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Denisova, Natalia Victorovna

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Total publications: 12
Scientific articles: 10

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Candidate of physico-mathematical sciences (1998)
Speciality: 01.02.01 (Theoretical mechanics)
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Keywords: dynamical systems, symmetries, the integrability problem for dynamics equations, hydrodynamics.

Subject:

V. V. Kozlov and N. V. Denisova studied the one-parameter groups of symmetries in the four-dimensional phase space that are generated by the vector fields commuting with the original Hamiltonian vector field whose Hamiltonian is quadratic in the momenta. Because of homogeneity, it is possible to restrict the consideration to polynomial fields of symmetries: their components are polynomials in the momenta. It was earlier established by V. V. Kozlov that if the genus of the configuration space surface is greater than one, then there exist no non-trivial symmetries. As proven by V. V. Kozlov and N. V. Denisova, for a surface of genus one (a two-dimensional torus), the first degree fields of symmetries are always Hamiltonian. Moreover, the latter fields are necessarily Noetherian and therefore a hidden cyclic coordinate exists. The second degree fields of symmetries are Hamiltonian only if the Gaussian curvature of the metric defined by the kinetic energy is not equal to zero. In this case, there is a quadratic integral, and, in the case of two degrees of freedom, the resulting equations are solved by using the method of separated variables. The structure of the symmetry fields of degree 3 and 4 is studied for Hamiltonian dynamical systems whose configuration space is a two-dimensional torus. A surprising connection between the degree of an irreducible additional integral and the topology of the configuration space of a mechanical system was discovered. The following hypotheses were stated. For the case of a two-dimensional sphere (its genus is equal to 0), the degree of an irreducible integral does not exceed 4. The integral of degree 3 corresponds to the Goryachev–Chaplygin case, and the integral of degree 4 is the Kowalevskaya integral, from the rigid body dynamics. For the case of a two-dimensional torus (its genus is equal to 1), the degree of an irreducible integral does not exceed 2. Notice that, as was earlier established by V. V. Kozlov, if the two-dimensional surface genus is greater than 1, then the mechanical system does not generally admit an additional non-constant integral. N. V. Denisova obtained constructive criteria for the existence of the conditional linear and quadratic integrals on a two-dimensional torus. The problem considered by N. V. Denisova is that of finding conditions ensuring that a reversible Hamiltonian system has integrals polynomial in the momenta. The kinetic energy is a zero-curvature Riemannian metric, and the potential is a smooth function on a two-dimensional torus.

Biography

1989–1994: student MSU, Dept. of Mechanics and Mathematics; 1994–1997: graduated MSU, Dept. of Mechanics and Mathematics (Ph. D. course); 1998: the Ph. D. thesis; 1997–1999: junior scientist of MSU, Dept. of Mechanics and Mathematics; 1999–present: assistant professor at the Chair of Mathematical Statistics and Random Processes Theory, Dept. of Mechanics and Mathematics, MSU. I teach the discussion section on the following subjects: probability theory, mathematical statistics, and random processes theory.

In 2001: the medal and premium of the Russian Academy of Sciences for young scientists, received for the publication cycle named "Symmetries and laws of conservation for dynamics equations".

   
Main publications:
  • Kozlov V. V., Denisova N. V. Simmetrii i topologiya dinamicheskikh sistem s dvumya stepenyami svobody // Matem. sb., 1993, 184(9), 125–148.
  • Kozlov V. V., Denisova N. V. Polinomialnye integraly geodezicheskikh potokov na dvumernom tore // Matem. sb., 1994, 185(12), 49–64.
  • Denisova N. V. O strukture polei simmetrii geodezicheskikh potokov na dvumernom tore // Matem. sb., 1997, 188(7), 107–122.
  • Denisova N. V. Polinomialnye po skorosti integraly dinamicheskikh sistem s dvumya stepenyami svobody i toricheskim konfiguratsionnom prostranstvom // Matem. zametki, 1998, 64(1), 37–44.
  • Denisova N. V., Kozlov V. V. Polinomialnye integraly obratimykh mekhanicheskikh sistem s konfiguratsionnym prostranstvom v vide dvumernogo tora // Matem. sb., 2000, 191(2), 43–63.

http://www.mathnet.ru/eng/person8533
List of publications on Google Scholar
List of publications on ZentralBlatt
https://mathscinet.ams.org/mathscinet/MRAuthorID/345904

