Fomenko-Zieschang Invariants Research Articles

Liouville classification of integrable Hamiltonian systems on surfaces of revolution

An algorithm for calculation of Fomenko-Zieschang invariants for Hamiltonian systems on 2-dimensional surfaces of revolution is described in the case of the potential V (r) = cos r. One typical example of such system is considered. Classic examples of systems equivalent to this system in the sense of Liouville are obtained. It is shown that a system studied on a surface of revolution is Liouville equivalent to a geodesic flow on this surface.

Topological classification of systems of Kovalevskaya-Yehia type

All the Fomenko-Zieschang invariants are calculated for the Kovalevskaya-Yehia problem, for all noncritical values of the parameters and , by constructing admissible systems of coordinates and determining the mutual disposition of the basis cycles. The family of Kovalevskaya-Yehia systems contains 29 pairwise Liouville non-equivalent foliations. These foliations include those that are equivalent to previously known foliations, which arose in the integrable cases of Kovalevskaya and of Kovalevskaya-Yehia for , in the Zhukovskiǐ case, and in the Goryachev-Chaplygin-Sretenskiǐ case. Eleven new foliations are included in the 29 foliations, new in the sense that they are not Liouville equivalent to any foliations discovered earlier which arose in the known integrable cases of the rigid body. The topological type of the Liouville foliation for the family of Kovalevskaya-Yehia systems stabilizes at large values of the energy , and this ‘high-energy’ system is roughly Liouville equivalent, at one of the energy levels, to the Goryachev-Chaplygin-Sretenskiǐ integrable case, which is already known.Bibliography: 29 titles.

Fomenko-Zieschang invariants of integrable systems with symplectic singularities

The paper is devoted to the study of integrable Hamiltonian systems which arise on symplectic manifolds with degenerate singularities. The results obtained extend the subject domain of the Fomenko-Zieschang theory of invariants.

A fine Liouville classification of the Kovalevskaya-Yehia integrable case

Methods and results of calculation of Fomenko-Zieschang invariants are presented for the Kovalevskaya-Yehia integrable case in the problem on heavy gyrostat motion in the gravitational field. All loop and isoenergy molecules of the system are presented.

Phase topology of Appelrot class I of a Kowalewski top in a magnetic field

We consider an integrable case generalizing the Appelrot class I of a Kowalewski top in a magnetic field. Its phase topology is investigated by means of Fomenko-Zieschang invariants. The offered method of approach to the calculation of marks completes Bolsinov’s method in the situation where it is not usable.

Calculation of the Fomenko-Zieschang invariants in the Kovalevskaya-Yehia integrable case

All the Fomenko-Zieschang invariants are calculated in the Kovalevskaya-Yehia integrable case of the problem of the motion of a heavy gyrostat in a gravitational field; the topological classification of the non-degenerate equilibria of this system is obtained and the loop molecules of all the singularities are calculated.Bibliography: 10 titles.

The Liouville classification of integrable systems of the Clebsch case

The Fomenko-Zieschang invariants for the integrable Clebsch case in the dynamics of a rigid body are calculated. Thus a complete description of the structure of the Liouville foliations on isoenergy surfaces of this integrable Hamiltonian system is obtained. As a corollary the Liouville equivalence of the Euler and Clebsch cases for certain values of the parameters is established.

Topological analysis of the two-centre problem on the two-dimensional sphere

The two-centre problem on the two-dimensional sphere (with the standard metric of constant positive curvature) is investigated from the topological point of view. The Fomenko-Zieschang invariants are constructed, which completely describe the topology of the Liouville foliations on isoenergy surfaces of this system. Various types of motion in the configuration space (regular motions and limit motions corresponding to bifurcations of Liouville tori) are described. The connection between Fomenko-Zieschang invariants (marked molecules) and various types of motion is considered.