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Соколов, В. В.; Tsiganov, A. V.

doi:10.1023/a:1015109904417

Cited by 30 publications

(14 citation statements)

References 5 publications

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“…This simplifies calculations significantly. In [1], for the system (1) with Hamiltonian function (2), V. V. Sokolov and A. V. Tsiganov gave a Lax representation with a spectral parameter and thereby proved the Liouville complete integrability of this system. This Lax representation generalizes the L-A pair for the Kowalevski gyrostat in a double field found by A. G. Reyman and M. A. Semenov-Tian-Shansky [3].…”

Section: Intrtoductionmentioning

confidence: 99%

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

Oshemkov

Ryabov

Sokolov

2017

Russ. J. Math. Phys.

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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. We find a set of points at which the momentum map has rank 1. This set consists of special periodic motions which correspond to the singular points of a bifurcation diagram on an iso-energetic surface. For such motions the phase variables can be expressed in terms of algebraic functions of a single auxiliary variable. These algebraic functions satisfy a differential equation integrable in elliptic functions of time. It is shown that the corresponding points in the three-dimensional space of the constants of the integrals belong to the intersection of two sheets of the discriminant surface of the Lax curve.

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Section: Intrtoductionmentioning

confidence: 99%

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

Oshemkov

Ryabov

Sokolov

2017

Russ. J. Math. Phys.

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“…According to [7,9], we can try to construct another commutative subalgebra generated by quadratic polynomials on coefficients of T ij (λ), which are integrals of motion for another integrable system associated with the same matrix T (λ).…”

Section: Generic Casementioning

confidence: 99%

“…Namely, using matrix T (λ) (3) and standard machinery of the reflection equation theory [7,9] we can construct another 2 × 2 matrix…”

Section: Generic Casementioning

confidence: 99%

A new integrable system onS²with the second integral quartic in the momenta

Tsiganov

2005

J. Phys. A: Math. Gen.

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“…In this note we consider XXX Heisenberg magnet with boundaries [6,7] using the Lax matrix for the standard two-site XXX Heisenberg magnet…”

Section: Introductionmentioning

confidence: 99%

“…Applying the standard machinery [6,7] to T (λ) (1.1) one gets another integrable systems with cubic and quartic additional integrals of motion. As a matter of fact, the construction leads automatically to the separated variables.…”

Section: Introductionmentioning

confidence: 99%