Euler Top Research Articles

Zhukovsky-Volterra top and quantisation ideals

In this letter, we revisit the quantisation problem for a fundamental model of classical mechanics—the Zhukovsky-Volterra top. We have discovered a four-parametric pencil of compatible Poisson brackets, comprising two quadratic and two linear Poisson brackets. Using the quantisation ideal method, we have identified two distinct quantisations of the Zhukovsky-Volterra top. The first type corresponds to the universal enveloping algebras of so(3), leading to Lie-Poisson brackets in the classical limit. The second type can be regarded as a quantisation of the four-parametric inhomogeneous quadratic Poisson pencil. We discuss the relationships between the quantisations obtained in our paper, Sklyanin's quantisation of the Euler top, and Levin-Olshanetsky-Zotov's quantisation of the Zhukovsky-Volterra top.

Improved Equations of the Lagrange Top and Examples of Analytical Solutions

Equations of a heavy rotating body with one fixed point can be deduced starting from a variational problem with holonomic constraints. When applying this formalism to the particular case of a Lagrange top, in the formulation with a diagonal inertia tensor the potential energy has a more complicated form as compared with that assumed in the literature on dynamics of a rigid body. This implies the corresponding improvements in equations of motion. Therefore, we revised this case, presenting several examples of analytical solutions to the improved equations. The case of precession without nutation has a surprisingly rich relationship between the rotation and precession rates, which is discussed in detail.

Asymmetric separation of variables for the extended Clebsch and Manakov models

In the present paper we develop a method of the vector fields Zi[8] of the bi-hamiltonian theory of separation of variables. Using a modification of this method we construct non-Stäckel asymmetric variable separation for three-dimensional extension of the Clebsch model, which is bi-hamiltonially equivalent to the system of interacting Manakov (Shottky-Frahm) and Euler tops. Our SoV is unusual: it is characterized by two curves of separation of different genus. We explicitly construct coordinates and momenta of separation, the corresponding reconstruction formulae and the Abel-type quadratures for the considered three-dimensional extension of the Clebsch and Manakov tops.

Tracking arbitrary free nutating trajectories of the Lagrange top using Floquet theory: linear feedback approach

We consider the problem of tracking arbitrary free nutating trajectories of the Lagrange top using linear feedback. The equations of motion of the top are linearised about the desired trajectories. Due to the inherent symmetries, this yields a system of linear equations with periodically varying coefficients governing the error dynamics. It is shown that the higher-order dynamics of the Lagrange top leads to nuances in the implementation of the control. Two different control strategies are implemented in the present work. These are applicable for impulsive and short-duration disturbances, respectively. The control effort is in the form of a body-fixed torque input about the symmetry axis and a vertical space-fixed force input. In both the proposed control strategies, the iterative application of the control effort over multiple periods is shown to stabilise the error. These strategies are shown to work for significantly large errors.

A Three-Dimensional Generalization of QRT Maps

We propose a geometric construction of three-dimensional birational maps that preserve two pencils of quadrics. The maps act as compositions of involutions, which, in turn, act along the straight line generators of the quadrics of the first pencil and are defined by the intersections with quadrics of the second pencil. On each quadric of the first pencil, the maps act as two-dimensional QRT maps. While these maps are of a pretty high degree in general, we find geometric conditions which guarantee that the degree is reduced to 3. The resulting degree 3 maps are illustrated by two known and two novel Kahan-type discretizations of three-dimensional Nambu systems, including the Euler top and the Zhukovski–Volterra gyrostat with two non-vanishing components of the gyrostatic momentum.

