On classical <Emphasis Type="Italic">r</Emphasis> matrix for the Kowalevski gyrostat on <Emphasis Type="Italic">so</Emphasis>(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). The text was submitted by the authors in English.     

Geometrization and Generalization of the Kowalevski Top

A new view on the Kowalevski top and the Kowalevski integration procedure is presented. For more than a century, the Kowalevski 1889 case, has attracted full attention of a wide community as the highlight of the classical theory of integrable systems. Despite hundreds of papers on the subject, the Kowalevski integration is still understood as a magic recipe, an unbelievable sequence of skillful tricks, unexpected identities and smart changes of variables. The novelty of our present approach is based on our four observations. The first one is that the so-called fundamental Kowalevski equation is an instance of a pencil equation of the theory of conics which leads us to a new geometric interpretation of the Kowalevski variables w, x ₁, x ₂ as the pencil parameter and the Darboux coordinates, respectively. The second is observation of the key algebraic property of the pencil equation which is followed by introduction and study of a new class of discriminantly separable polynomials. All steps of the Kowalevski integration procedure are now derived as easy and transparent logical consequences of our theory of discriminantly separable polynomials. The third observation connects the Kowalevski integration and the pencil equation with the theory of multi-valued groups. The Kowalevski change of variables is now recognized as an example of a two-valued group operation and its action. The final observation is surprising equivalence of the associativity of the two-valued group operation and its action to the n = 3 case of the Great Poncelet Theorem for pencils of conics.     

Kowalevski top in quantum mechanics

The quantum mechanical Kowalevski top is studied by the direct diagonalization of the Hamiltonian. The spectra show different behaviors depending on the region divided by the bifurcation sets of the classical invariant tori. Some of these spectra are nearly degenerate due to the multiplicity of the invariant tori. The Kowalevski top has several symmetries and symmetry quantum numbers can be assigned to the eigenstates. We have also carried out the semiclassical quantization of the Kowalevski top by the EBK formulation. It is found that the semiclassical spectra are close to the exact values, thus the eigenstates can be also labeled by the integer quantum numbers. The symmetries of the system are shown to have close relations with the semiclassical quantum numbers and the near-degeneracy of the spectra.     

Explicit integration of a generic Hénon-Heiles system with quartic potential

There are seven time independent, integrable, Hénon-Heiles systems: three with cubic and four with quartic potential. The cubic and one of the quartic cases have been separated in the last decades. The other three cases 1:6:1, 1:6:8 and 1:12:16 have resisted several attempts in the last years. In this paper we focus our attention on the 1:12:16 case whose equations of motion have been separated only in the degenerate case ab = 0. We give here the separation coordinates for the generic case using a method introduced by Franco Magri in 2005 under the name of Kowalevski’s Conditions.     

Painlevé VI, Rigid Tops and Reflection Equation

We show that the Painlevé VI equation has an equivalent form of the non-autonomous Zhukovsky-Volterra gyrostat. This system is a generalization of the Euler top in and includes the additional constant gyrostat momentum. The quantization of its autonomous version is achieved by the reflection equation. The corresponding quadratic algebra generalizes the Sklyanin algebra. As by product we define integrable XYZ spin chain on a finite lattice with new boundary conditions.     

Non-commutative generalization of integrable quadratic ODE systems

Letters in Mathematical Physics - We find all homogeneous quadratic systems of ODEs with two dependent variables that have polynomial first integrals and satisfy the Kowalevski–Lyapunov test....     

