1989, 1992], Kruse, Marsden, and Scheurle , and other references cited therein.
Converse to the Energy—Momentum Method.Because of the block structure mentioned, it has also been possible to prove, in a sense, a converse of the energy-momentum method. That is, if the second variation is indefinite, then the system is unstable. One cannot, of course, hope to do this literally as stated, since there are many systems (e.g., gyroscopic system mentioned earlier – an explicit example is given in Exercise 1.7-4) that are formally unstable, and yet their linearizations have eigenvalues lying on the imaginary axis. Most of these are presumably unstable due to "Arnold diffusion," but of course this is a very delicate situation to prove analytically. Instead, the technique is to show that with the addition of dissipation, the system is destabilized. This idea of dissipation-induced instability goes back to Thomson and Tait in the last century. In the context of the energy-momentum method, Bloch, Krishnaprasad, Marsden, and Ratiu [1994, 1996] show that with the addition of appropriate dissipation, the indefiniteness of the second variation is sufficient to induce linear instability in the problem.
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