Topology and stability of integrable Hamiltonian systems

There is a new survey paper by A. Bolsinov, A. Borisov and I. Mamaev about the topology and stability of integrable Hamiltonian systems in Russian Mathematical Surveys (Vol. 65, 2010, No. 2, pp  259–318). A copy of it can be downloaded here. In this paper, the authors discuss the topology of (finite dimensional) integrable systems, and its applications to the problem of finding stable special (periodic) solutions. Many examples are given in this paper (spinning tops, rigid bodies moving in a fluid, expanding gaseous ellipsoids).

I’m very flattered that in this paper they called my work on topology of integrable Hamiltonian systems “fundamental” (don’t know if anyone else calls it such way). And I find their analysis of the Gaffet system a very interesting application of the topological theory to “real-world” problems (here in astrophysics).

Apparently, even though the topological theory of integrable Hamiltonian systems is more or less well-established now, and can be used to easily check topological properties (e.g. stability) of special solutions of such systems, many people working in mechanics still don’t know it, and have to make heavy computations every time they want to check that some solution is stable (by analytical methods). So the aim of this paper, according to its authors, it to attract people in mechanics to topological methods which can solve their problems more easily.

Some excerpts from the introduction of the paper:

… our principal goal was to present a mathematically rigorous exposition of the general theory and results that is accessible to a wide circle of researchers …

… Poincaré́ was the first to deeply realize the conceptual significance of qualitative topological methods in dynamics on the whole …

… topological methods going back to Poincaré́ were developed by Smale in the paper “Topology and mechanics” (1970) …

… The questions discussed here originate historically from the work of the seminars conducted in the 1980s in the Mechanics-Mathematics Faculty of Moscow State University under the supervision of V. V. Kozlov and A. T. Fomenko…

… Kozlov proposed constructing more invariant objects based on bifurcation diagrams …

… an independent development of questions in the topological analysis of integrable systems was begun by Kharlamov and Pogosian …

… a general ‘Morse theory of integrable systems’  … [first studied by Fomenko and his collaborators]

… Nevertheless, it should be pointed out that so far this theory has not found systematic, effective applications among mechanicians and other specialists in the applied sciences …

… a number of authors who have in our opinion made the greatest contributions to the development of this direction: Duistermaat [27], Eliasson [28], Lerman and Umanskii [29], Cushman and Bates [30], and Delzant [31]. We point out especially the fundamental results of Zung [32], [33] on singularities of integrable (including multidimensional) systems, which go far beyond the framework of the ideas of the theory of topological invariants of integrable systems that are discussed below …

… On the example of the Gaffet system our method was tested in full scope. An additional integral in this problem may be either of third or of sixth degree; it was found analytically by Gaffet. These integrals are so complicated that it is extremely difficult to obtain qualitative information about the solution of the system. By using our method we found new periodic solutions of this problem, which describe pulsating expanding gaseous ellipsoids and are possibly interesting in astrophysics. Furthermore, stability of these solutions was thoroughly investigated …

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