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2013
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# 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. Integrable Discretization and Deformation of the Nonholonomic Chaplygin Ball 2017, vol. 22, no. 4, pp.  353-367 Abstract The rolling of a dynamically balanced ball on a horizontal rough table without slipping was described by Chaplygin using Abel quadratures. We discuss integrable discretizations and deformations of this nonholonomic system using the same Abel quadratures. As a by-product one gets a new geodesic flow on the unit two-dimensional sphere whose additional integrals of motion are polynomials in the momenta of fourth order. Keywords: nonholonomic systems, Abel quadratures, arithmetic of divisors Citation: Tsiganov A. V.,  Integrable Discretization and Deformation of the Nonholonomic Chaplygin Ball, Regular and Chaotic Dynamics, 2017, vol. 22, no. 4, pp. 353-367 DOI:10.1134/S1560354717040025
 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