Andrey Tsiganov
ul. Ulyanovskaya 1, Petrodvorets, 198504, St. Petersburg
St. Petersburg State University
Professor at the Department of Computational Physics, Faculty of Physics, Saint Petersburg State University
Born: March 28, 1963
1980 – 1986: student of the Saint Petersburg State University (SPbSU);
1993 – Candidate of Science (Ph.D.). Thesis title: "Toda Lattices and Some Tops in the Quantum Inverse Scattering Method" (Saint Petersburg State University);
2003 Doctor of Science. Thesis title: “Finite-Dimensional Integrable Systems of Classical Mechanics in the Method of Separation of Variables” (Saint Petersburg State University);
1986 – 1989 Researcher at the D.V. Efremov Scientific Research Institute of Electrophysical Apparatus, Russia, St. Petersburg;
1989 – 1993 Senior Programmer at the Department of Earth Physics, Institute of Physics, Saint Petersburg State University;
1994 – 1997 Researcher at the Department of Mathematical and Computational Physics, Institute of Physics, Saint Petersburg State University;
1998 – 2004 Associate Professor at the Department of Computational Physics, St. Petersburg University;
Since 2004 Professor at the Department of Computational Physics, Saint Petersburg State University;
Member of the editorial boards of the international scientific journals “Regular and Chaotic Dynamics” and “Russian Journal of Nonlinear Dynamics”.
Publications:
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Tsiganov A. V.
Integrable Systems on a Sphere, an Ellipsoid and a Hyperboloid
2023, vol. 28, no. 6, pp. 805-821
Abstract
Affine transformations in Euclidean space generate a correspondence between
integrable systems on cotangent bundles to a sphere, ellipsoid and hyperboloid embedded in
$R^n$. Using this correspondence and the suitable coupling constant transformations, we can get
real integrals of motion in the hyperboloid case starting with real integrals of motion in the
sphere case. We discuss a few such integrable systems with invariants which are cubic, quartic
and sextic polynomials in momenta.
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Tsiganov A. V.
Reduction of Divisors and the Clebsch System
2022, vol. 27, no. 3, pp. 307-319
Abstract
There are a few Lax matrices of the Clebsch system. Poles of the Baker – Akhiezer function determine the class of equivalent divisors on the corresponding spectral curves. According to the Riemann – Roch theorem, each class has a unique reduced representative. We discuss properties of such a reduced divisor on the spectral curve of $3\times 3$ Lax matrix having a natural generalization to $gl^*(n)$ case.
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Borisov A. V., Tsiganov A. V.
On the Nonholonomic Routh Sphere in a Magnetic Field
2020, vol. 25, no. 1, pp. 18-32
Abstract
This paper is concerned with the motion of an unbalanced dynamically symmetric
sphere rolling without slipping on a plane in the presence of an external magnetic field. It
is assumed that the sphere can consist completely or partially of dielectric, ferromagnetic,
superconducting and crystalline materials. According to the existing phenomenological theory,
the analysis of the sphere’s dynamics requires in this case taking into account the Lorentz torque,
the Barnett – London effect and the Einstein – de Haas effect. Using this mathematical model,
we have obtained conditions for the existence of integrals of motion which allow one to reduce
integration of the equations of motion to a quadrature similar to the Lagrange quadrature for
a heavy rigid body.
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Borisov A. V., Tsiganov A. V.
On the Chaplygin Sphere in a Magnetic Field
2019, vol. 24, no. 6, pp. 739-754
Abstract
We consider the possibility of using Dirac’s ideas of the deformation of Poisson
brackets in nonholonomic mechanics. As an example, we analyze the composition of external
forces that do no work and reaction forces of nonintegrable constraints in the model of
a nonholonomic Chaplygin sphere on a plane. We prove that, when a solenoidal field is
applied, the general mechanical energy, the invariant measure and the conformally Hamiltonian
representation of the equations of motion are preserved. In addition, we consider the case of
motion of the nonholonomic Chaplygin sphere in a constant magnetic field taking dielectric
and ferromagnetic (superconducting) properties of the sphere into account. As a by-product
we also obtain two new integrable cases of the Hamiltonian rigid body dynamics in a constant
magnetic field taking the magnetization by rotation effect into account.
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Tsiganov A. V.
The Kepler Problem: Polynomial Algebra of Nonpolynomial First Integrals
2019, vol. 24, no. 4, pp. 353-369
Abstract
The sum of elliptic integrals simultaneously determines orbits in the Kepler problem and the addition of divisors on elliptic curves. Periodic motion of a body in physical space is defined by symmetries, whereas periodic motion of divisors is defined by a fixed point on the curve. The algebra of the first integrals associated with symmetries is a well-known mathematical object, whereas the algebra of the first integrals associated with the coordinates of fixed points is unknown. In this paper, we discuss polynomial algebras of nonpolynomial first integrals of superintegrable systems associated with elliptic curves.
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Tsiganov A. V.
Hamiltonization and Separation of Variables for a Chaplygin Ball on a Rotating Plane
2019, vol. 24, no. 2, pp. 171-186
Abstract
We discuss a non-Hamiltonian vector field appearing in considering the partial motion of a Chaplygin ball rolling on a horizontal plane which rotates with constant angular velocity. In two partial cases this vector field is expressed via Hamiltonian vector fields using a nonalgebraic deformation of the canonical Poisson bivector on $e^*(3)$. For the symmetric ball we also calculate variables of separation, compatible Poisson brackets, the algebra of Haantjes operators and $2\times2$ Lax matrices.
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Tsiganov A. V.
On Discretization of the Euler Top
2018, vol. 23, no. 6, pp. 785-796
Abstract
The application of intersection theory to construction of $n$-point finite-difference equations associated with classical integrable systems is discussed. As an example, we present a few new discretizations of motion of the Euler top sharing the integrals of motion with the continuous time system and the Poisson bracket up to the integer scaling factor.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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Tsiganov A. V., Khudobakhshov V. A.
Integrable Systems on the Sphere Associated with Genus Three Algebraic Curves
2011, vol. 16, nos. 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.
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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.
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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, nos. 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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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 $so(4)$.
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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.
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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.
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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.
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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.
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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.
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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.
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