Papers by Andrey Tsiganov
We introduce a family of compatible Poisson brackets on the space of 2× 2 polynomial matrices, wh... more We introduce a family of compatible Poisson brackets on the space of 2× 2 polynomial matrices, which contains the reflection equation algebra bracket. Then we use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes the XXX Heisenberg magnet with boundary conditions, the generalized Toda lattices and the Kowalevski top.
bi-hamiltonian geometry of some integrable systems on
Cubic invariants for two-dimensional degenerate Hamiltonian systems are considered by using varia... more Cubic invariants for two-dimensional degenerate Hamiltonian systems are considered by using variables of separation of the associated Stäckel problems with quadratic integrals of motion. For the superintegrable Stäckel systems the cubic invariant is shown to admit new algebro-geometric representation that is far more elementary than the all the known representations in physical variables. A complete list of all known systems on the plane which admit a cubic invariant is discussed. 1
For the Kowalevski gyrostat change of variables similar to that of the Kowalevski top is done. We... more 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 gyrostat practically coincide with elliptic coordinates on sphere for the Clebsch case. Equivalence of considered integrable systems allows to construct two Lax matrices for the gyrostat using known rational and elliptic Lax matrices for the Clebsch model. Associated with these matrices solutions of the Clebsch system and, therefore, of the Kowalevski gyrostat problem are discussed. The Kötter solution of the Clebsch system in modern notation is presented in detail.
Regular and Chaotic Dynamics, 2000
Regular and Chaotic Dynamics, 2003
Theoretical and Mathematical Physics - THEOR MATH PHYS-ENGL TR, 2000
We investigate the explicit construction of a canonical transformation of the time variable and t... more We investigate the explicit construction of a canonical transformation of the time variable and the Hamiltonian whereby a given completely integrable system is mapped into another integrable system. The change of time induces a transformation of the equations of motion and of their solutions, the integrals of motion, the methods of separation of variables, the Lax matrices, and the correspondingr-matrices. For several specific families of integrable systems (Toda chains, Holt systems, and Stckel-type systems), we construct canonical transformations of time in the extended phase space that preserve the integrability property.
An integrable Hamiltonian system on a Poisson manifold consists of a Lagrangian foliation and a H... more An integrable Hamiltonian system on a Poisson manifold consists of a Lagrangian foliation and a Hamilton function H . The invariant separated variables are independent on values of integrals of motion and Casimir functions. It means that they are invariant with respect to abelian group of symplectic diffeomorphisms of and belong to the invariant intersection of all the subfoliations of . In this paper we show that for many known integrable systems this invariance property allows us to calculate their separated variables explicitly.
Let M be a 2n-dimensional symplectic manifold (phase space) with coordinates {pj, qj}j=1. The Ham... more Let M be a 2n-dimensional symplectic manifold (phase space) with coordinates {pj, qj}j=1. The Hamilton function H(p, q) defines the hamiltonian dynamical system on M. Here p and q denote p1, . . . , pn and q1, . . . , qn, respectively. By adding to M the time qn+1 = t and the Hamiltonian pn+1 = −H one gets 2n + 2dimensional extended phase space ME of the given hamiltonian system [9]. Canonical functional S on ME has the following completely symmetric form
arXiv: Exactly Solvable and Integrable Systems, 2001
A polynomial deformation of the Kowalewski top is considered. This deformation includes as a dege... more 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 $5\times 5$ matrix Lax pair for the deformed Kowalewski top is proposed. Also deformations of the two-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 Semenov-Tian {S}hansky. In addition, a similar deformation of the Goryachev-Chaplygin top and its $3\times 3$ matrix Lax representation is constructed.