Growth and Integrability of Some Birational Maps in Dimension Three

Motivated by the study of the Kahan–Hirota–Kimura discretisation of the Euler top, we characterise the growth and integrability properties of a collection of elements in the Cremona group of a complex projective 3-space using techniques from algebraic geometry. This collection consists of maps obtained by composing the standard Cremona transformation c3∈Bir(P3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{c}\\,}}_3\\in {{\\,\ extrm{Bir}\\,}}(\\mathbb {P}^3)$$\\end{document} with projectivities that permute the fixed points of c3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{c}\\,}}_3$$\\end{document} and the points over which c3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{c}\\,}}_3$$\\end{document} performs a divisorial contraction. Specifically, we show that three behaviour are possible: (A) integrable with quadratic degree growth and two invariants, (B) periodic with two-periodic degree sequences and more than two invariants, and (C) non-integrable with submaximal degree growth and one invariant.

Quantum Bianchi-VII problem, Mathieu functions and arithmetic

The geodesic problem on the compact threefolds with the Riemannian metric of Bianchi-VII0 type is studied in both classical and quantum cases. We show that the problem is integrable and describe the eigenfunctions of the corresponding Laplace-Beltrami operators explicitly in terms of the Mathieu functions with parameter depending on the lattice values of some binary quadratic forms. We use the results from number theory to discuss the level spacing statistics in relation with the Berry-Tabor conjecture and compare the situation with Bianchi-VI0 case (Sol-case in Thurston's classification) and with Bianchi-IX case, corresponding to the classical Euler top.

Interconnection of Lagrange–Dirac systems through nonstandard interaction structures

The Lagrange–Dirac interconnection theory has been developed for primitive subsystems coupled by a standard interaction Dirac structure, i.e. a structure of the form [Formula: see text], where [Formula: see text] is a regular distribution, [Formula: see text] is its annihilator and [Formula: see text] is the configuration manifold of the theory. In this work, we extend this theory to allow for parameter-dependent subsystems coupled by nonstandard interaction Dirac structures. This is done, first, by using the Dirac tensor product and, then, by using interaction forces. Both approaches are shown to be equivalent, and also equivalent to a variational principle. After that, we demonstrate the relevance of this generalization by investigating three applications. First, an electromechanical system is modeled; namely, a piston driven by an ideal DC motor through a scotch-yoke mechanism. Second, we relate the interconnection theory to the Euler–Poincaré–Suslov reduction. More specifically, we show that the reduced system may be regarded as an interconnected Lagrange–Dirac system with parameters. The nonholonomic Euler top is presented as a particular instance of this situation. Lastly, control interconnected systems are defined and a control for a planar rigid body with wheels is designed.

Elliptic Gaudin-type model in an external magnetic field and modified algebraic Bethe ansatz

We consider the elliptic Gaudin-type model in an external magnetic field [11–13,16,15,22] associated with non-skew-symmetric elliptic r-matrix [11] defined on 4:1 unramified covering of the Weierstrass cubic y2=(u+j1)(u+j2)(u+j3). We develop a modified algebraic Bethe ansatz for the considered elliptic r-matrix and for the algebra generated by the entries of the corresponding Lax operator with the aim of obtaining the spectra of the relevant Gaudin-type Hamiltonians in terms of solutions of modified Bethe equations. The applications of the obtained result to the diagonalization of the anisotropic quantum Euler top, quantum Zhukovsky-Volterrra top, quantum Steklov and Rubanovsky tops are given.

Bifurcation Diagram of the Model of a Lagrange Top with a Vibrating Suspension Point

The article considers a model system that describes a dynamically symmetric rigid body in the Lagrange case with a suspension point that performs high-frequency oscillations. This system, reduced to axes rigidly connected to the body, after the averaging procedure, has the form of the Hamilton equations with two degrees of freedom and has the Liouville integrability property of a Hamiltonian system with two degrees of freedom, which describes the dynamics of a Lagrange top with an oscillating suspension point. The paper presents a bifurcation diagram of the moment mapping. Using the bifurcation diagram, we presented in geometric form the results of the study of the problem of stability of singular points, in particular, singular points of rank zero and rank one.

