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Nelin. Dinam., 2010, Volume 6, Number 4, Pages 769–805 (Mi nd5)  

This article is cited in 10 scientific papers (total in 10 papers)

Topological analysis and Boolean functions. I. Methods and application to classical systems

M. P. Kharlamov

Volgograd Academy of Public Administration

Abstract: We aim to completely formalize the rough topological analysis of integrable Hamiltonian systems admitting analytical solutions such that the initial phase variables along with the time derivatives of the auxiliary variables are expressed as rational functions (in fact, as polynomials) in some set of radicals depending on one variable each. We suggest a method to define the admissible regions in the integral constants space, the segments of oscillation of the separated variables and the number of connected components of integral manifolds and critical integral surfaces. This method is based on some algorithms of processing the tables of some Boolean vector-functions and of reducing the matrices of linear Boolean vector-functions to some canonical form. From this point of view we consider here the topologically richest classical problems of the rigid body dynamics. The article will be continued with the investigation of some new integrable problems.

Keywords: algebraic separation of variables, integral manifolds, Boolean functions, topological analysis, algorithms.

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UDC: 517.938.5:531.38+519.6
MSC: 70E17, 70G40
Received: 27.05.2010

Citation: M. P. Kharlamov, “Topological analysis and Boolean functions. I. Methods and application to classical systems”, Nelin. Dinam., 6:4 (2010), 769–805

Citation in format AMSBIB
\Bibitem{Kha10}
\by M.~P.~Kharlamov
\paper Topological analysis and Boolean functions. I.~Methods and application to classical systems
\jour Nelin. Dinam.
\yr 2010
\vol 6
\issue 4
\pages 769--805
\mathnet{http://mi.mathnet.ru/nd5}


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    This publication is cited in the following articles:
    1. Kharlamov M.P., Savushkin A.Yu., “Geometricheskii podkhod k razdeleniyu peremennykh v mekhanicheskikh sistemakh”, Vestn. Volgogradskogo gos. un-ta. Ser. 1: Matem. Fiz., 2010, no. 13, 47–74  elib
    2. M. P. Kharlamov, “Topologicheskii analiz i bulevy funktsii: II. Prilozheniya k novym algebraicheskim resheniyam”, Nelineinaya dinam., 7:1 (2011), 25–51  mathnet
    3. Ryabov P.E., “Explicit integration and topology of D. N. Goryachev case”, Dokl. Math., 84:1 (2011), 502–505  crossref  mathscinet  zmath  isi  elib  elib  scopus
    4. S. S. Nikolaenko, “The number of connected components in the preimage of a regular value of the momentum mapping for the geodesic flow on ellipsoid”, Moscow University Mathematics Bulletin, 68:5 (2013), 241–245  mathnet  crossref  mathscinet
    5. S. S. Nikolaenko, “A topological classification of the Chaplygin systems in the dynamics of a rigid body in a fluid”, Sb. Math., 205:2 (2014), 224–268  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. P. E. Ryabov, “The phase topology of a special case of Goryachev integrability in rigid body dynamics”, Sb. Math., 205:7 (2014), 1024–1044  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. S. S. Nikolaenko, “Topological classification of the Goryachev integrable case in rigid body dynamics”, Sb. Math., 207:1 (2016), 113–139  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    8. Mikhail P. Kharlamov, Pavel E. Ryabov, Alexander Yu. Savushkin, “Topological Atlas of the KowalevskiSokolov Top”, Regul. Chaotic Dyn., 21:1 (2016), 24–65  mathnet  crossref  mathscinet  zmath
    9. Nikolaenko S.S., “Topological Classification of the Goryachev Integrable Systems in the Rigid Body Dynamics: Non-Compact Case”, Lobachevskii J. Math., 38:6 (2017), 1050–1060  crossref  mathscinet  zmath  isi  scopus
    10. I. F. Kobtsev, “Geodesic flow of a 2D ellipsoid in an elastic stress field: topological classification of solutions”, Moscow University Mathematics Bulletin, 73:2 (2018), 64–70  mathnet  crossref  isi
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