Kovalevskaya Case Research Articles

ON OPTIMAL STABILIZATION OF PART OF VARIABLES OF ROTARY MOVEMENT OF A RIGID BODY WITH ONE FIXED POINT IN THE CASE OF SOPHIA KOVALEVSKAYA

An optimal stabilization problem for part of variables of rotary movement of a rigid body with one fixed point in the Sophia Kovalevskaya's case is discussed in this work. The differential equations of motion of the system are given and it is shown that the system may rotate around Ox with a constant angular velocity. Taking this motion as unexcited, the differential equations for the corresponding excited motion were drawn up. Then the system was linearized and a control action was introduced along one of the generalized coordinates. The optimal stabilization problem for part of the variables was posed and solved. The graphs of optimal trajectories and optimal control were constructed.

Nonintegrability of the Problem of the Motion of an Ellipsoidal Body with a Fixed Point in a Flow of Particles

The problem of the motion, in the free molecular flow of particles, of a rigid body with a fixed point bounded by the surface of an ellipsoid of revolution is considered. This problem is similar in many aspects to the classical problem of the motion of a heavy rigid body about a fixed point. In particular, this problem possesses the integrable cases corresponding to the classical Euler – Poinsot, Lagrange and Hess cases of integrability of the equations of motion of a heavy rigid body with a fixed point. A natural question arises about the existence of analogues of other integrable cases that exist in the problem of motion of a heavy rigid body with a fixed point (Kovalevskaya case, Goryachev – Chaplygin case, etc) for the system considered. Using the standard Euler angles as generalized coordinates, the Hamiltonian function of the system is derived. Equations of motion of the body in the flow of particles are presented in Hamiltonian form. Using the theorem on the Liouville-type nonintegrability of Hamiltonian systems near elliptic equilibrium positions, which has been proved by V. V. Kozlov, necessary conditions for the existence in the problem under consideration of an additional analytic first integral independent of the energy integral are presented. We have proved that the necessary conditions obtained are not fulfilled for a rigid body with a mass distribution corresponding to the classical Kovalevskaya integrable case in the problem of the motion of a heavy rigid body with a fixed point. Thus, we can conclude that this system does not possess an integrable case similar to the Kovalevskaya integrable case in the problem of the motion of a heavy rigid body with a fixed point.

Noncompactness Property of Fibers and Singularities of Non-Euclidean Kovalevskaya System on Pencil of Lie Algebras

It is shown that Liouville foliations of the family of non-Euclidean analogs of Kovalevskaya integrable system on a pencil of Lie algebras have both compact and noncompact fibers. There exists a bifurcation of their compact common level surface into a noncompact one that has a noncompact singular fiber. In particular, this is true for the non-Euclidean $$e(2,1)$$ -analog of the Kovalevskaya case of rigid body dynamics. In the case of nonzero area integral, an effective criterion for existence of a noncompact connected component of the common level surface of first integrals and Casimir functions is proved.

Remarks on the Covering of the Possible Motion Area by Solutions in Rigid Body Systems

Abstract The problem of motion of a rigid body with a fixed point is considered. We study qualitatively the solutions of the system after Routh reduction. For the Lagrange integrable case, we show that the trajectories of solutions starting at the boundary of a possible motion area can both cover and not cover the entire possible motion area. It distinguishes these systems from the systems without gyroscopic forces, where the trajectories always cover the possible motion area. We also present some numerical and analytical results on the same matter for the Kovalevskaya case.

Linear orbital stability analysis of the pendulum-type motions of a Kovalevskaya top with a suspension point vibrating horizontally

The motion of a heavy rigid body is considered, with a point (suspension point) performing high-frequency horizontal vibration. The center of mass of the body is on the principal axis of inertia for this point. Within the framework of an approximate autonomous system, pendulum-type motions of the body are studied, they occur in a vertical plane containing the axis of vibration or in a plane perpendicular to it. For a body with a mass geometry corresponding to S. V. Kovalevskaya case, a linear orbital stability analysis of these motions is performed.