Topological analysis of pseudo-Euclidean Euler top for special values of the parameters

An analogue of the Euler top is considered for a pseudo-Euclidean space is under consideration. In the cases when the geometric integral or area integral vanishes the bifurcation diagrams of the moment map are constructed and the homeomorphism class of each leaf of the Liouville foliation is determined. For each arc of the bifurcation diagram, for one of the two possible cases of the mutual arrangement of the moments of inertia, the types of singularities in the preimage of a small neighbourhood of this arc (analogues of Fomenko 3-atoms) are determined, and for nonsingular isoenergy and isointegral surfaces an invariant of rough Liouville equivalence (an analogue of a rough molecule) is constructed. The pseudo-Euclidean Euler system turns out to have noncompact noncritical bifurcations. Bibliography: 23 titles.

Euler top and freedom in supersymmetrization of one-dimensional mechanics

In this paper we formulate the Euler top as a system on the complex projective plane, playing the role of phase space, i.e., as a one-dimensional mechanical system. Then we suggest the supersymmetrization scheme of the generic one-dimensional systems with positive Hamiltonian which yields á priori integrable family of N=2k supersymmetric Hamiltonians parameterized by N/2 arbitrary real functions.

Application of the Kovacic algorithm for the investigation of motion of a heavy rigid body with a fixed point in the Hess case

AbstractIn 1890, Hess found new special case of integrability of Euler–Poisson equations of motion of a heavy rigid body with a fixed point. In 1892, Nekrasov proved that the solution of the problem of motion of a heavy rigid body with a fixed point under Hess conditions reduces to integrating the second‐order linear differential equation. In this paper, the corresponding linear differential equation is derived and its coefficients are presented in the rational form. Using the Kovacic algorithm, we proved that the Liouvillian solutions of the corresponding second‐order linear differential equation exist only in the case, when the moving rigid body is the Lagrange top, or in the case, when the constant of the area integral is zero.

The half-period addition formulae for genus two hyperelliptic ℘ functions and the Sp(4, ) Lie group structure

In the previous study, by using the two-flows Kowalevski top, we have demonstrated that the genus two hyperelliptic functions provide the Sp(4, )/ ≅ SO(3, 2) Lie algebra structure. In this study, by directly using the differential equations of the genus two hyperelliptic ℘ functions instead of using integrable models, we demonstrate that the half-period addition formula for the genus two hyperelliptic functions provides the order two Sp(4, ) Lie group structure.

Rigid fibers of integrable systems on cotangent bundles

(Non-)displaceability of fibers of integrable systems has been an important problem in symplectic geometry. In this paper, for a large class of classical Liouville integrable systems containing the Lagrangian top, the Kovalevskaya top and the C. Neumann problem, we find a non-displaceable fiber for each of them. Moreover, we show that the non-displaceable fiber which we detect is the unique fiber which is non-displaceable from the zero-section. As a special case of this result, we also show the existence of a singular level set of a convex Hamiltonian, which is non-displaceable from the zero-section. To prove these results, we use the notion of superheaviness introduced by Entov and Polterovich.

The Harmonic Lagrange Top and the Confluent Heun Equation

The harmonic Lagrange top is the Lagrange top plus a quadratic (harmonic) potential term. We describe the top in the space fixed frame using a global description with a Poisson structure on \(T^{*}S^{3}\). This global description naturally leads to a rational parametrisation of the set of critical values of the energy-momentum map. We show that there are 4 different topological types for generic parameter values. The quantum mechanics of the harmonic Lagrange top is described by the most general confluent Heun equation (also known as the generalised spheroidal wave equation). We derive formulas for an infinite pentadiagonal symmetric matrix representing the Hamiltonian from which the spectrum is computed.