Lagrangian relative equilibria for a gyrostat in the three-body problem

In this paper we consider the noncanonical Hamiltonian dynamics of a gyrostat in the three-body problem. By means of geometric mechanics methods, we study the approximate Poisson dynamics that arise when we develop the potential of the system in Legendre series and truncate this to an arbitrary order k. After reduction of the dynamics by means of the two symmetries of the system, we consider the existence and number of equilibria which we denominate of Lagrangian type, in analogy with classic results on the topic. Necessary and sufficient conditions are established for their existence in an approximate dynamics of order k, and explicit expressions for these equilibria are given, this being useful for the subsequent study of their stability. The number of Lagrangian equilibria is thoroughly studied in approximate dynamics of orders zero and one. The main result of this work indicates that the number of Lagrangian equilibria in an approximate dynamics of order k for k ≥1 is independent of the order of truncation of the potential, if the gyrostat S ₀ is almost spherical. In relation to the stability of these equilibria, necessary and sufficient conditions are given for linear stability of Lagrangian equilibria when the gyrostat is almost spherical. In this way, we generalize the classical results on equilibria of the three-body problem and many results provided by other authors using more classical techniques for the case of rigid bodies.      

Diagonalization of three-dimensional pseudo-Riemannian metrics

We give a short proof of the fact that any Riemannian or Lorentzian real analytic metric in dimension 3 can be locally adapted to the diagonal form. We use the classical Cauchy–Kowalevski Theorem to this purpose.     

CHAOS, CHAOS CONTROL AND SYNCHRONIZATION OF ELECTRO-MECHANICAL GYROSTAT SYSTEM

The dynamic behavior of electro-mechanical gyrostat system subjected to external disturbance is studied in this paper. By applying numerical results, phase diagrams, power spectrum, Period-T maps, and Lyapunov exponents are presented to observe periodic and chaotic motions. The effect of the parameters changed in the system can be found in the bifurcation and parametric diagrams. Several methods, the delayed feedback control, adaptive control algorithm (ACA) control are used to control chaos effectively. Anticontrol of chaos destroyed the periodic motions and replaced by chaotic motion effectively by adding constant motor torque and adding periodic motor torque. Finally, synchronization of chaos in the electro-mechanical gyrostat system is studied.     

Systems of Hess-Appel’rot Type

We construct higher-dimensional generalizations of the classical Hess- Appel’rot rigid body system. We give a Lax pair with a spectral parameter leading to an algebro-geometric integration of this new class of systems, which is closely related to the integration of the Lagrange bitop performed by us recently and uses Mumford relation for theta divisors of double unramified coverings. Based on the basic properties satisfied by such a class of systems related to bi-Poisson structure, quasi-homogeneity, and conditions on the Kowalevski exponents, we suggest an axiomatic approach leading to what we call the “class of systems of Hess-Appel’rot type”.     

CHAOS, CHAOS CONTROL AND SYNCHRONIZATION OF A GYROSTAT SYSTEM

The dynamic behavior of a gyrostat system subjected to external disturbance is studied in this paper. By applying numerical results, phase diagrams, power spectrum, period-T maps, and Lyapunov exponents are presented to observe periodic and choatic motions. The effect of the parameters changed in the system can be found in the bifurcation and parametric diagrams. For global analysis, the basins of attraction of each attractor of the system are located by employing the modified interpolated cell mapping (MICM) method. Several methods, the delayed feedback control, the addition of constant torque, the addition of periodic force, the addition of periodic impulse torque, injection of dither signal control, adaptive control algorithm (ACA) control and bang-bang control are used to control chaos effectively. Finally, synchronization of chaos in the gyrostat system is studied.     

New truncated q-GC gyrostat and its physical implication

A new form of the truncated global transfer matrix T(x) is given for the 6-vertex R(x)-matrix. It is a q-deformation of the new Goryachev-Chaplygin gyrostat. The derived Hamiltonian gives the correct form of the q-top obeying the RTT relation. Physically, the q-GC top can be viewed as a model including quadratic terms in the magnetic translation operators.     

New invariant relations for the generalized two-field gyrostat

In the paper, we consider a completely integrable Hamiltonian system with three degrees of freedom found by V.V. Sokolov and A.V. Tsiganov. This system is known as the generalized two-field gyrostat. For the case of only gyroscopic forces present, we find new invariant four-dimensional submanifolds such that the induced dynamical systems are almost everywhere Hamiltonian with two degrees of freedom.     

A generalization of the Kovalevskaya top

Integrals of motion of the Kovalevskaya gyrostat on Lie algebras O(4), e(3), o(3, 1) are constructed both in classical and quantum mechanics. The additional term to the classical Kovalevskaya integral on e(3) is shown to be proportional to the integral of the Goryachev-Chaplygin gryostat.     

