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On integrable system on S 2 with the second integral quartic in the momenta.
, 2004
"... We consider integrable system on the sphere S 2 with an additional integral of fourth order in the momenta. At the special values of parameters this system coincides with the KowalevskiGoryachevChaplygin system. 1 ..."
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We consider integrable system on the sphere S 2 with an additional integral of fourth order in the momenta. At the special values of parameters this system coincides with the KowalevskiGoryachevChaplygin system. 1
On the Lax pairs for the generalized Kowalewski and GoryachevChaplygin tops, Theor
 Math. Phys
"... A polynomial deformation of the Kowalewski top is considered. This deformation includes as a degeneration a new integrable case for the Kirchhoff equations found recently by one of the authors. A 5 × 5 matrix Lax pair for the deformed Kowalewski top is proposed. Also deformations of the twofield Ko ..."
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Cited by 6 (5 self)
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field Kowalewski gyrostat and the so(p,q) Kowalewski top are found. All our Lax pairs are deformations of the corresponding Lax representations found by Reyman and SemenovTian Shansky. In addition, a similar deformation of the GoryachevChaplygin top and its 3 × 3 matrix Lax representation is constructed. 1
Yangian, Truncated Yangian and Quantum Integrable Models
, 1995
"... Based on the RTT relation for the given rational Rmatrix the Yangian and truncated Yangian are discussed. The former can be used to generate the longrange interaction models, whereas the latter can be related to the GoryachevChaplygin(GC) gyrostat. The trigonometric extension of the GoryachevChap ..."
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Based on the RTT relation for the given rational Rmatrix the Yangian and truncated Yangian are discussed. The former can be used to generate the longrange interaction models, whereas the latter can be related to the GoryachevChaplygin(GC) gyrostat. The trigonometric extension of the GoryachevChaplygin
Commutative Poisson subalgebras for the Sklyanin brackets and deformations of some known integrable models, Theor
 Math. Phys
"... A hierarchy of commutative Poisson subalgebras for the Sklyanin bracket is proposed. Each of the subalgebras provides a complete set of integrals in involution with respect to the Sklyanin bracket. Using different representations of the bracket, we find some integrable models and a separation of var ..."
Abstract

Cited by 5 (3 self)
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of variables for them. The models obtained are deformations of known integrable systems like the GoryachevChaplygin top, the Toda lattice and the Heisenberg model. 1 Introduction.
EXPLICIT INTEGRABLE SYSTEMS ON TWO DIMENSIONAL MANIFOLDS WITH A CUBIC FIRST INTEGRAL
, 2010
"... A few years ago Selivanova gave an existence proof for some integrable models, in fact geodesic flows on two dimensional manifolds, with a cubic first integral. However the explicit form of these models hinged on the solution of a nonlinear third order ordinary differential equation which could not ..."
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) containing as special cases examples dueto Goryachev, Chaplygin, Dullin, Matveev and Tsiganov. Integrable systems 1 1
On Integrals of Third Degree in Momenta
, 1999
"... Consider a Riemannian metric on a surface, and let the geodesic flow of the metric have a second integral that is a third degree polynomial in the momenta. Then we can naturally construct a vector field on the surface. We show that the vector field preserves the volume of the surface, and therefor ..."
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, and therefore is a Hamiltonian vector field. As examples we treat the GoryachevChaplygin top, the Toda lattice and the CalogeroMoser system, and construct their global Hamiltonians. We show that the simplest choice of Hamiltonian leads to the Toda lattice. 1
On Integrals of Third Degree in Momenta
"... Consider a Riemannian metric on a surface, and let the geodesic flow of the metric have a second integral that is a third degree polynomial in the momenta. Then we can naturally construct a vector field on the surface. We show that the vector field preserves the volume of the surface, and therefore ..."
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Cited by 2 (2 self)
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is a Hamiltonian vector field. As examples we treat the GoryachevChaplygin top, the Toda lattice and the CalogeroMoser system, and construct their global Hamiltonians. We show that the simplest choice of Hamiltonian leads to the Toda lattice. 1 Introduction Simple questions about the existence
I.S.MAMAEV
, 2000
"... Generalizations of the Kovalevskaya, Chaplygin, Goryachev–Chaplygin and Bogoyavlensky systems on a bundle are considered in this paper. Moreover, a method of introduction of separating variables and action–angle variables is described. Another integration method for the Kovalevskaya top on the bundl ..."
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Generalizations of the Kovalevskaya, Chaplygin, Goryachev–Chaplygin and Bogoyavlensky systems on a bundle are considered in this paper. Moreover, a method of introduction of separating variables and action–angle variables is described. Another integration method for the Kovalevskaya top
A family of the Poisson brackets compatible with the
, 2007
"... We introduce a family of compatible Poisson brackets on the space of 2×2 polynomial matrices, which contains the Sklyanin bracket, and use it to derive a multiHamiltonian structure for a set of integrable systems that includes XXX Heisenberg magnet, the open and periodic Toda lattices, the discrete ..."
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, the discrete selftrapping model and the GoryachevChaplygin gyrostat. 1 Introduction. The ingenious discovery of Magri [6, 7] that integrable Hamiltonian systems usually prove to be biHamiltonian, and vice versa, leads us to the following fundamental problem: given a dynamical system which is Hamiltonian
TOPOLOGICAL ANALYSIS OF CLASSICAL INTEGRABLE SYSTEMS IN THE DYNAMICS OF THE RIGID BODY 1
, 906
"... The equations of motion of a rigid body around a fixed point ˙ν = ν × ω, A ˙ω + ω × Aω = e × ν (1) (in which A is the inertia tensor, ω is the angular velocity, ν is the unit vertical vector, and e is the radiusvector of the center of mass, and all the quantities are related to the moving coordinat ..."
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Cited by 2 (1 self)
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(ν, ω). (3) Of special interest are the general integrability cases: the solutions of Lagrange, Euler, Kovalevskaya, and GoryachevChaplygin. The first two can be included in Smale’s scheme for studying the phase topology of natural systems [4]; their analysis is carried out in [5]. The topology
Results 1  10
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12