Related papers: The Clebsch System
This is the first part of a two-part paper describing a new concept of separation of variables applied to the Clebsch integrable case of the Kirchhoff equations. There are two principal novelties: 1) Separating coordinates are constructed…
This is an expository presentation of a completely integrable Hamiltonian system of Clebsch top under a special condition introduced by Weber. After a brief account on the geometric setting of the system, the structure of the Poisson…
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…
The complete variables separation is given for one Hamiltonian system with two degrees of freedom arising in the motion of the Kowalevski type top in two constant fields.
Constrained Hamiltonian systems are investigated by using the Hamilton-Jacobi method. Integration of a set of equations of motion and the action function is discussed. It is shown that we have two types of integrable systems: a) ${\it…
In 1890 German mathematician and physicist W. Hess found new special case of integrability of Euler - Poisson equations of motion of a heavy rigid body with a fixed point. In 1892 P. A. Nekrasov proved that the solution of the problem of…
We propose a systematic procedure called the Clebsch canonization for obtaining a canonical Hamiltonian system that is related to a given Lie-Poisson equation via a momentum map. We describe both coordinate and geometric versions of the…
We present an algebraic geometrical and analytical description of the Goryachev case of rigid body motion. It belongs to a family of systems sharing the same properties: although completely integrable, they are not algebraically integrable,…
The paper is a commentary of one section of the celebrated paper by Sophie Kowalewski on the motion of a rigid body with a fixed point. Its purpose is to show that the results of Kowalewski may be recovered by using the separability…
R. Hirota and K. Kimura discovered integrable discretizations of the Euler and the Lagrange tops, given by birational maps. Their method is a specialization to the integrable context of a general discretization scheme introduced by W. Kahan…
A heavy top with a fixed point and a rigid body in an ideal fluid are important examples of Hamiltonian systems on a dual to the semidirect product Lie algebra $e(n)=so(n)\ltimes\mathbb R^n$. We give a Lagrangian derivation of the…
This is the second part of a paper describing a new concept of separation of variables applied to the classical Clebsch integrable case. The quadratures obtained in Part I (also uploaded in arXiv.org) lead to a new type of the Abel map…
This work investigates the dynamics of closed quantum systems in the Bloch vector representation using methods from rigid body dynamics and the theory of integrable systems. To this end, equations of motion for Bloch components are derived…
The contributions of Sophya Kowalewski to the integrability theory of the equations for the heavy top extend to a larger class of Hamiltonian systems on Lie groups; this paper explains these extensions, and along the way reveals further…
A new view on the Kowalevski top and the Kowalevski integration procedure is presented. For more than a century, the Kowalevski 1889 case, attracts full attention of a wide community as the highlight of the classical theory of integrable…
We consider the problem of motion of several rigid bodies immersed in a perfect compressible fluid. Using the method of convex integration we establish the existence of infinitely many weak solutions with {\it a priori} prescribed motion of…
In this paper we consider cases of existence of invariant measure, additional first integrals, and Poisson structure in a problem of rigid body's rolling without sliding on plane and sphere. The problem of rigid body's motion on plane was…
In this study we work on a novel Hamiltonian system which is Liouville integrable. In the integrable Hamiltonian model, conserved currents can be represented as Binomial polynomials in which each order corresponds to the integral of motion…
In the present paper we derive two well-known integrable cases of rigid body dynamics (the Lagrange top and the Clebsch system) performing an algebraic contraction on the two-body Lax matrices governing the (classical) su(2) Gaudin models.…
The Clebsch method provides a unifying approach for deriving variational principles for continuous and discrete dynamical systems where elements of a vector space are used to control dynamics on the cotangent bundle of a Lie group…