Alexey Bolsinov
D.Sc., Professor
Member of the editorial board of «Regular & Chaotic Dynamics»
Publications:
Bolsinov A. V., Bao J.
A Note about Integrable Systems on Lowdimensional Lie Groups and Lie Algebras
2019, vol. 24, no. 3, pp. 266280
Abstract
The goal of the paper is to explain why any leftinvariant Hamiltonian system on (the cotangent bundle of) a 3dimensonal Lie group $G$ is Liouville integrable. We derive this property from the fact that the coadjoint orbits of $G$ are twodimensional so that the integrability of leftinvariant systems is a common property of all such groups regardless their dimension.
We also give normal forms for leftinvariant Riemannian and subRiemannian metrics on 3dimensional 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. 
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. 571579
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 2tori; 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.

Bolsinov A. V., Borisov A. V., Mamaev I. S.
The Bifurcation Analysis and the Conley Index in Mechanics
2012, vol. 17, no. 5, pp. 457478
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.

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. 443464
Abstract
The problem of Hamiltonization of nonholonomic systems, both integrable and nonintegrable, 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.

Bolsinov A. V., Oshemkov A. A.
BiHamiltonian structures and singularities of integrable systems
2009, vol. 14, nos. 45, pp. 431454
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 biHamiltonian 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 biHamiltonian 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 biHamiltonian 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.

Bolsinov A. V., Borisov A. V., Mamaev I. S.
Lie algebras in vortex dynamics and celestial mechanics — IV
1999, vol. 4, no. 1, pp. 2350
Abstract
1.Classificaton of the algebra of $n$ vortices on a plane
2.Solvable problems of vortex dynamics 3.Algebraization and reduction in a threebody 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 threebody problem. 
Bolsinov A. V., Dullin H. R.
On Euler Case in Rigid Body Dynamics and Jacobi Problem
1997, vol. 2, no. 1, pp. 1325
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.
