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2013
Impact Factor

Andrey Tsiganov

Andrey Tsiganov
ul. Ulyanovskaya 1, Petrodvorets, 198504, St. Petersburg
St. Petersburg State University

Professor V.A. Folk Institute of Physics St Petersburg State University

Publications:

Tsiganov A. V.
Bäcklund Transformations for the Nonholonomic Veselova System
2017, vol. 22, no. 2, pp.  163-179
Abstract
We present auto and hetero Bäcklund transformations of the nonholonomic Veselova system using standard divisor arithmetic on the hyperelliptic curve of genus two. As a by-product one gets two natural integrable systems on the cotangent bundle to the unit two-dimensional sphere whose additional integrals of motion are polynomials in the momenta of fourth order.
Keywords: nonholonomic dynamical system, bi-Hamiltonian geometry, Bäcklund transformations
Citation: Tsiganov A. V.,  Bäcklund Transformations for the Nonholonomic Veselova System, Regular and Chaotic Dynamics, 2017, vol. 22, no. 2, pp. 163-179
DOI:10.1134/S1560354717020058
Tsiganov A. V.
Killing Tensors with Nonvanishing Haantjes Torsion and Integrable Systems
2015, vol. 20, no. 4, pp.  463-475
Abstract
The second-order integrable Killing tensor with simple eigenvalues and vanishing Haantjes torsion is the key ingredient in construction of Liouville integrable systems of Stäckel type. We present two examples of the integrable systems on three-dimensional Euclidean space associated with the second-order Killing tensors possessing nontrivial torsion. Integrals of motion for these integrable systems are the second- and fourth-order polynomials in momenta, which are constructed using a special family of the Killing tensors.
Keywords: Killing tensors, integrable systems, separation of variables
Citation: Tsiganov A. V.,  Killing Tensors with Nonvanishing Haantjes Torsion and Integrable Systems, Regular and Chaotic Dynamics, 2015, vol. 20, no. 4, pp. 463-475
DOI:10.1134/S156035471504005X
Tsiganov A. V.
Simultaneous Separation for the Neumann and Chaplygin Systems
2015, vol. 20, no. 1, pp.  74-93
Abstract
The Neumann and Chaplygin systems on the sphere are simultaneously separable in variables obtained from the standard elliptic coordinates by the proper Bäcklund transformation. We also prove that after similar Bäcklund transformations other curvilinear coordinates on the sphere and on the plane become variables of separation for the system with quartic potential, for the Hénon–Heiles system and for the Kowalevski top. This allows us to speak about some analog of the hetero Bäcklund transformations relating different Hamilton–Jacobi equations.
Keywords: bi-Hamiltonian geometry, Bäcklund transformations, separation of variables
Citation: Tsiganov A. V.,  Simultaneous Separation for the Neumann and Chaplygin Systems, Regular and Chaotic Dynamics, 2015, vol. 20, no. 1, pp. 74-93
DOI:10.1134/S1560354715010062
Tsiganov A. V.
On the Lie Integrability Theorem for the Chaplygin Ball
2014, vol. 19, no. 2, pp.  185-197
Abstract
The necessary number of commuting vector fields for the Chaplygin ball in the absolute space is constructed. We propose to get these vector fields in the framework of the Poisson geometry similar to Hamiltonian mechanics.
Keywords: nonholonomic dynamical system, Poisson bracket, Lie theorem, Chaplygin ball
Citation: Tsiganov A. V.,  On the Lie Integrability Theorem for the Chaplygin Ball, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 185-197
DOI:10.1134/S1560354714020038
Tsiganov A. V.
On the Poisson Structures for the Nonholonomic Chaplygin and Veselova Problems
2012, vol. 17, no. 5, pp.  439-450
Abstract
We discuss a Poisson structure, linear in momenta, for the generalized nonholonomic Chaplygin sphere problem and the $LR$ Veselova system. Reduction of these structures to the canonical form allows one to prove that the Veselova system is equivalent to the Chaplygin ball after transformations of coordinates and parameters.
Keywords: nonholonomic mechanics, Poisson brackets
Citation: Tsiganov A. V.,  On the Poisson Structures for the Nonholonomic Chaplygin and Veselova Problems, Regular and Chaotic Dynamics, 2012, vol. 17, no. 5, pp. 439-450
DOI:10.1134/S1560354712050061
Tsiganov A. V.
