Mikhail Kharlamov

This work continues the author’s article in Rus. J. Nonlinear Dynamics (2010, v. 6, N. 4) and contains applications of the Boolean functions method to investigation of the admissible regions and the phase topology of three algebraically solvable systems in the problem of motion of the Kowalevski top in the double force field.

We aim to completely formalize the rough topological analysis of integrable Hamiltonian systems admitting analytical solutions such that the initial phase variables along with the time derivatives of the auxiliary variables are expressed as rational functions (in fact, as polynomials) in some set of radicals depending on one variable each. We suggest a method to define the admissible regions in the integral constants space, the segments of oscillation of the separated variables and the number of connected components of integral manifolds and critical integral surfaces. This method is based on some algorithms of processing the tables of some Boolean vector-functions and of reducing the matrices of linear Boolean vector-functions to some canonical form. From this point of view we consider here the topologically richest classical problems of the rigid body dynamics. The article will be continued with the investigation of some new integrable problems.

The Kowalevski gyrostat in two constant fields is known as the unique profound example of an integrable Hamiltonian system with three degrees of freedom not reducible to a family of systems in fewer dimensions. As the first approach to topological analysis of this system we find the critical set of the integral map; this set consists of the trajectories with number of frequencies less than three. We obtain the equations of the bifurcation diagram in three-dimensional space of the first integrals constants.

We consider the analogue of the 4th Appelrot class of the Kowalevskaya top for the case of double force field. The trajectories of this family fill the 4-dimensional surface in the 6-dimensional phase space. We point out two almost everywhere independent partial integrals that give the regular parametrization of the corresponding sheet of the bifurcation diagram in the complete problem. Projections of the Liouville tori onto the plane of auxiliary variables are investigated. The bifurcation diagram of the partial integrals is found. The region of existence of motions in terms of the integral constants is established. We introduce the change of variables that separate the system of differential equations for this case.

The motion of a rigid body about a fixed point in a double constant force field is governed by a Hamiltonian system with three degrees of freedom. We consider the general case when there are no one-dimensional symmetry groups. We point out the critical points of the Hamilton function and corresponding critical values of energy. Using the Morse theory, we have found the smooth type of non-degenerate five-dimensional iso-energetic levels and find their projections onto the configuration space, diffeomorphic to a three-dimensional projective space. The analogs of classical motion possibility regions, the projections of iso-energetic manifolds onto one of the Poisson spheres, are studied.