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 Mat. Sb., 1993, Volume 184, Number 9, Pages 125–148 (Mi msb1015)

Symmetries and the topology of dynamical systems with two degrees of freedom

V. V. Kozlov, N. V. Denisova

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The problem of geodesic curves on a closed two-dimensional surface and some of its generalizations related with the addition of gyroscopic forces are considered. The authors study one-parameter groups of symmetries in the four-dimensional phase space that are generated by vector fields commuting with the original Hamiltonian vector field. If the genus of the surface is greater than one, then there are no nontrivial symmetries. For a surface of genus one (a two-dimensional torus) it is established that if there is an additional integral polynomial in the velocities, even or odd with respect to each component of the velocity, then there is a polynomial integral of degree one or two. For a surface of genus zero examples of nontrivial integrals of degree three and four are given. Fields of symmetries of first and second degree are studied. The presence of such symmetries is related to the existence of ignorable cyclic coordinates and separated variables. The influence of gyroscopic forces on the existence of fields of symmetries with polynomial components is studied.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1995, 80:1, 105–124

Bibliographic databases:

Document Type: Article
UDC: 517.9+531.01
MSC: Primary 70H33, 70H05; Secondary 70E15, 58F17, 58F05

Citation: V. V. Kozlov, N. V. Denisova, “Symmetries and the topology of dynamical systems with two degrees of freedom”, Mat. Sb., 184:9 (1993), 125–148; Russian Acad. Sci. Sb. Math., 80:1 (1995), 105–124

Citation in format AMSBIB
\Bibitem{KozDen93} \by V.~V.~Kozlov, N.~V.~Denisova \paper Symmetries and the~topology of dynamical systems with two degrees of freedom \jour Mat. Sb. \yr 1993 \vol 184 \issue 9 \pages 125--148 \mathnet{http://mi.mathnet.ru/msb1015} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1257339} \zmath{https://zbmath.org/?q=an:0818.70018} \transl \jour Russian Acad. Sci. Sb. Math. \yr 1995 \vol 80 \issue 1 \pages 105--124 \crossref{https://doi.org/10.1070/SM1995v080n01ABEH003516} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995QH35500006} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. V. Kozlov, N. V. Denisova, “Polynomial integrals of geodesic flows on a two-dimensional torus”, Russian Acad. Sci. Sb. Math., 83:2 (1995), 469–481
2. A. V. Bolsinov, V. V. Kozlov, A. T. Fomenko, “The Maupertuis principle and geodesic flows on the sphere arising from integrable cases in the dynamics of a rigid body”, Russian Math. Surveys, 50:3 (1995), 473–501
3. Bolotin S., Kozlov V., “Symmetry Fields of Geodesic Flows”, Russ. J. Math. Phys., 3:3 (1995), 279–295
4. Kozlov V., “Symmetries and Regular Behavior of Hamiltonian Systems”, Chaos, 6:1 (1996), 1–5
5. N. V. Denisova, “The structure of infinitesimal symmetries of geodesic flows on a two-dimensional torus”, Sb. Math., 188:7 (1997), 1055–1069
6. Anikeev P., “On the Second Degree Fields of Symmetry for an Impulse of Geodesic Flows on the Two-Dimensional Sphere”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1997, no. 4, 29–32
7. Denisova N., “Polynomial Fields of the Third Degree Symmetries of Geodesic Flows on a Two-Dimensional Torus”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1998, no. 2, 48–53
8. V. V. Ten, “Polynomial first integrals for systems with gyroscopic forces”, Math. Notes, 68:1 (2000), 135–138
9. V. S. Kalnitsky, “Symmetries of a flat cosymbol algebra of the differential operators”, J. Math. Sci. (N. Y.), 222:4 (2017), 429–436
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