Geodesic
Topology of domains of possible motions of integrable systems
Sbornik. Mathematics, Tome 187 (1996) no. 5, pp. 679-684 
V. V. Kozlov; V. V. Ten. Topology of domains of possible motions of integrable systems. Sbornik. Mathematics, Tome 187 (1996) no. 5, pp. 679-684. http://geodesic.mathdoc.fr/item/SM_1996_187_5_a2/
@article{SM_1996_187_5_a2,
     author = {V. V. Kozlov and V. V. Ten},
     title = {Topology of domains of possible motions of integrable systems},
     journal = {Sbornik. Mathematics},
     pages = {679--684},
     year = {1996},
     volume = {187},
     number = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1996_187_5_a2/}
}
TY  - JOUR
AU  - V. V. Kozlov
AU  - V. V. Ten
TI  - Topology of domains of possible motions of integrable systems
JO  - Sbornik. Mathematics
PY  - 1996
SP  - 679
EP  - 684
VL  - 187
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/SM_1996_187_5_a2/
LA  - en
ID  - SM_1996_187_5_a2
ER  - 
%0 Journal Article
%A V. V. Kozlov
%A V. V. Ten
%T Topology of domains of possible motions of integrable systems
%J Sbornik. Mathematics
%D 1996
%P 679-684
%V 187
%N 5
%U http://geodesic.mathdoc.fr/item/SM_1996_187_5_a2/
%G en
%F SM_1996_187_5_a2

Voir la notice de l'article provenant de la source Math-Net.Ru

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 $k\geqslant 1$.

[1] Kozlov V. V., “Variatsionnoe ischislenie v tselom i klassicheskaya mekhanika”, UMN, 40:2 (1985), 33–60 | MR | Zbl

[2] Kozlov V. V., “Topologicheskie prepyatstviya k integriruemosti naturalnykh mekhanicheskikh sistem”, DAN SSSR, 249:6 (1979), 1299–1302 | MR | Zbl

[3] Kolokoltsov V. N., “Geodezicheskie potoki na dvumernykh mnogoobraziyakh s dopolnitelnym po skorosti pervym integralom”, Izv. AN SSSR. Ser. matem., 46:5 (1982), 994–1010 | MR | Zbl

[4] Taimanov I. A., “Topologicheskie prepyatstviya k integriruemosti geodezicheskikh potokov na neodnosvyaznykh mnogoobraziyakh”, Izv. AN SSSR. Ser. matem., 51:2 (1987), 429–435 | MR

[5] Bolotin S. V., “Dvoyakoasimptoticheskie traektorii minimalnykh geodezicheskikh”, Vestn. MGU. Ser. 1. Matem., mekh., 1992, no. 1, 92–96 | MR

[6] Paternain G. P., “On the topology of manifolds with completely integrable geodesic flows”, Ergod. Th. and Dynam. Sys., 12 (1992), 109–121 | DOI | MR

[7] Bolotin S. V., “Neintegriruemost zadachi $n$ tsentrov pri $n>2$”, Vestn. MGU. Ser. 1. Matem., mekh., 1984, no. 3, 65–68 | MR

[8] Kozlov V. V., “Integriruemost i neintegriruemost v gamiltonovoi mekhanike”, UMN, 38:1 (1983), 3–67 | MR | Zbl

[9] Abrarov D. L., “Topologicheskie prepyatstviya k suschestvovaniyu uslovno-lineinykh integralov”, Vestn. MGU. Ser. 1. Matem., mekh., 1984, no. 6, 72–75 | MR | Zbl

[10] Bolotin S. V., “O pervykh integralakh sistem s giroskopicheskimi silami”, Vestn. MGU. Ser. 1. Matem., mekh., 1984, no. 6, 75–82 | MR | Zbl