Goryachev Chaplygin Research Articles

Stability of Stationary Motions of Mechanical Systems with a Particular Goryachev–Chaplygin Integral

Stability of Stationary Motions of Mechanical Systems with a Particular Goryachev–Chaplygin Integral

Analysis of the orbital stability of periodic pendulum motions of a heavy rigid body with a fixed point under the Goryachev–Chaplygin condition

In this paper the motion of a rigid body with a fixed point in a uniform gravity field is considered. It is assumed that the main moments of inertia of the body correspond to the case of Goryachev–Chaplygin, i.e., they are in the ratio of 1:1:4. In contrast to the integrable case of Goryachev–Chaplygin, there are no restrictions imposed on the position of the center of mass of the body. The problem of the orbital stability of periodic pendulum motions of a body (oscillations and rotations) is investigated. The equations of perturbed motion were obtained and the problem of orbital stability was reduced to the stability problem of the equilibrium position of a second-order linear system with 2π-periodic coefficients, the right-hand sides of which depend on two parameters. Based on the analysis of the linearized system, it has been established that the rotations are orbitally unstable for all possible values of the parameters. Moreover, a diagram of stability of pendulum oscillations has been constructed, on which the regions of orbital instability (parametric resonance) and regions of orbital stability in the linear approximation are indicated. At small values of oscillations amplitudes, a nonlinear analysis was performed. Based on the analysis of the coefficients of the normalized Hamilton function using theorems of the KAM theory, rigorous conclusions on the orbital stability of pendulum oscillations with small amplitudes were obtained.

Reducing the Degree of Integrals of Hamiltonian Systems by Using Billiards

In the theory of integrable Hamiltonian systems with two degrees of freedom, widely known are integrable systems having integrals of high degrees, namely, 3 and 4. Examples are the Kovalevskaya system and its generalizations—the Kovalevskaya–Yehia system and the Kovalevskaya system on the Lie algebra so(4) the Goryachev–Chaplygin–Sretensky, Sokolov, and Dullin–Matveev systems. It is shown that, at a number of isoenergy 3-surfaces, the third and fourth degrees of integrals of these systems can be reduced by using integrable billiards bounded by arcs of confocal quadrics. Moreover, the integrals of degree 3 and 4 are reduced to the same canonical quadratic integral on a billiard.

Tensor invariants and integration of differential equations

The connection between tensor invariants of systems of differential equations and explicit integration of them is discussed. A general result on the integrability of dynamical systems admitting a complete set of integral invariants in the sense of Cartan is proved. The existence of an invariant -form is related to the representability of the dynamical system in Hamiltonian form (with a symplectic structure which may be degenerate). This general idea is illustrated using an example of linear systems of differential equations. A general concept of flags of tensor invariants is introduced. General relations between the Kovalevskaya exponents of quasi-homogeneous systems of differential equations and flags of quasi-homogeneous tensor invariants having a certain structure are established. Results of a general nature are applied, in particular, to show that the general solution of the equations of rotation for a rigid body is branching in the Goryachev–Chaplygin case. Bibliography: 50 titles.

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.

Abel’s theorem and Bäcklund transformations for the Hamilton-Jacobi equations

We consider an algorithm for constructing auto-Backlund transformations for finitedimensional Hamiltonian systems whose integration reduces to the inversion of the Abel map. In this case, using equations of motion, one can construct Abel differential equations and identify the sought Backlund transformation with the well-known equivalence relation between the roots of the Abel polynomial. As examples, we construct Backlund transformations for the Lagrange top, Kowalevski top, and Goryachev–Chaplygin top, which are related to hyperelliptic curves of genera 1 and 2, as well as for the Goryachev and Dullin–Matveev systems, which are related to trigonal curves in the plane.

On bi-Hamiltonian geometry of some integrable systems on the sphere with cubic integral of motion

We obtain the bi-Hamiltonian structure for a family of integrable systems on the sphere with an additional integral of third order in momenta. These results are applied to the Goryachev system and Goryachev–Chaplygin top for which we give an explicit procedure to find the separated coordinates and the separated relations.

A family of the Poisson brackets compatible with the Sklyanin bracket

We introduce a family of compatible Poisson brackets on the space of 2 × 2 polynomial matrices, which contains the Sklyanin bracket, and use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes the XXX Heisenberg magnet, the open and periodic Toda lattices, the discrete self-trapping model and the Goryachev–Chaplygin gyrostat.

Stability problem for pendulum-type motions of a rigid body in the Goryachev-Chaplygin case

We consider the motion of a rigid body with a single fixed point in a homogeneous gravity field. The body mass geometry and the initial conditions for its motion correspond to the case of Goryachev—Chaplygin integrability. We study the orbital stability problem for periodic motions corresponding to vibrations and rotations of the rigid body rotating about the equatorial axis of the inertia ellipsoid.