One Invariant Measure and Different Poisson Brackets for Two Non-Holonomic Systems
2012, vol. 17, no. 1, pp.  72-96
Abstract
We discuss the non-holonomic Chaplygin and the Borisov–Mamaev–Fedorov systems, for which symplectic forms are different deformations of the square root from the corresponding invariant volume form. In both cases second Poisson bivectors are determined by $L$-tensors with non-zero torsion on configuration space, in contrast with the well-known Eisenhart–Benenti and Turiel constructions.
Keywords: non-holonomic mechanics, Chaplygin’s rolling ball, Poisson brackets
Citation: Tsiganov A. V.,  One Invariant Measure and Different Poisson Brackets for Two Non-Holonomic Systems, Regular and Chaotic Dynamics, 2012, vol. 17, no. 1, pp. 72-96
DOI:10.1134/S1560354712010078
Tsiganov A. V., Khudobakhshov V. A.
Integrable Systems on the Sphere Associated with Genus Three Algebraic Curves
2011, vol. 16, no. 3-4, pp.  396-414
Abstract
New variables of separation for few integrable systems on the two-dimensional sphere with higher order integrals of motion are considered in detail. We explicitly describe canonical transformations of initial physical variables to the variables of separation and vice versa, calculate the corresponding quadratures and discuss some possible integrable deformations of initial systems.
Keywords: integrable systems, separation of variables, Abel equations
Citation: Tsiganov A. V., Khudobakhshov V. A.,  Integrable Systems on the Sphere Associated with Genus Three Algebraic Curves, Regular and Chaotic Dynamics, 2011, vol. 16, no. 3-4, pp. 396-414
DOI:10.1134/S1560354711030117
Tsiganov A. V.
New variables of separation for particular case of the Kowalevski top
2010, vol. 15, no. 6, pp.  659-669
Abstract
We discuss the polynomial bi-Hamiltonian structures for the Kowalevski top in special case of zero square integral. An explicit procedure to find variables of separation and separation relations is considered in detail.
Keywords: Kowalevski top, separation of variables, bi-Hamiltonian geometry, differential geometry, algebraic curves
Citation: Tsiganov A. V.,  New variables of separation for particular case of the Kowalevski top, Regular and Chaotic Dynamics, 2010, vol. 15, no. 6, pp. 659-669
DOI:10.1134/S156035471006002X
Tsiganov A. V.
Change of the time for the periodic Toda lattices and natural systems on the plane with higher order integrals of motion
2009, vol. 14, no. 4-5, pp.  541-549
Abstract
We discuss some special classes of canonical transformations of the time variable, which relate different integrable systems. Such dual systems have different integrals of motion, Lax equations, separated variables and bi-hamiltonian structures. As an example the two-dimensional periodic Toda lattices associated with the classical root systems and the dual natural systems on the palne are considered in detail.
Keywords: Toda lattices, change of time
Citation: Tsiganov A. V.,  Change of the time for the periodic Toda lattices and natural systems on the plane with higher order integrals of motion, Regular and Chaotic Dynamics, 2009, vol. 14, no. 4-5, pp. 541-549
DOI:10.1134/S1560354709040108
Tsiganov A. V.
Leonard Euler: Addition Theorems and Superintegrable Systems
2009, vol. 14, no. 3, pp.  389-406
Abstract
We consider the Euler approach to constructing to investigating of the superintegrable systems related to the addition theorems. As an example we reconstruct Drach systems and get some new two-dimensional superintegrable Stäckel systems.
Keywords: superintegrable systems, addition theorems
Citation: Tsiganov A. V.,  Leonard Euler: Addition Theorems and Superintegrable Systems, Regular and Chaotic Dynamics, 2009, vol. 14, no. 3, pp. 389-406
DOI:10.1134/S1560354709030034
Kostov N. A., Tsiganov A. V.
