Kowalevski Gyrostat Research Articles

Bifurcation diagrams and critical subsystems of the Kowalevski gyrostat in two constant fields

The Kowalevski gyrostat in two constant fields is known as the unique example of an integrable rigid body problem described by the Hamiltonian system with three degrees of freedom not reducible to a family of systems in fewer dimensions. The practical explicit integration of this system can hardly be obtained by the existing techniques. Then the challenging problem becomes to fulfill the qualitative investigation based on the study of the Liouville foliation of the phase space. As the first approach to topological analysis of this system we find the stratified critical set of the momentum map; this set is represented as the union of manifolds with induced almost Hamiltonian systems having less than three degrees of freedom. We obtain the equations of the bifurcation diagram in three-dimensional space. These equations have the form convenient for the classification of the bifurcation sets arising on 5-dimensional iso-energetic levels.

Periodic motions of the Kowalevski gyrostat in two constant fields

The case of the gyrostat motion found by A G Reyman and M A Semenov-Tian-Shansky is known as the Liouville integrable Hamiltonian system with three degrees of freedom without symmetry groups. We find the set of points at which the integral map has rank 1. This set consists of special periodic motions generating the singular points of bifurcation diagrams on iso-energetic surfaces. For such motions, all phase variables are expressed as algebraic functions of one auxiliary variable satisfying the differential equation integrable in elliptic functions of time. It is shown that the corresponding points in three-dimensional space of the integral constants belong to the intersection of two sheets of the discriminant surface of the Lax curve.

On classical r matrix for the Kowalevski gyrostat on so(4)

We present the trigonometric Lax matrix and classical r matrix for the Kowalevski gyrostat on so(4) algebra by using auxiliary matrix algebras so(3,2) or sp(4).

Критические подсистемы гиростата Ковалевской в двух постоянных полях

Критические подсистемы гиростата Ковалевской в двух постоянных полях

On Classical r-Matrix for the Kowalevski Gyrostat on so(4)

We present the trigonometric Lax matrix and classical r-matrix for the Kowalevski gyrostat on so(4) algebra by using the auxiliary matrix algebras so(3,2) or sp(4).

On a trajectory isomorphism of the Kowalevski gyrostat and the Clebsch problem

For the Kowalevski gyrostat change of variables similar to that of the Kowalevski top is done. We establish one to one correspondence between the Kowalevski gyrostat and the Clebsch system and demonstrate that Kowalevski variables for the gyrostat practically coincide with elliptic coordinates on sphere for the Clebsch case. Equivalence of considered integrable systems allows to construct two Lax matrices for the gyrostat using known rational and elliptic Lax matrices for the Clebsch model. Associated with these matrices solutions of the Clebsch system and, therefore, of the Kowalevski gyrostat problem are discussed. The Kotter solution of the Clebsch system in modern notation is presented in detail.

Simultaneous separation for the Kowalevski and Goryachev$ndash$Chaplygin gyrostats

In the special case of zero square integral the Kowalevski gyrostat and Goryachev-Chaplygin gyrostat share a simple separation of variables originated from the 4x4 Lax matrix.

The Kowalevski top as a reduction of a Hamiltonian system on $\mathfrak {sp}(4, \mathbb {R})^*$

We show that the Kowalevski top and Kowalevski gyrostat are obtained as a reduction of a Hamiltonian system on sp(4,R)*. Therefore the Lax-pair representations for the Kowalevski top and Kowalevski gyrostat are obtained via a direct method by transforming the canonical Lax-pair representation of a system on sp(4,R)*. Also we show that the nontrivial integral of motion of the Kowalevski top comes from a Casimir function of the Lie-Poisson algebra sp(4,R)*.

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.

Remarks on Kowalevski's top

We present a review of some results on Kowalevski's top (KT) in classical and quantum mechanics. The following items are considered: (i) the generalized KT (GKT) on the Lie algebras so(4), e(3) and so(3,1); (ii) Kowalevski's gyrostat on these algebras; (iii) action of the GKT; (iv) quantum counterparts of the KT; (v) semiclassical quantization of the GKT; (vi) generalization of the KT by Chaplygin and Goryachev at l = 0 and its Lax representation. Unsolved questions are also discussed.

Lax representation and Poisson geometry of the Kowalevski top

We discuss the Poisson structure underlying the two-field Kowalevski gyrostat and the Kowalevski top. We start from their Lax structure and construct a suitable pencil of Poisson brackets which endows these systems with the structure of bi-Hamiltonian completely integrable systems. We study the Casimir functions of such pencils, and show how it is possible to frame the Kowalevski systems within the so-called Gel'fand-Zakharevich bi-Hamiltonian setting for integrable systems.

The Kowalevski Top: its r-Matrix Interpretation and Bihamiltonian Formulation

The Lax pair representation for the Kowalevski top and for the Kowalevski gyrostat are obtained directly via the r-matrix method. The construction of Reyman and Semenov–Tian-Shansky is then used to give a bihamiltonian description of the Kowalevski gyrostat