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2013
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# Alexey Bolsinov

Ashby Road, Loughborough, LE11 3TU, UK
Loughborough University

D.Sc., Professor

Member of the editorial board of «Regular & Chaotic Dynamics»

## Publications:

 Bolsinov A. V., Bao J. A Note about Integrable Systems on Low-dimensional Lie Groups and Lie Algebras 2019, vol. 24, no. 3, pp.  266-280 Abstract The goal of the paper is to explain why any left-invariant Hamiltonian system on (the cotangent bundle of) a 3-dimensonal Lie group $G$ is Liouville integrable. We derive this property from the fact that the coadjoint orbits of $G$ are two-dimensional so that the integrability of left-invariant systems is a common property of all such groups regardless their dimension. We also give normal forms for left-invariant Riemannian and sub-Riemannian metrics on 3-dimensional Lie groups focusing on the case of solvable groups, as the cases of $SO(3)$ and $SL(2)$ have been already extensively studied. Our description is explicit and is given in global coordinates on $G$ which allows one to easily obtain parametric equations of geodesics in quadratures. Keywords: Integrable systems, Lie groups, geodesic flow, left-invariant metric, sub-Riemannian structure Citation: Bolsinov A. V., Bao J.,  A Note about Integrable Systems on Low-dimensional Lie Groups and Lie Algebras, Regular and Chaotic Dynamics, 2019, vol. 24, no. 3, pp. 266-280 DOI:10.1134/S156035471903002X
 Bolsinov A. V., Borisov A. V., Mamaev I. S. Rolling of a Ball without Spinning on a Plane: the Absence of an Invariant Measure in a System with a Complete Set of Integrals 2012, vol. 17, no. 6, pp.  571-579 Abstract In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles. Keywords: non-holonomic constraint, Liouville foliation, invariant torus, invariant measure, integrability Citation: Bolsinov A. V., Borisov A. V., Mamaev I. S.,  Rolling of a Ball without Spinning on a Plane: the Absence of an Invariant Measure in a System with a Complete Set of Integrals, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 571-579 DOI:10.1134/S1560354712060081
 Bolsinov A. V., Borisov A. V., Mamaev I. S. The Bifurcation Analysis and the Conley Index in Mechanics 2012, vol. 17, no. 5, pp.  457-478 Abstract The paper is devoted to the bifurcation analysis and the Conley index in Hamiltonian dynamical systems. We discuss the phenomenon of appearance (disappearance) of equilibrium points under the change of the Morse index of a critical point of a Hamiltonian. As an application of these techniques we find new relative equilibria in the problem of the motion of three point vortices of equal intensity in a circular domain. Keywords: Morse index, Conley index, bifurcation analysis, bifurcation diagram, Hamiltonian dynamics, fixed point, relative equilibrium Citation: Bolsinov A. V., Borisov A. V., Mamaev I. S.,  The Bifurcation Analysis and the Conley Index in Mechanics, Regular and Chaotic Dynamics, 2012, vol. 17, no. 5, pp. 457-478 DOI:10.1134/S1560354712050073
 Bolsinov A. V., Borisov A. V., Mamaev I. S. Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds 2011, vol. 16, no. 5, pp.  443-464 Abstract The problem of Hamiltonization of nonholonomic systems, both integrable and non-integrable, is considered. This question is important in the qualitative analysis of such systems and it enables one to determine possible dynamical effects. The first part of the paper is devoted to representing integrable systems in a conformally Hamiltonian form. In the second part, the existence of a conformally Hamiltonian representation in a neighborhood of a periodic solution is proved for an arbitrary (including integrable) system preserving an invariant measure. Throughout the paper, general constructions are illustrated by examples in nonholonomic mechanics. Keywords: conformally Hamiltonian system, nonholonomic system, invariant measure, periodic trajectory, invariant torus, integrable system Citation: Bolsinov A. V., Borisov A. V., Mamaev I. S.,  Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 443-464 DOI:10.1134/S1560354711050030
 Bolsinov A. V., Oshemkov A. A. Bi-Hamiltonian structures and singularities of integrable systems 2009, vol. 14, no. 4-5, pp.  431-454 Abstract A Hamiltonian system on a Poisson manifold $M$ is called integrable if it possesses sufficiently many commuting first integrals $f_1, \ldots f_s$ which are functionally independent on $M$ almost everywhere. We study the structure of the singular set $K$ where the differentials $df_1, \ldots, df_s$ become linearly dependent and show that in the case of bi-Hamiltonian systems this structure is closely related to the properties of the corresponding pencil of compatible Poisson brackets. The main goal of the paper is to illustrate this relationship and to show that the bi-Hamiltonian approach can be extremely effective in the study of singularities of integrable systems, especially in the case of many degrees of freedom when using other methods leads to serious computational problems. Since in many examples the underlying bi-Hamiltonian structure has a natural algebraic interpretation, the technology developed in this paper allows one to reformulate analytic and topological questions related to the dynamics of a given system into pure algebraic language, which leads to simple and natural answers. Keywords: integrable Hamiltonian systems, compatible Poisson structures, Lagrangian fibrations, bifurcations, semisimple Lie algebras Citation: Bolsinov A. V., Oshemkov A. A.,  Bi-Hamiltonian structures and singularities of integrable systems, Regular and Chaotic Dynamics, 2009, vol. 14, no. 4-5, pp. 431-454 DOI:10.1134/S1560354709040029
 Bolsinov A. V., Borisov A. V., Mamaev I. S. Lie algebras in vortex dynamics and celestial mechanics — IV 1999, vol. 4, no. 1, pp.  23-50 Abstract 1.Classificaton of the algebra of $n$ vortices on a plane 2.Solvable problems of vortex dynamics 3.Algebraization and reduction in a three-body problem The work [13] introduces a naive description of dynamics of point vortices on a plane in terms of variables of distances and areas which generate Lie–Poisson structure. Using this approach a qualitative description of dynamics of point vortices on a plane and a sphere is obtained in the works [14,15]. In this paper we consider more formal constructions of the general problem of n vortices on a plane and a sphere. The developed methods of algebraization are also applied to the classical problem of the reduction in the three-body problem. Citation: Bolsinov A. V., Borisov A. V., Mamaev I. S.,  Lie algebras in vortex dynamics and celestial mechanics — IV, Regular and Chaotic Dynamics, 1999, vol. 4, no. 1, pp. 23-50 DOI:10.1070/RD1999v004n01ABEH000097
 Bolsinov A. V., Dullin H. R. On Euler Case in Rigid Body Dynamics and Jacobi Problem 1997, vol. 2, no. 1, pp.  13-25 Abstract Using two classical integrable problems, we demonstrate some methods of a new theory of orbital classification for integrable Hamiltonian systems with two degrees of freedom. We show that the Liouville foliations (i.e., decompositions of the phase space into Liouville tori) of the two systems under consideration are diffeomorphic. Moreover, these systems are orbitally topologically equivalent, but this equivalence cannot be made smooth. Citation: Bolsinov A. V., Dullin H. R.,  On Euler Case in Rigid Body Dynamics and Jacobi Problem, Regular and Chaotic Dynamics, 1997, vol. 2, no. 1, pp. 13-25 DOI:10.1070/RD1997v002n01ABEH000022