New Lax pair for restricted multiple three wave interaction system, quasiperiodic solutions and bi-Hamiltonian structure
2008, vol. 13, no. 6, pp.  593-601
Abstract
We study restricted multiple three wave interaction system by the inverse scattering method. We develop the algebraic approach in terms of classical $r$-matrix and give an interpretation of the Poisson brackets as linear $r$-matrix algebra. The solutions are expressed in terms of polynomials of theta functions. In particular case for $n = 1$ in terms of Weierstrass functions.
Keywords: Lax pair, bi-Hamiltonian structure, three wave interaction system
Citation: Kostov N. A., Tsiganov A. V.,  New Lax pair for restricted multiple three wave interaction system, quasiperiodic solutions and bi-Hamiltonian structure, Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 593-601
DOI:10.1134/S1560354708060099
Tsiganov A. V.
On Maximally Superintegrable Systems
2008, vol. 13, no. 3, pp.  178-190
Abstract
Locally any completely integrable system is maximally superintegrable system since we have the necessary number of the action-angle variables. The main problem is the construction of the single-valued additional integrals of motion on the whole phase space by using these multi-valued action-angle variables. Some constructions of the additional integrals of motion for the Stäckel systems and for the integrable systems related with two different quadratic $r$-matrix algebras are discussed. Among these system there are the open Heisenberg magnet and the open Toda lattices associated with the different root systems.
Keywords: superintegrable systems, Toda lattices, Stackel systems
Citation: Tsiganov A. V.,  On Maximally Superintegrable Systems, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 178-190
DOI:10.1134/S1560354708030040
Tsiganov A. V.
The Poisson Bracket Compatible with the Classical Reflection Equation Algebra
2008, vol. 13, no. 3, pp.  191-203
Abstract
We introduce a family of compatible Poisson brackets on the space of $2 \times 2$ 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.
Keywords: Poisson bracket, bi-hamiltonian structure, reflection equation algebra
Citation: Tsiganov A. V.,  The Poisson Bracket Compatible with the Classical Reflection Equation Algebra, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 191-203
DOI:10.1134/S1560354708030052
Kostko A. L., Tsiganov A. V.
On the Bi-Hamiltonian Structures for the Goryachev–Chaplygin Top
2008, vol. 13, no. 1, pp.  37-44
Abstract
The bi-hamiltonian structures for the Goryachev–Chaplygin top are constructed by using the Chaplygin variables and the Sklyanin bracket.
Keywords: integrable and bi-hamiltonian system, the Goryachev–Chaplygin top
Citation: Kostko A. L., Tsiganov A. V.,  On the Bi-Hamiltonian Structures for the Goryachev–Chaplygin Top, Regular and Chaotic Dynamics, 2008, vol. 13, no. 1, pp. 37-44
DOI:10.1134/S156035470801005X
Tsiganov A. V.
Towards a classification of natural integrable systems
2006, vol. 11, no. 3, pp.  343-362
Abstract
For construction and classification of the natural integrable systems we propose to use a criterion of separability in Darboux–Nijenhuis coordinates, which can be tested without an a priori explicit knowledge of these coordinates. As an example we apply this method for the search of integrable systems on the plane.
Keywords: integrable systems, bi-Hamiltonian geometry, separation of variables
Citation: Tsiganov A. V.,  Towards a classification of natural integrable systems , Regular and Chaotic Dynamics, 2006, vol. 11, no. 3, pp. 343-362
DOI:10.1070/RD2006v011n03ABEH000358
Grigoryev Y. A., Tsiganov A. V.
Symbolic software for separation of variables in the Hamilton–Jacobi equation for the $L$-systems
2005, vol. 10, no. 4, pp.  413-422
Abstract
We discuss a computer implementation of the known algorithm of finding separation coordinates for a special class of orthogonal separable systems called $L$-systems or Benenti systems.
Keywords: integrable systems, Hamilton–Jacobi equation, separation of variables
Citation: Grigoryev Y. A., Tsiganov A. V.,  Symbolic software for separation of variables in the Hamilton–Jacobi equation for the $L$-systems , Regular and Chaotic Dynamics, 2005, vol. 10, no. 4, pp. 413-422
DOI:10.1070/RD2005v010n04ABEH000323
Tsiganov A. V.
