Euler Top Research Articles

Canonical System on Elliptic Curves

We deduce a canonical algebraic complete integrable system using the representation of the Heisenberg group. This system is shown to have solutions equivalent to those of the rigid body motion on SO(3) (Euler Top).

Poisson structure of dynamical systems with three degrees of freedom

It is shown that the Poisson structure of dynamical systems with three degrees of freedom can be defined in terms of an integrable one-form in three dimensions. Advantage is taken of this fact and the theory of foliations is used in discussing the geometrical structure underlying complete and partial integrability. Techniques for finding Poisson structures are presented and applied to various examples such as the Halphen system which has been studied as the two-monopole problem by Atiyah and Hitchin. It is shown that the Halphen system can be formulated in terms of a flat SL(2,R)-valued connection and belongs to a nontrivial Godbillon–Vey class. On the other hand, for the Euler top and a special case of three-species Lotka–Volterra equations which are contained in the Halphen system as limiting cases, this structure degenerates into the form of globally integrable bi-Hamiltonian structures. The globally integrable bi-Hamiltonian case is a linear and the SL(2,R) structure is a quadratic unfolding of an integrable one-form in 3+1 dimensions. It is shown that the existence of a vector field compatible with the flow is a powerful tool in the investigation of Poisson structure and some new techniques for incorporating arbitrary constants into the Poisson one-form are presented herein. This leads to some extensions, analogous to q extensions, of Poisson structure. The Kermack–McKendrick model and some of its generalizations describing the spread of epidemics, as well as the integrable cases of the Lorenz, Lotka–Volterra, May–Leonard, and Maxwell–Bloch systems admit globally integrable bi-Hamiltonian structure.

Bifurcation of liquid drops

The reduced energy momentum method of Simo et al. (1991) may be used to locate bifurcations of steady motions of conservative systems from symmetric steady motions. A family of functionals parametrized by the isotropy algebra can be used to characterize a symmetric relative equilibrium. A variation of the Equivariant Branching Lemma is used to construct a scalar equilibrium equation on an appropriate submanifold of the configuration manifold. This equation can be solved for the free parameter associated to the isotropy subalgebra to obtain stability and bifurcation results. Applications to three mechanical systems are considered: the heavy Lagrange top, the Riemann ellipsoids, and planar incompressible free boundary flow. The bifurcation analyses of the heavy top and the Riemann ellipsoids recapture the classical results of Routh and Riemann. The analysis of swirling planar flows establishes the existence of nonrigid steady motions of asymmetric fluid regions bifurcating from a rigidly rotating fluid disc and improves an existing formal stability result for the rigidly rotating disc. A common feature of these three examples is the existence of a range of angular velocities for which the symmetric steady motions are both (formally) orbitally and linearly stable and yet bifurcations to asymmetric steady motions occur.

Canonical system on elliptic curves

We deduce a canonical algebraic complete integrable system using the representation of the Heisenberg group. This system is shown to have solutions equivalent to those of the rigid body motion on SO(3) (Euler Top).

Cyclic coverings of abelian varieties and the Goryachev-Chaplygin top

Cyclic coverings of abelian varieties and the Goryachev-Chaplygin top

Separable coordinates, integrability and the Niven equations

Finite (polynomial) solutions of Laplace's equation are investigated. The unifying features of this study are the so-called Niven equations which yield the dimension of the space of such solutions. In carrying out this study complete sets of solutions are obtained on the n-dimensional sphere in terms of ellipsoidal coordinates. This corresponds to an integrable system having all the integrals of motion given by quadratic orbits of the universal enveloping algebra of O(n+1). They call this system the n-dimensional Euler top. The spectrum of the integrals of motion has been recently computed for n=3 by Komarov and Kuznetsov (1991) using results originally due to Niven (1891). These calculations are extended to arbitrary dimension.

The heavy top: a geometric treatment

The authors consider the steady group motions of a rigid body with a fixed point moving in a gravitational field. For an asymmetric top, rotation about the axis of gravity is the only permissible group motion; for a Lagrange top, simultaneous rotation about the axis of gravity and spin about the axis of symmetry of the top is permissible. The analysis of the heavy top follows the reduced energy momentum method of Simo et al. (1991), which is applicable to a wide range of conservative systems with symmetry. Steady group motions are characterized as solutions of a variational problem on the configuration space; local minima of the amended potential correspond to nonlinearly orbitally stable steady motions. The combination of a low-dimensional configuration space and a relatively large number of parameters that produce substantial qualitative changes in the dynamics makes possible a thorough, detailed analysis, which not only reproduces the classical results for this well known system, but leads to some results which we believe are new. They determine general equilibrium and nonlinear stability conditions for steady group motions of a heavy top with a fixed point. They rederive the classical equilibrium and stability conditions for sleeping tops and precessing Lagrange tops, analyse in detail the stability of a family of steady rotations of tilted tops which bifurcate from the branch of sleeping tops parametrized by angular velocity, and classify the possible stability transitions of an arbitrary top as its angular velocity is increased. They obtain a simple, general expression for the characteristic polynomial of the linearized equations of motion and analyse the linear stability of both sleeping tops and the family of tilted top motions previously mentioned. Finally, they demonstrate the coexistence of stable branches of steadily precessing tops that bifurcate from the branch of sleeping Lagrange tops throughout the range of angular velocities for which the sleeping top is stable.