Evolution of motion of a rigid body similar to Lagrange top under the influence of slowly time varying torques

Perturbed rotational rigid body motions similar to the case of Lagrange top subjected to restoring and perturbation slowly time-varying torques of forces are studied in this paper. The restoring torque also depends on the small angle of nutation. The following problem is formulated in the current research: analyzing the solution’s behavior of system of motion equations for the values of small parameter different from zero on a sufficiently large interval of time. An averaging method is used for solving the problem. We have established the terms for feasibility to average (with respect to phase of nutation angle) the equations of rigid body motion related to Lagrange case. The averaged system of equations is received in the first approximation. Asymptotic approach permits to obtain some qualitative results and to describe evolution of motion using simplified averaged equations. In the case of rotational motion of a body immersed in the linear-dissipative medium, the numerical integration of the averaged system of equations is conducted. A new class of rotational motions of the Lagrange top is studied for an unsteady perturbation torque, as well as restoring torque that slowly varies with time and depends on the small nutation angle. These results are an extension of our former works (in the absence of slow time τ in the expressions for μ and M i or μ, where M i are dependent on τ only).

PROBLEMS OF EVOLUTION OF RIGID BODY MOTION SIMILAR TO LAGRANGE TOP

The problem of evolution of the rigid body rotations about a fixed point continues to attract the attention of researches. In many cases, the motion in the Lagrange case can be regarded as a generating motion of the rigid body. In this case the body is assumed to have a fixed point and to be in the gravitational field, with the center of mass of the body and the fixed point both lying on the dynamic symmetry axes of the body. A restoring torque, analogues to the moment of the gravity forces, is created by the aerodynamic forces acting on the body in the gas flow. Therefore, the motions, close to the Lagrange case, have been investigated in a number of works on the aircraft dynamics, where various perturbation torques were taken into account in addition to the restoring torque. Many works have studied the rotational motion of a heavy rigid body about a fixed point under the action of perturbation and restoring torques. The correction of the studied models is carried out by taking into account external and internal perturbation factors of various physical nature as well as relevant assumptions according to unperturbed motion. The results of reviewed works may be of interest to specialists in the field of rigid body dynamics, gyroscopy, and applications of asymptotic methods. The authors of this papers present a new approach for the investigation of perturbed motions of Lagrange top for perturbations which assumes averaging with respect to the phase of the nutation angle. Nonlinear equations of motions are simplified and solved explicitly, so that the description of motion is obtained. Asymptotic approach permits to obtain some qualitative results and to describe evolution of rigid body motion using simplified averaged equations. Thus it is possible to avoid numerical integration. The authors present a unified approach to the dynamics of angular motions of rigid bodies subject to perturbation torques of different physical nature. These papers contains both the basic foundations of the rigid body dynamics and the application of the asymptotic method of averaging. The approach based on the averaging procedure is applicable to rigid bodies closed to Lagrange gyroscope. The presented brief survey does not purport to be complete and can be expanded. However, it is clear from this survey that there is an literature on the dynamics of rigid body moving about a fixed point under the influence of perturbation torques of various physical nature. The research in this area is in connection with the problems of motion of flying vehicles, gyroscopes, and other objects of modern technology

Structurally Stable Nondegenerate Singularities of Integrable Systems

In this paper, we study singularities of the Lagrangian fibration given by a completely integrable system. We prove that a non-degenerate singular fibre satisfying the so-called connectedness condition is structurally stable under (small enough) real-analytic integrable perturbations of the system. In other words, the topology of the fibration in a neighbourhood of such a fibre is preserved after any such perturbation. As an illustration, we show that a saddle-saddle singularity of the Kovalevskaya top is structurally stable under real-analytic integrable perturbations, but structurally unstable under $C^\infty$ smooth integrable perturbations.

Two flows Kowalevski top as the full genus two Jacobi’s inversion problem and Sp(4, ) lie group structure

By using the first and second flows of the Kowalevski top, we can recreate the Kowalevski top into two−flows Kowalevski top, which has two−time variables. Then, we demonstrate that equations of the two−flows Kowalevski top become those of the full genus two Jacobi inversion problem. In addition to the Lax pair for the first flow, we construct a Lax pair for the second flow. Using the first and second flows, we demonstrate that the Lie group structure of these two Lax pairs is Sp(4, )/. With the two−flows Kowalevski top, we can conclude that the Lie group structure of the genus two hyperelliptic function is Sp(4, )/.