New invariant relations for one critical subsystem of a generalized two-field gyrostat

In this paper new invariant relations for one critical subsystem of a completely integrable Hamiltonian system with three degrees of freedom found by V.V. Sokolov and A.V. Tsyganov, known as a generalized two-field gyrostat, are obtained. The dynamic system that is induced on the invariant four-dimensional submanifolds is almost everywhere Hamiltonian with two degrees of freedom. The type of system motions on this invariant manifold is determined.     

Qualitative analysis of the phase flow of an integrable approximation of a generalized roto-translatory problem

In this paper, we consider an integrable approximation of the planar motion of a gyrostat in Newtonian interaction with a spherical rigid body. We then describe the Hamiltonian dynamics, in the fibers of constant total angular momentum vector of an invariant manifold of motion. Finally, using the Liouville-Arnold theorem and a particular analysis of the momentum map in its critical points, we obtain a complete topological classification of the different invariant sets of the phase flow of this problem. The results can be applied to study two-body roto-translatory problems where the rotation of one of them has a strong influence on the orbital motion of the system.      

Explicit determination of certain periodic motions of a generalized two-field gyrostat

The case of motion of a generalized two-field gyrostat found by V. V. Sokolov and A.V. Tsiganov is known as a Liouville integrable Hamiltonian system with three degrees of freedom. For this system, we find some special periodic motions at which the momentum mapping has rank 1. For such motions, all phase variables can be expressed in terms of algebraic functions of a single auxiliary variable and a set of constants. This auxiliary variable satisfies a differential equation which can be integrated in elliptic functions of time. As an application, the explicit formulas of characteristic exponents for determining the Williamson type of the special periodic motions are obtained.     

杠杆式回转仪定量演示方法研究

在对杠杆式回转仪运动现象进行理论分析的基础上,通过测量出圆轮自转角速度,旋转轴进动角速度、进动角度的参数,定量研究了力矩与进动速度间关系、角速度与进动速度间关系、角动量定理的实验验证,最后将测量值与理论计算值进行了比较。将杠杆式回转仪运动状态的定量分析引入大学物理实验,培养了学生实际动手能力,加深了学生对角动量定理,力...     

Yields of Charmed and Beautiful Mesons in e+e－ Annihilation at s=10—178GeV 的e+e-

Based on the electroweak unified standard model,the probability P_i for the production of i-quark jet (i=u,d,c and b)in e⁺e^－ annihilation has been studied.The probability P_i is a function of energy s.The calculated component ratios of P_u:P_d:P_s:P_c:P_b are 4:1:1:4:1 for s<30GeV.7:9:9:7:9 for s=m_z=91.16GeV and 1:1:1:1:1 for s=78 and 117GeV,respectively.Using P_c and P_b,we whave calculated the yields of charmed and beautiful mesons in e⁺e^－ annihilation at s=10—178GeV.     

Newly developed highly sensitive LiF:Mg,Cu,Si TL discs with good stability to heat treatment

The preparation method and some dosimetric properties of the new LiF:Mg,Cu,Si discs are presented. The effect of heat treatments on LiF:Mg,Cu,Si was investigated. The shape of the glow curve for LiF:Mg,Cu,Si is similar to that for standard LiF:Mg,Cu,P (GR-200A), and shows minimal differences when annealed in the range from 260 °C to 290 °C for 10 min. The TL sensitivity for LiF:Mg,Cu,Si is much lower than that for GR-200A, but is 35 times larger than that for TLD-100 and is slightly higher than that for HMCP. The height of the high-temperature peaks for LiF:Mg,Cu,Si is not only lower than that for GR-200A, but also lower than that for HMCP. The glow curve shape of LiF:Mg,Cu,Si annealed at 260 °C for different times shows minimal differences and TL response remains stable. These results indicate that the new LiF:Mg,Cu,Si disc has a good stability to thermal treatments and a lower residual TL signal.