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.
Abstract
A new approach to exact discretization of the Duffing equation is presented. Integrable discrete maps are obtained by using well-studied operations from the elliptic curve cryptography.
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Tsiganov A. V.
Abstract
In the framework of the Jacobi method we obtain a new integrable system on the plane with a natural Hamilton function and a second integral of motion which is a polynomial of sixth order in momenta. The corresponding variables of separation are images of usual parabolic coordinates on the plane after a suitable Bäcklund transformation. We also present separated relations and prove that the corresponding vector field is bi-Hamiltonian.
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Grigoryev Y. A., Sozonov A. P., Tsiganov A. V.
Abstract
We discuss an algorithmic construction of the auto Bäcklund transformations of Hamilton–Jacobi equations and possible applications of this algorithm to finding new integrable systems with integrals of motion of higher order in momenta. We explicitly present Bäcklund transformations for two Hamiltonian systems on the plane separable in parabolic and elliptic coordinates.
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Tsiganov A. V.
Abstract
We show how to to get variables of separation for the Chaplygin system on the sphere with velocity dependent potential using relations of this system with other integrable system separable in sphero-conical coordinates on the sphere.
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Vershilov A. V., Grigoryev Y. A., Tsiganov A. V.
Abstract
We discuss an application of the Poisson brackets deformation theory to the construction of the integrable perturbations of the given integrable systems. The main examples are the known integrable perturbations of the Kowalevski top for which we get new bi-Hamiltonian structures in the framework of the deformation theory.
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Tsiganov A. V.
Abstract
We discuss an application of the Lie integrability theorem to the nonholonomic system describing the rolling of a dynamically balanced ball on horizontal absolutely rough table without slipping or sliding.
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Bizyaev I. A., Tsiganov A. V.
Abstract
We discuss an embedding of the vector field associated with the nonholonomic Routh sphere in subgroup of the commuting Hamiltonian vector fields associated with this system. We prove that the corresponding Poisson brackets are reduced to canonical ones in the region without of homoclinic trajectories.
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Tsiganov A. V.
Abstract
We prove the trajectory equivalence of the Chaplygin sphere problem, the Veselova system on $e^*(3)$ and a Hamiltonian system on two-dimensional sphere with the non-standard metric.
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Tsiganov A. V.
Abstract
Construction of the Poisson structures for the nonholonomic Chaplygin and Borisov–Mamaev–Fedorov systems is discussed. The corresponding vector fields are conformally Hamiltonian and generalized conformally Hamiltonian vector fields with respect to the linear in momenta Poisson brackets. We suppose that this difference is closely related with the non-trivial deformation of canonical Poisson bivector, which appears in the Borisov–Mamamev–Fedorov case.
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Tsiganov A. V.
Abstract
The main aim of the second part of the paper is a construction of the rational potentials, which may be added to the Hamiltonians of the Chaplygin and Borisov–Mamaev–Fedorov systems without loss of integrability. All these potentials may be considered as natural nonholonomic generalizations of the standard separable potentials associated with an elliptic (or sphero-conical) coordinate system on the sphere.
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Tsiganov A. V.
Abstract
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Tsiganov A. V.
Abstract
We discuss the nonholonomic Chaplygin and the Borisov–Mamaev–Fedorov systems when the corresponding phase space is equivalent to cotangent bundle to dwo-dimensional sphere. In both cases Poisson bivectors are determined by L-tensors with non-zero torsion on the configurational space, in contrast with the well known Eisenhart–Benenti and Turiel constructions.
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Khudobakhshov V. A., Tsiganov A. V.
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.
Abstract
We discuss the polynomial bi-Hamiltonian structures for the Kowalewski 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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Grigoryev Y. A., Tsiganov A. V.
Abstract
The paper deals with superintegrable $N$-degree-of-freedom systems of Richelot type, for which $n\leqslant N$ equations of motion are the Abel equations on a hyperelliptic curve of genus $n−1$. The corresponding additional integrals of motion are second-order polynomials in momenta.
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Vershilov A. V., Tsiganov A. V.
Abstract
We classify quadratic Poisson structures on $so^*(4)$ and
$e^*(3)$, which have the same foliations by symplectic leaves as canonical
Lie-Poisson tensors. The separated variables for some of the corresponding
bi-integrable systems are constructed
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Tsiganov A. V.
Abstract
Elliptic coordinates on the dual space to the Lie algebra $e(3)$ are introduced. On the zero level of the Casimir function, these coordinates coincide with the standard elliptic coordinates on the cotangent bundle to the two-dimensional sphere. The possibility of use of these coordinates in the theory of integrable systems is discussed.
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Grigoryev Y. A., Tsiganov A. V.
Abstract
We discuss an algorithm for calculating of the separated variables for the Hamilton-Jacobi equation for the wide class of the so-called L-systems on the Riemann manifolds of the constant curvature. We suggest a software implementation of this algorithm in the system of symbolic computations Maple and consider several examples.
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