The search for integrable dynamical systems is one of the most fascinating branches of classical ... more The search for integrable dynamical systems is one of the most fascinating branches of classical physics. Integrable systems are quite rare and still only a few examples are known. In this paper we will present a new direct method for construction of natural integrable systems on ωN bi-Hamiltonian manifolds. A bi-Hamiltonian manifold is a smooth manifold M endowed with a pair of compatible Poisson tensors P and P ′ associated with the Poisson brackets {., .} and {., .}′ respectively. The class of manifolds we will consider are particular bi-Hamiltonian manifolds where one of the two Poisson tensors in non degenerate and thus defines a symplectic form ω = P −1 and a recursion operator N = P ′P−1. Dynamical integrable system on M with functionally independent integrals of motion H1, . . . ,Hn in the bi-involution {Hi,Hj} = 0 and {Hi,Hj} = 0 for all i, j (1.1)
One of the cornerstones of the quantum and classical inverse scattering method is the r-matrix al... more One of the cornerstones of the quantum and classical inverse scattering method is the r-matrix algebras with a spectral parameter dependence. The rich structure of such algebras allows us to cover with this formalism a variety of known integrable models and to find new ones. In this paper we consider representation of the Sklyanin r-matrix algebra on symplectic leaves of the Lie algebra so(4) = so(3) ⊕ so(3) using standard techniques of the reflection equation theory. As a matter of fact, the construction leads automatically to an integrable system with additional quartic integral of motion. Let si, ti, i = 1, 2, 3, be coordinates on the algebra so(4) = so(3) ⊕ so(3) with the following Lie-Poisson brackets { si , sj } = εijk sk , { si , tj } = 0 , { ti , tj } = εijktk , (1.1)
A hierarchy of commutative Poisson subalgebras for the Sklyanin bracket is proposed. Each of the ... more 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 variables for them. The models obtained are deformations of known integrable systems like the Goryachev-Chaplygin top, the Toda lattice and the Heisenberg model.
arXiv: Exactly Solvable and Integrable Systems, 2004
An integrable deformation of the known integrable model of two interacting p-dimensional and q-di... more An integrable deformation of the known integrable model of two interacting p-dimensional and q-dimensional spherical tops is considered. After reduction this system gives rise to the generalized Lagrange and the Kowalevski tops. The corresponding Lax matrices and classical r-matrices are calculated.
arXiv: Exactly Solvable and Integrable Systems, 2000
We discuss some special classes of canonical transformations of the extended phase space, which r... more We discuss some special classes of canonical transformations of the extended phase space, which relate integrable systems with a common Lagrangian submanifold. Various parametric forms of trajectories are associated with different integrals of motion, Lax equations, separated variables and action-angles variables. In this review we will discuss namely these induced transformations instead of the various parametric form of the geometric objects.
Symmetry, Integrability and Geometry: Methods and Applications
The counterparts of the Bertrand-Darboux equation for the two-dimensional nonholonomic systems ar... more The counterparts of the Bertrand-Darboux equation for the two-dimensional nonholonomic systems are discussed. We directly prove that integrable potentials for the nonholonomic Suslov problem, for the Veselova system and the Chaplygin ball on the plane can be recovered from the well-known integrable potentials, which have been obtained many years ago in the framework of Hamiltonian mechanics.
Journal of Physics A: Mathematical and Theoretical
The perturbed Kepler problem is shown to be a bi-Hamiltonian system in spite of the fact that the... more The perturbed Kepler problem is shown to be a bi-Hamiltonian system in spite of the fact that the graph of the Hamilton function is not a hypersurface of translation, which is against a necessary condition for the existence of the bi-Hamiltonian structure according to the Fernandes theorem. In fact, both the initial and perturbed Kepler systems are isochronous systems and, therefore, the Fernandes theorem cannot be applied to them.
Theoretical and Mathematical Physics, Jan 3, 2002
A polynomial deformation of the Kowalewski top is considered. This deformation includes as a dege... more 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 two-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 Semenov-Tian Shansky. In addition, a similar deformation of the Goryachev-Chaplygin top and its 3 × 3 matrix Lax representation is constructed.
Eprint Arxiv Solv Int 9705004, May 6, 1997
In this work we consider superintegrable systems in the classical r-matrix method. By using other... more In this work we consider superintegrable systems in the classical r-matrix method. By using other authomorphisms of the loop algebras we construct new superintegrable systems with rational potentials from geodesic motion on R 2n .
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Papers by Andrey Tsiganov