Speciality:
01.02.01 (Theoretical mechanics)
E-mail: 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.
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.