Topological Analysis of the Liouville Foliation for the Kovalevskaya Integrable Case on the Lie Algebra so(4)

In this paper we study the topology of the Liouville foliation for the integrable case of Euler’s equations on the Lie algebra so(4) discovered by I.V. Komarov, which is a generalization of the Kovalevskaya integrable case in rigid body dynamics. We generalize some results by A.V. Bolsinov, P.H. Richter, and A.T. Fomenko about the topology of the classical Kovalevskaya case. We also show how the Fomenko–Zieschang invariant can be calculated for every admissible curve in the image of the momentum map.

Classical perturbation theory and resonances in some rigid body systems

We consider the system of a rigid body in a weak gravitational field on the zero level set of the area integral and study its Poincare sets in integrable and nonintegrable cases. For the integrable cases of Kovalevskaya and Goryachev–Chaplygin we investigate the structure of the Poincare sets analytically and for nonintegrable cases we study these sets by means of symbolic calculations. Based on these results, we also prove the existence of periodic solutions in the perturbed nonintegrable system. The Chaplygin integrable case of Kirchhoff’s equations is also briefly considered, for which it is shown that its Poincare sets are similar to the ones of the Kovalevskaya case.

On the Motion of a Gyro Close to Kovalevskaya's Case

On the Motion of a Gyro Close to Kovalevskaya's Case

Generalizations of the Kovalevskaya case and quaternions

This paper provides a detailed description of various reduction schemes in rigid body dynamics. The analysis of one of such nontrivial reductions makes it possible to put the cases already found in order and to obtain new generalizations of the Kovalevskaya case to e(3). Note that the indicated reduction allows one to obtain in a natural way some singular additive terms that were proposed earlier by D.N. Goryachev.

Representation of solutions to the problem of the motion of a heavy rigid body in the Kovalevskaya case in terms of Weierstrass - and -functions and nonintegrability of the Hess case by quadratures

A method for the representation of Delaunay's solutions and some other particular solutions to the problem of the motion of a heavy rigid body in the Kovalevskaya case in terms of the Weierstrass - and -functions is put forward. The Hess case in the problem of the motion of a heavy rigid body is shown to be nonintegrable by quadratures. Bibliography: 24 titles.

The topology of the analog of Kovalevskaya integrability case on the Lie algebra so(4) under zero area integral

The topology of the space of closures of solutions to an integrable system on the Lie algebra so(4) being an analogue of the Kovalevskaya case has been studied. Fomenko-Zieschang invariants are calculated for this purpose in the case of zero area integral, which classify isoenergetic 3-surfaces and the corresponding Liouville foliations.

Representation of solutions to the problem of the motion of a heavy rigid body in the Kovalevskaya case in terms of Weierstrass $\zeta$- and $\wp$-functions and nonintegrability of the Hess case by quadratures

Предложен способ представления решений Делоне и некоторых других частных решений задачи о движении тяжелого твердого тела в случае Ковалевской в $\zeta$- и $\wp$-функциях Вейерштрасса. Для случая Гесса задачи о движении тяжелого твердого тела доказана невозможность решения в квадратурах. Библиография: 24 названия.

The topology of the Liouville foliation for the Kovalevskaya integrable case on the Lie algebra

In this paper we study topological properties of an integrable case for Euler's equations on the Lie algebra , which can be regarded as an analogue of the classical Kovalevskaya case in rigid body dynamics. In particular, for all values of the parameters of the system under consideration, the bifurcation diagrams of the momentum mapping are constructed, the types of critical points of rank are determined, the bifurcations of Liouville tori are described, and the loop molecules are computed for all singular points of the bifurcation diagrams. It follows from the obtained results that some topological properties of the classical Kovalevskaya case can be obtained from the corresponding properties of the considered integrable case on the Lie algebra by taking a natural limit. Bibliography: 21 titles.