Invariant sets in the Goryachev–Chaplygin problem: existence, stability and branching

The existence, stability and branching of invariant sets in the problem of the motion of a heavy rigid body with a fixed point, which satisfies the Goryachev–Chaplygin conditions, are discussed. Both trivial invariant sets, in which the pendulum-like motions of a Goryachev–Chaplygin spinning top lie, as well as non-trivial invariant sets, in which the motion of the top is described by elliptic functions of time, are investigated.

Separation of Variables in the Kovalevskaya–Goryachev–Chaplygin Gyrostat

We construct separation variables for the Kovalevskaya–Goryachev–Chaplygin gyrostat for arbitrary values of the parameters. We show that different separation variables can be constructed for the same integrable system if different integrals of motion are chosen.

Commutative Poisson Subalgebras for Sklyanin Brackets and Deformations of Some Known Integrable Models

We construct hierarchies of commutative Poisson subalgebras for Sklyanin brackets. Each of the subalgebras is generated by a complete set of integrals in involution. Some new integrable systems and schemes for separation of variables for them are elaborated using various well-known representations of the brackets. The constructed models include deformations for the Goryachev–Chaplygin top, the Toda chain, and the Heisenberg model.

Lax Pairs for the Deformed Kowalevski and Goryachev–Chaplygin Tops

We consider a quadratic deformation of the Kowalevski top. This deformation includes a new integrable case for the Kirchhoff equations recently found by one of the authors as a degeneration. A 5×5 matrix Lax pair for the deformed Kowalevski top is proposed. We also find similar deformations of the two-field Kowalevski gyrostat and the so(p,q) Kowalevski top. All our Lax pairs are deformations of the corresponding Lax representations found by Reyman and Semenov-Tian-Shansky. A similar deformation of the Goryachev–Chaplygin top and its 3×3 matrix Lax representation is also constructed.

On certain integrable motions of a rigid body acted upon by gravity and magnetic fields

The motion of a gyrostat about a fixed point in uniform gravity and magnetic fields is considered. The gyrostatic moment is directed along a principal axis of inertia while the centre of mass and the magnetic moment lie in the orthogonal principal plane. We show that the conditions imposed by Dragovic (Mat. Vesnik 49 (1997) 279–281) on the motion in the integrable case pointed out by him and considered as a generalisation of the famous Goryachev–Chaplygin case (Mathematical Reviews: MR #99d: 70014) lead only to a pendulum-like motion about a fixed axis, in which the presence of the magnetic field is not significant. We show also that this motion remains possible even when the Goryachev conditions on the moments of inertia are completely removed.

New Examples of Integrable Conservative Systems on S 2 and the Case of Goryachev-Chaplygin

In this paper we propose new examples of conservative systems on S 2 possessing cubic integrals in momenta. We obtain a classification of a large class of such systems. We show that this class contains new families of integrable systems as well as the classical case of Goryachev–Chaplygin.

Integrable Models, SUSY Gauge Theories and String Theory

We consider the close relation between duality in N = 2 SUSY gauge theories and integrable models. Various integrable models ranging from Toda lattices, Calogero models, spinning tops, and spin chains are related to the quantum moduli space of vacua of N = 2 SUSY gauge theories. In particular, SU(3) gauge theories with two flavors of massless quarks in the fundamental representation can be related to the spectral curve of Goryachev–Chaplygin top, which is a Nahm's equation in disguise. This can be generalized to the cases with massive quarks, and Nf = 0,1,2, where a system with seven dimensional phase space has the relevant hyperelliptic curve appear in the Painlevè test. To understand the string origin of the integrability of these theories we obtain exact nonperturbative point particle limit of type II string compactified on a Calabi–Yau manifold, which gives the hyperelliptic curve of SU(2) QCD with Nf = 1 hypermultiplet.

Truncated Yangian for T(5) = 0 and the generalized G-C gyrostat

By solving the RTT relation for the truncation T (5) = 0, where the transfer matrix T( u) = σ n = 0 4 u − n T ( n) , a set of algebraic relations for the truncated Yangian is presented. In comparison with the Goryachev-Chaplygin (G-C) gyrostat there appear two new operators D and B 3. The algebraic relations satisfied by them are explicitly given. With the help of this closed algebra we then derive the extended G-C gyrostat in an algebraic form.

Truncated Yangian and Goryachev-Chaplygin gyrostat

The quantum Goryachev-Chaplygin (GC) gyrostat is shown to be related to the “truncated” Yangian. A more general solution and the related constraint on the transfer matrix T( u) are derived.

The Goryachev-Chaplygin gyrostat in quantum mechanics

The authors investigate the quantum mechanical analogue of the classical integrable system named in the title. The Goryachev-Chaplygin (GC) gyrostat is a generalisation of the GC top, where the Coriolis interaction is taken into account. The problem is formulated in terms of the Euclid E(3) group. The integrals of motion are derived. The separation of variables is based on the connection between the degenerated representation of the E(3) group and the special representation of the SO(3,2) group. Some numerical results on the spectra of energy and separation constant are presented. The strong field limit for the integrals of motion is considered in detail, as well as a correlation diagram connecting the states in the limits of weak and strong fields. It appears that the dependence of energy and separation constant on the strength of the Coriolis interaction has the zone behaviour.