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    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


    Tsiganov A. V.
    Duffing Oscillator and Elliptic Curve Cryptography
    2018, Vol. 14, no. 2, pp.  235-241
    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.
    Keywords: integrable maps, divisor arithmetic
    Citation: Tsiganov A. V.,  Duffing Oscillator and Elliptic Curve Cryptography, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 2, pp. 235-241
    Tsiganov A. V.
    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.
    Keywords: finite-dimensional integrable systems, separation of variables, Bäcklund transformations
    Citation: Tsiganov A. V.,  On an integrable system on a plane with an integral of motion of sixth order in momenta, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 1, pp. 117-127
    Grigoryev Y. A., Sozonov A. P., Tsiganov A. V.
    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.
    Keywords: integrable systems, separation of variables, velocity-dependent potentials
    Citation: Grigoryev Y. A., Sozonov A. P., Tsiganov A. V.,  On an integrable system on the plane with velocity-dependent potential, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 3, pp. 355-367
    Tsiganov A. V.
    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.
    Keywords: integrable systems, separation of variables, velocity dependent potentials
    Citation: Tsiganov A. V.,  Separation of variables for some generalization of the Chaplygin system on a sphere, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 1, pp. 179-185
    Vershilov A. V., Grigoryev Y. A., Tsiganov A. V.
    On an integrable deformation of the Kowalevski top
    2014, Vol. 10, No. 2, pp.  223-236
    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.
    Keywords: Poisson geometry, Kowalevski top
    Citation: Vershilov A. V., Grigoryev Y. A., Tsiganov A. V.,  On an integrable deformation of the Kowalevski top, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 2, pp. 223-236
    Tsiganov A. V.
    On the absolute dynamics of the Chaplygin ball
    2013, Vol. 9, No. 4, pp.  711-719
    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.
    Keywords: nonholonomic mechanics, integrable systems, Poisson geometry
    Citation: Tsiganov A. V.,  On the absolute dynamics of the Chaplygin ball, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 4, pp. 711-719
    Tsiganov A. V.
    On the nonholonomic Veselova and Chaplygin systems
    2012, Vol. 8, No. 3, pp.  541-547
    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.
    Keywords: nonholonomic systems, Poisson brackets
    Citation: Tsiganov A. V.,  On the nonholonomic Veselova and Chaplygin systems, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 541-547
    Bizyaev I. A., Tsiganov A. V.
    On the Routh sphere
    2012, Vol. 8, No. 3, pp.  569-583
    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.
    Keywords: nonholonomic mechanics, Routh sphere, Poisson brackets
    Citation: Bizyaev I. A., Tsiganov A. V.,  On the Routh sphere, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 569-583
    Tsiganov A. V.
    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.
    Keywords: nonholonomic mechanics, Chaplygin sphere, Poisson brackets
    Citation: Tsiganov A. V.,  On the Poisson structures for the Chaplygin ball and its generalizations, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 2, pp. 345-353
    Tsiganov A. V.
    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.
    Keywords: nonholonomic mechanics, Chaplygin sphere, Poisson brackets
    Citation: Tsiganov A. V.,  On the bi-Hamiltonian structure of the Chaplygin and Borisov–Mamaev–Fedorov systems at a zero constant of areas. II, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp. 43-55
    Tsiganov A. V.
    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.
    Keywords: nonholonomic mechanics, Chaplygin sphere, Poisson brackets
    Citation: Tsiganov A. V.,  On deformations of the canonical Poisson bracket for the nonholonomic Chaplygin and the Borisov–Mamaev–Fedorov systems on zero-level of the area integral I, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 577-599
    Tsiganov A. V.
    Citation: Tsiganov A. V.,  Comments on P.E. Ryabov «Explicit integration and topology of D.N. Goryachev case», Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp. 715-717
    Khudobakhshov V. A., Tsiganov A. V.
    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: Khudobakhshov V. A., Tsiganov A. V.,  On quadratures of integrable systems on a sphere with higher degree integrals of motion, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 1, pp. 53-74
    Tsiganov A. V.
    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.
    Keywords: Kowalewski top, separation of variables, bi-Hamiltonian geometry, differential geometry, algebraic curves
    Citation: Tsiganov A. V.,  New variables of separation for particular case of the Kowalewski top, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 3, pp. 639-652
    Grigoryev Y. A., Tsiganov A. V.
    On the Abel equations and the Richelot integrals
    2009, Vol. 5, No. 4, pp.  463-478
    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.
    Keywords: superintegrable systems, separation of variables, Abel equations
    Citation: Grigoryev Y. A., Tsiganov A. V.,  On the Abel equations and the Richelot integrals, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 4, pp. 463-478
    Vershilov A. V., Tsiganov A. V.
    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
    Keywords: integrable system, bi-hamiltonian geometry, separation of variables
    Citation: Vershilov A. V., Tsiganov A. V.,  On the Darboux-Nijenhuis Variables on the Poisson Manifold $so^*(4)$, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 2, pp. 141-155
    Tsiganov A. V.
    On elliptic coordinates on the Lie algebra $e(3)$
    2006, Vol. 2, No. 3, pp.  347-352
    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.
    Keywords: elliptic coordinates, integrable systems, separation of variables
    Citation: Tsiganov A. V.,  On elliptic coordinates on the Lie algebra $e(3)$, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 3, pp. 347-352
    Grigoryev Y. A., Tsiganov A. V.
    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.
    Keywords: integrable systems, Hamilton-Jacobi equation, separation of variables
    Citation: Grigoryev Y. A., Tsiganov A. V.,  Computing of the separated variables for the Hamilton-Jacobi equation on a computer, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 2, pp. 163-179

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