The stability of the plane periodic motions of a symmetrical rigid body with a fixed point

A rigorous non-linear analysis of the orbital stability of plane periodic motions (pendulum oscillations and rotations) of a dynamically symmetrical heavy rigid body with one fixed point is carried out. It is assumed that the principal moments of inertia of the rigid body, calculated for the fixed point, are related by the same equation as in the Kovalevskaya case, but here no limitations are imposed on the position of the mass centre of the body. In the case of oscillations of small amplitude and in the case of rotations with high angular velocities, when it is possible to introduce a small parameter, the orbital stability is investigated analytically. For arbitrary values of the parameters, the non-linear problem of orbital stability is reduced to an analysis of the stability of a fixed point of the simplectic mapping, generated by the system of equations of perturbed motion. The simplectic mapping coefficients are calculated numerically, and from their values, using well-known criteria, conclusions are drawn regarding the orbital stability or instability of the periodic motion. It is shown that, when the mass centre lies on the axis of dynamic symmetry (the case of Lagrange integrability), the well-known stability criteria are inapplicable. In this case, the orbital instability of the periodic motions is proved using Chetayev's theorem. The results of the analysis are presented in the form of stability diagrams in the parameter plane of the problem.

Absolute and relative choreographies in rigid body dynamics

For the classical problem of motion of a rigid body about a fixed point with zero area integral, we present a family of solutions that are periodic in the absolute space. Such solutions are known as choreographies. The family includes the well-known Delone solutions (for the Kovalevskaya case), some particular solutions for the Goryachev-Chaplygin case, and the Steklov solution. The “genealogy” of solutions of the family naturally appearing from the energy continuation and their connection with the Staude rotations are considered. It is shown that if the integral of areas is zero, the solutions are periodic with respect to a coordinate frame that rotates uniformly about the vertical (relative choreographies).

A method for obtaining first integrals and integrating factors of autonomous systems and application to Euler-Poisson equations

A method for obtaining first integrals and integrating factors of n-th order autonomous systems is proposed. The search for first integrals and integrating factors can be reduced to the search for a class of invariant manifolds of the systems. Finally, the proposed method is applied to Euler-Poisson equations (gyroscope system), and the fourth first integral of the system in general Kovalevskaya case can be obtained.

Techniques for searching first integrals by Lie group and application to gyroscope system

In the paper, the methods of finding first integrals of an autonomous system using one-parameter Lie groups are discussed. A class of nontrivial one-parameter Lie groups admitted by the classical gyroscope system is found, and based on the properties of first integral determined by the one-parameter Lie group, the fourth first integral of the gyroscope system in Euler case, agrange case and Kovalevskaya case can be obtained in a uniform idea. An error on the fourth first integral in general Kovalevskaya case A = B =2 C , z G =0), which appeared in literature is found and corrected.

Абсолютные и относительные хореографии в динамике твердого тела

For the classical problem of motion of a rigid body about a fixed point with zero area integral, we present a family of solutions that are periodic in the absolute space. Such solutions are known as choreographies. The family includes the well-known Delone solutions (for the Kovalevskaya case), some particular solutions for the Goryachev-Chaplygin case, and the Steklov solution. The “genealogy” of solutions of the family naturally appearing from the energy continuation and their connection with the Staude rotations are considered. It is shown that if the integral of areas is zero, the solutions are periodic with respect to a coordinate frame that rotates uniformly about the vertical (relative choreographies).

The boundedness of trajectories in the neighbourhood of the orbitally unstable periodic motion of a Hamiltonian system

An autonomous Hamiltonian system with two degrees of freedom is considered. It is assumed that a periodic motion and a second-order resonance (parametric resonance) exist in the system. The unperturbed periodic motion is orbitally stable or unstable. However, even in the case of instability, the trajectories of the perturbed motion may remain in a bounded neighbourhood of the unperturbed trajectory for all values of the time. An asymptotic estimate of the size of this neighbourhood is given for the case when the Hamiltonian depends on a small parameter. The results are applied to the problem of the non-local stability of fast planar rotations of a heavy rigid body in the Kovalevskaya case, and to the problem of the stability of periodic Poincaré motions of the first kind in the restricted three-body problem, for one special case of second-order resonance.

The twisted top

We describe a new type of top, the twisted top, obtained by appending a cocycle to the Lie–Poisson bracket for the charged heavy top, thus breaking its semidirect product structure. The twisted top has an integrable case that corresponds to the Lagrange (symmetric) top. We give a canonical description of the twisted top in terms of Euler angles. We also show by a numerical calculation of the largest Lyapunov exponent that the Kovalevskaya case of the twisted top is chaotic.