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 Mat. Sb., 1996, Volume 187, Number 5, Pages 59–64 (Mi msb127)

Topology of domains of possible motions of integrable systems

V. V. Kozlov, V. V. Ten

M. V. Lomonosov Moscow State University

Abstract: A study is made of analytic invertible systems with two degrees of freedom on a fixed three-dimensional manifold of level of the energy integral. It is assumed that the manifold in question is compact and has no singular points (equilibria of the initial system). The natural projection of the energy manifold onto the two-dimensional configuration space is called the domain of possible motion. In the orientable case it is sphere with $k$ holes and $p$ attached handles. It is well known that for $k=0$ and $p\geqslant 2$, the system possesses no non-constant analytic integrals on the corresponding level of the energy integral. The situation in the case of domains of possible motions with a boundary turns out to be very different. The main result can be stated as follows: there are examples of analytically integrable systems with arbitrary values of $p$ and $k\geqslant 1$.

DOI: https://doi.org/10.4213/sm127

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English version:
Sbornik: Mathematics, 1996, 187:5, 679–684

Bibliographic databases:

UDC: 517.9+531.01
MSC: Primary 58F05; Secondary 70G25, 70F10

Citation: V. V. Kozlov, V. V. Ten, “Topology of domains of possible motions of integrable systems”, Mat. Sb., 187:5 (1996), 59–64; Sb. Math., 187:5 (1996), 679–684

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• http://mi.mathnet.ru/eng/msb127
• https://doi.org/10.4213/sm127
• http://mi.mathnet.ru/eng/msb/v187/i5/p59

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This publication is cited in the following articles:
1. Bolotin, SV, “A variational construction of chaotic trajectories for a Hamiltonian system on a torus”, Bollettino Della Unione Matematica Italiana, 1B:3 (1998), 541
2. Bertotti, ML, “Chaotic trajectories for natural systems on a torus”, Discrete and Continuous Dynamical Systems, 9:5 (2003), 1343
3. Rudnev, M, “Integrability versus topology of configuration manifolds and domains of possible motions”, Archiv der Mathematik, 86:1 (2006), 90
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