Publications in Math-Net.Ru
2020
1. N. V. Denisova, “On Momentum-Polynomial Integrals of a Reversible Hamiltonian System of a Certain Form”, Trudy Mat. Inst. Steklova, 310 (2020),  143–148  mathnet  mathscinet  elib; Proc. Steklov Inst. Math., 310 (2020), 131–136  isi  scopus
2012
2. N. V. Denisova, V. V. Kozlov, D. V. Treschev, “Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space”, Izv. RAN. Ser. Mat., 76:5 (2012),  57–72  mathnet  mathscinet  zmath  elib; Izv. Math., 76:5 (2012), 907–921  isi  elib  scopus
2002
3. N. V. Denisova, “Tensor invariants of dynamical systems, and steady flows of a viscous fluid”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2002, 3,  40–45  mathnet  mathscinet  zmath
2000
4. N. V. Denisova, V. V. Kozlov, “Polynomial integrals of reversible mechanical systems with a two-dimensional torus as the configuration space”, Mat. Sb., 191:2 (2000),  43–63  mathnet  mathscinet  zmath; Sb. Math., 191:2 (2000), 189–208  isi  scopus
1999
5. N. V. Denisova, V. V. Kozlov, “On the chaotization of the oscillations of coupled pendulums”, Dokl. Akad. Nauk, 367:2 (1999),  191–193  mathnet  mathscinet; Dokl. Math., 44:7 (1999), 466–468  isi
1998
6. N. V. Denisova, “Integrals polynomial in velocity for two-degrees-of-freedom dynamical systems whose configuration space is a torus”, Mat. Zametki, 64:1 (1998),  37–44  mathnet  mathscinet  zmath; Math. Notes, 64:1 (1998), 31–37  isi
7. N. V. Denisova, “Polynomial fields of third degree symmetries of geodesic flows on a two-dimensional torus”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1998, 2,  48–53  mathnet  mathscinet  zmath
1997
8. N. V. Denisova, “The structure of infinitesimal symmetries of geodesic flows on a two-dimensional torus”, Mat. Sb., 188:7 (1997),  107–122  mathnet  mathscinet  zmath; Sb. Math., 188:7 (1997), 1055–1069  isi  scopus
1995
9. N. V. Denisova, “Polynomial integrals and the branching of solutions of reversible dynamical systems on a sphere”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1995, 2,  79–82  mathnet  mathscinet  zmath
1994
10. V. V. Kozlov, N. V. Denisova, “Polynomial integrals of geodesic flows on a two-dimensional torus”, Mat. Sb., 185:12 (1994),  49–64  mathnet  mathscinet  zmath; Russian Acad. Sci. Sb. Math., 83:2 (1995), 469–481  isi
1993
11. 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  mathnet  mathscinet  zmath; Russian Acad. Sci. Sb. Math., 80:1 (1995), 105–124  isi

2011
12. I. V. Astashova, A. V. Borovskikh, V. V. Bykov, A. Yu. Goritskii, N. V. Denisova, V. V. Zhikov, Yu. S. Ilyashenko, T. O. Kapustina, V. V. Kozlov, A. A. Kon'kov, I. V. Matrosov, E. V. Radkevich, O. S. Rozanova, È. R. Rozendorn, N. Kh. Rozov, M. S. Romanov, I. N. Sergeev, I. V. Filimonova, A. V. Filinovskii, G. A. Chechkin, A. S. Shamaev, T. A. Shaposhnikova, “Olga Arsenjevna Oleinik”, Tr. Semim. im. I. G. Petrovskogo, 28 (2011),  5–7  mathnet  elib; J. Math. Sci. (N. Y.), 179:3 (2011), 345–346
2007
13. I. V. Astashova, L. A. Bagirov, A. V. Borovskikh, V. V. Bykov, A. N. Vetokhin, A. Yu. Goritskii, G. V. Grishina, N. V. Denisova, Yu. S. Ilyashenko, T. O. Kapustina, V. V. Kozlov, A. A. Kon'kov, I. V. Matrosov, V. M. Millionshchikov, V. A. Nikishkin, E. V. Radkevich, O. S. Rozanova, È. R. Rozendorn, N. Kh. Rozov, V. A. Sadovnichii, V. S. Samovol, I. N. Sergeev, I. V. Filimonova, A. V. Filinovskii, A. F. Filippov, T. S. Khachlaev, G. A. Chechkin, A. S. Shamaev, T. A. Shaposhnikova, “Vladimir Alexandrovich Kondratiev on the 70th anniversary of his birth”, Tr. Semim. im. I. G. Petrovskogo, 26 (2007),  5–28  mathnet  mathscinet; J. Math. Sci. (N. Y.), 143:4 (2007), 3183–3197

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