On the Steklov–Lyapunov case of the rigid body motion
2004, vol. 9, no. 2, pp.  77-89
Abstract
We construct a Poisson map between manifolds with linear Poisson brackets corresponding to the two samples of Lie algebra $e(3)$. Using this map we establish equivalence of the Steklov–Lyapunov system and the motion of a particle on the surface of the sphere under the influence of the fourth order potential. To study separation of variables for the Steklov case on the Lie algebra $so(4)$ we use the twisted Poisson map between the bi-Hamiltonian manifolds $e(3)$ and $so(4)$.
Citation: Tsiganov A. V.,  On the Steklov–Lyapunov case of the rigid body motion, Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 77-89
DOI:10.1070/RD2004v009n02ABEH000267
Komarov I. V., Tsiganov A. V.
On integration of the Kowalevski gyrostat and the Clebsch problems
2004, vol. 9, no. 2, pp.  169-187
Abstract
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.
Citation: Komarov I. V., Tsiganov A. V.,  On integration of the Kowalevski gyrostat and the Clebsch problems, Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 169-187
DOI:10.1070/RD2004v009n02ABEH000274
Kostko A. L., Tsiganov A. V.
Noncanonical transformations of the spherical top
2003, vol. 8, no. 2, pp.  143-154
Abstract
We discuss noncanonical transformations connecting different integrable systems on the symplectic leaves of the Poisson manifolds. The special class of transformations, which consists of the symplectic mappings of symplectic leaves and of the parallel transports induced by diffeomorphisms in the base of symplectic foliation, is considered. As an example, we list integrable systems associated with the spherical top. The corresponding additional integrals of motion are second, third and six order polynomials in momenta.
Citation: Kostko A. L., Tsiganov A. V.,  Noncanonical transformations of the spherical top, Regular and Chaotic Dynamics, 2003, vol. 8, no. 2, pp. 143-154
DOI:10.1070/RD2003v008n02ABEH000233
Tsiganov A. V.
On Integrable Deformation of the Poincaré System
2002, vol. 7, no. 3, pp.  331-336
Abstract
An integrable deformation of the Poincaré system is considered. This system is derived by using the general $r$-matrix theory.
Citation: Tsiganov A. V.,  On Integrable Deformation of the Poincaré System, Regular and Chaotic Dynamics, 2002, vol. 7, no. 3, pp. 331-336
DOI:10.1070/RD2002v007n03ABEH000215
Tsiganov A. V.
On the Invariant Separated Variables
2001, vol. 6, no. 3, pp.  307-326
Abstract
An integrable Hamiltonian system on a Poisson manifold consists of a Lagrangian foliation $\mathscr{F}$ 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 $\mathscr{F}$ and belong to the invariant intersection of all the subfoliations of $\mathscr{F}$. In this paper we show that for many known integrable systems this invariance property allows us to calculate their separated variables explicitly.
Citation: Tsiganov A. V.,  On the Invariant Separated Variables, Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 307-326
DOI:10.1070/RD2001v006n03ABEH000179
Tsiganov A. V.
The Kepler Canonical Transformations of the Extended Phase Space
2000, vol. 5, no. 1, pp.  117-127
Abstract
We consider compositions of the transformations of the time variable and canonical transformations of the other coordinates, which map a completely integrable system into another completely integrable system. Change of the time gives rise to transformations of the integrals of motion and the Lax pairs, transformations of the corresponding spectral curves and $R$-matrices.
Citation: Tsiganov A. V.,  The Kepler Canonical Transformations of the Extended Phase Space, Regular and Chaotic Dynamics, 2000, vol. 5, no. 1, pp. 117-127
DOI:10.1070/RD2000v005n01ABEH000128
Tsiganov A. V.
Lax Representation for an Integrable Motion on the Sphere With a Qubic Second Invariant
1999, vol. 4, no. 3, pp.  21-29
Abstract
The motion on the sphere in a potential $V \simeq (x_1x_2x_3)^{-2/3}$ is considered. The physical origin, the Lax representation and the linearization procedure for this two-dimensional integrable system are considered.
Citation: Tsiganov A. V.,  Lax Representation for an Integrable Motion on the Sphere With a Qubic Second Invariant, Regular and Chaotic Dynamics, 1999, vol. 4, no. 3, pp. 21-29
DOI:10.1070/RD1999v004n03ABEH000112

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