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2013
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# Dmitriy Zotev

Department of Mathematics, Volgograd State Pedagogical University

## Publications:

 Zotev D. B. On a Partial Integral which can be Derived from Poisson Matrix 2007, vol. 12, no. 1, pp.  81-85 Abstract Consider a surface which is a common level of some functions. Suppose that this surface is invariant under a Hamiltonian system. The question is if a partial integral can be derived explicitly from the Poisson matrix of these functions. In some cases such an integral is equal to the determinant of the matrix. This paper establishes a necessary and sufficient condition for this to hold true. The partial integral that results is not trivial if the induced Poisson structure is non-degenerate at one point at least. Therefore, the invariant surface must be even-dimensional. Keywords: Hamiltonian system, invariant submainfold, partial integral, Poisson matrix determinant, trace matrix Citation: Zotev D. B.,  On a Partial Integral which can be Derived from Poisson Matrix, Regular and Chaotic Dynamics, 2007, vol. 12, no. 1, pp. 81-85 DOI:10.1134/S1560354707010078
 Kharlamov M. P., Zotev D. B. Non-degenerate energy surfaces of rigid body in two constant fields 2005, vol. 10, no. 1, pp.  15-19 Abstract The problem of motion of a rigid body in two constant fields is considered. The motion is described by the Hamiltonian system with three degrees of freedom. This system in general case does not have any explicit symmetry groups and, therefore, cannot be reduced to a family of systems with two degrees of freedom. The critical points of the energy integral are found. It appeared that the energy of the system is a Morse function and has exactly four distinct critical points with different critical values and Morse indexes 0,1,2,3. In particular, the body has four equilibria, only one of which is stable. Basing on the Morse theory the smooth type of 5-dimensional non-degenerate iso-energetic manifolds is pointed out. Keywords: rigid body, two constant fields, iso-energetic manifolds Citation: Kharlamov M. P., Zotev D. B.,  Non-degenerate energy surfaces of rigid body in two constant fields, Regular and Chaotic Dynamics, 2005, vol. 10, no. 1, pp. 15-19 DOI:10.1070/RD2005v010n01ABEH000296
 Zotev D. B. Fomenko–Zieschang Invariant in the Bogoyavlenskyi Integrable Case 2000, vol. 5, no. 4, pp.  437-457 Abstract The topology of an integrable Hamiltonian system with two degrees of freedom, occuring in dynamics of the magnetic heavy body with a fixed point [1], is explored. The equations of critical submanifolds of the supplementary integral $f$, restricted to arbitrary isoenergy surface $Q_h^3$, are obtained. In particular, all the phase trajectories of a stable periodic motion are found. It is proved, that $f$ is a Bottean integral. The bifurcation diagram, full Fomenko–Zieschang invariant and the topology of each regular isoenergy surface $Q_h^3$ are calculated, as well as the topology of phase manifold $M^4$, which has a degenerate peculiarity of the symplectic structure. This peculiarity did not appear in dynamics before. A method of the computer visualization of Liouville tori bifurcations is offering. Citation: Zotev D. B.,  Fomenko–Zieschang Invariant in the Bogoyavlenskyi Integrable Case, Regular and Chaotic Dynamics, 2000, vol. 5, no. 4, pp. 437-457 DOI:10.1070/RD2000v005n04ABEH000158