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Alexander Y. Savushkin

Publications and source records attributed to Alexander Y. Savushkin.

6 recordsLinked to original sources
arXiv2015

Phase topology of the Kowalevski-Sokolov top

The phase topology of the integrable Hamiltonian system on $e(3)$ found by V.V.Sokolov (2001) and generalizing the Kowalevski case is investigated. The generalization contains, along with a homogeneous potential force field, gyroscopic forces depending on the configurational variables. Relative equilibria are classified, their type is calculated and the character of stability is defined. The Smale diagrams of the case are found and the classification of iso-energy manifolds of the reduced systems with two degrees of freedom is given. The set of critical points of the complete momentum map is represented as a union of critical subsystems; each critical subsystem is a one-parameter family of almost Hamiltonian systems with one degree of freedom. For all critical points we explicitly calculate the characteristic values defining their type. We obtain the equations of the surfaces bearing the bifurcation diagram of the momentum map. We give examples of the existing iso-energy diagrams with a complete description of the corresponding rough topology (of the regular Liouville tori and their bifurcations).

nlin.SI↗
arXiv2013

Bifurcation diagrams of the integral mappings of a top with singular symmetry

In general case, a Hamiltonian system with three degrees of freedom describing the motion of a rigid body in two constant fields does not admit any symmetry groups. H.Yehia has found conditions under which the equations of motion of the Kovelevskaya type gyrostat in such a field have, in addition to the energy integral, an integral linear with respect to angular velocities. Later it was noticed that this integral under the conditions of Yehia exists for arbitrary axially symmetric gyrostat having the centers of the fields applications in the equatorial plane. The corresponding natural mechanical system has an $S^1$-symmetry which has a non-empty set of fixed points. Smale's program of topological analysis can be fulfilled with some modifications considering the singularity of the symmetry action. We construct the bifurcation diagrams of the momentum map for a family of systems with a singular symmetry and investigate the dependency on one essential parameter expressing the ratio of the axial and equatorial inertia moments.

nlin.SI↗
arXiv2012

Geometrical approach to separation of variables in mechanical systems

The article presents a compact review of the analytical results (2002-2009) in the study of the system describing the motion of a top in two constant fields. The Liouville integrability of this system under certain condition of the Kowalevski type was established by A.G.Reyman and M.A.Semenov-Tian-Shansky. We present some geometrical foundations of finding separations of variables. Two systems of local planar coordinates are introduced leading to separation of variables for two subsystems with two degrees of freedom in the dynamics of the generalized Kowalevski top.

math.DS↗
arXiv2008

Separation of variables and integral manifolds in one problem of motion of generalized Kowalevski top

In the phase space of the integrable Hamiltonian system with three degrees of freedom used to describe the motion of a Kowalevski-type top in a double constant force field, we point out the four-dimensional invariant manifold. It is shown that this manifold consists of critical motions generating a smooth sheet of the bifurcation diagram, and the induced dynamic system is Hamiltonian with certain subset of points of degeneration of the symplectic structure. We find the transformation separating variables in this system. As a result, the solutions can be represented in terms of elliptic functions of time. The corresponding phase topology is completely described.

nlin.SI↗
arXiv2008

Explicit integration of one problem of motion of the generalized Kowalevski top

In the problem of motion of the Kowalevski top in a double force field the 4-dimensional invariant submanifold of the phase space was pointed out by M.P.Kharlamov (Mekh. Tverd. Tela, 32, 2002). We show that the equations of motion on this manifold can be separated by the appropriate change of variables, the new variables s1, s2 being elliptic functions of time. The natural phase variables (components of the angular velocity and the direction vectors of the forces with respect to the movable basis) are expressed via s1, s2 explicitly in elementary algebraic functions.

nlin.SI↗