Reductions of self-dual Yang-Mills fields and classical systems.

One-dimensional reductions of the self-dual Yang-Mills equations yield various classical systems depending on the choice of the Lie algebra associated with the self-dual fields. Included are the Euler-Arnold equations for rigid bodies in n dimensions, the Kovalevskaya top, and a generalization of the Nahm equation which is related to a classical third-order differential equation possessing a movable natural boundary in the complex plane.

Generalization of the Goryachev-Chaplygin gyrostat in quantum mechanics

A generalization of the Goryachev-Chaplygin top, which admits the application of the quantum method of the inverse scattering problem, is constructed. The additional parameter introduced plays the same role as the spin of a node in spin chains.

Lax representation and new formulae for the Goryachev-Chaplygin top

A Lax pair for the Goryachev-Chaplygin top (GCT) is presented. The pair is obtained from that for the Kowalewski top found by Reyman and Semenov-Tian-Shansky (1987). Explicit formulae are constructed in terms of theta functions for the dynamical variables.

Lax representation with a spectral parameter for the kowalevski top and its generalizations

Lax representation with a spectral parameter for the kowalevski top and its generalizations

The solutions of the equations of motion of the kovalevskaya top in finite form

Elementary transformations of phase variables are used to obtain several novel forms of the system of Euler-Poisson (EP) equations with Kovalevskaya conditions /1/. It is shown that the use of such equations makes possible not only the detection, but also the construction in a finite explicit form, of a solution for all four classes of degenerate motions mentioned by Appel'rot in /4/, and inadequately studied up to now, without using Kovalevskaya quadratures /2, 3/. In particular, an explicit solution is given in a novel form for the third class. The new forms of the equations of motion are used in a unique manner to study some particular results of investigation of degenerate solutions obtained by various methods /5–8/.

A Lax pair for Kowalevski's top

We show that on each level surface of the invariants, the equations of the Kowalevski top are equivalent to a Neumann system describing the motion of a mass point on the sphere S 2:| p|=1 under the influence of force − Qp. This allows us to write a global Lax pair for the Kowalevski system and to show that Kowalevski's original reduction of the integration of the equations of motion to Jacobi's inversion theorem is identical to the introduction of “elliptical spherical coordinates” for integrating C. Neumann's system.

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.

The Goryachev-Chaplygin top and the Toda lattice

The Goryachev-Chaplygin top is a rigid body rotating about a fixed point with principal moments of inertiaA, B, C satisfyingA=B=4C and with center of mass lying in the equatorial plane. The problem is algebraically completely integrable as a linear flow on a hyperelliptic Jacobian, only upon putting the principal angular momentum in the horizontal plane. This system admits asymptotic solutions with fractional powers int and depending on 4 degrees of freedom. As a consequence, the affine invariant surfaces of the Chaplygin top are double covers of the hyperelliptic Jacobian above, ramified along two translates of the theta divisor, touching in one point. This system is an instance of a (master) system of differential equations in 7 unknowns having 5 quadratic constants of motion; a careful analysis of this system reveals an intimate (rational) relationship with the 3-body periodic Toda lattice.

An elegant integrable system

A set of evolution equations of the form dY dt =Y 2 (for off-diagonal elements) where Y is an n× n symmetric matrix with vanishing diagonal elements, is shown to admit a class of integrable solutions, which are related to a generalisation of the Euler top.

Goryachev-Chaplygin top and the inverse scattering method

It is shown that the inverse scattering method applies to both the classical and the quantum Goryachev-Chaplygin top. A new method, based on theR-matrix formalism, is proposed for deriving the equations determining the spectrum of the quantum integrals of motion. This method is of a rather general nature and may serve as an alternative to the so-called algebraic Bethe Ansatz.

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.

Quantum-mechanical Goryachev-Chaplygin top in a strong field

Quantum-mechanical Goryachev-Chaplygin top in a strong field

Euler-Poisson equations on Lie algebras and the N -dimensional heavy rigid body

The classical Euler-Poisson equations describing the motion of a heavy rigid body about a fixed point are generalized to arbitrary Lie algebras as Hamiltonian systems on coad-joint orbits of a tangent bundle Lie group. the N-dimensional Lagrange and symmetric heavy top are thereby shown to be completely integrable.