Inna Basak

Barcelona, E-08028 Spain
Department de Matematica Aplicada I, Universitat Politecnica de Catalunya


Fedorov Y. N., Basak I.
The paper revises the explicit integration of the classical Steklov–Lyapunov systems via separation of variables, which had been first made by F. Kötter in 1900, but was not well understood until recently. We give a geometric interpretation of the separating variables and then, applying the Weierstrass hyperelliptic root functions, obtain explicit theta-function solution to the problem. We also analyze the structure of poles of the solution on the Jacobian on the corresponding hyperelliptic curve. This enables us to obtain a solution for an alternative set of phase variables of the systems that has a specific compact form.
In conclusion we discuss the problem of integration of the Rubanovsky gyroscopic generalizations of the above systems.
Keywords: Steklov–Lyapunov system, explicit solution, separation of variables, algebraic integrability
Citation: Fedorov Y. N., Basak I.,  Separation of Variables and Explicit Theta-function Solution of the Classical Steklov–Lyapunov Systems: A Geometric and Algebraic Geometric Background, Regular and Chaotic Dynamics, 2011, vol. 16, nos. 3-4, pp. 374-395
Basak I.
The main goal of this paper consists of bifurcation analysis of classical integrable Zhukovskii–Volterra system. We use the fact that the ZV system is bi-Hamiltonian and apply new techniques [1] for analysis of singularities of bi-Hamiltonian systems, which can be formulated as follows: the structure of singularities of a bi-Hamiltonian system is determined by that of the corresponding compatible Poisson brackets.
Keywords: integrable Hamiltonian sistems, compatible Poisson structures, bifurcations, semisimple Lie algebras
Citation: Basak I.,  Bifurcation analysis of the Zhukovskii–Volterra system via bi-Hamiltonian approach, Regular and Chaotic Dynamics, 2010, vol. 15, no. 6, pp. 677-684
Basak I.
Explicit Solution of the Zhukovski–Volterra Gyrostat
2009, vol. 14, no. 2, pp.  223-236
The paper is devoted to explicit integration of the classical generalization of the Euler top: the Zhukovski–Volterra system describing the free motion of a gyrostat. We revise the solution for the components of the angular momentum first obtained by Volterra in [1] and present an alternative solution based on an algebraic parametrization of the invariant curves. This also enables us to derive an effective description of the motion of the body in space.
Keywords: rigid body dynamics, explicit integration, elliptic curves
Citation: Basak I.,  Explicit Solution of the Zhukovski–Volterra Gyrostat, Regular and Chaotic Dynamics, 2009, vol. 14, no. 2, pp. 223-236

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