# Volume 16, Number 5, 2011 Proceeding of GDIS 2010, Belgrade

 Benenti S. Abstract An example of physically realizable non-linear non-holonomic mechanical system is proposed. The dynamical equations are written following a general method proposed in an earlier paper. In order to make this paper self-contained, an improved and shortened approach to the dynamics of non-holonomic systems is illustrated in preliminary sections. Keywords: non-holonomic systems, dynamical systems Citation: Benenti S., The Non-holonomic Double Pendulum: an Example of Non-linear Non-holonomic System, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 417-442 DOI:10.1134/S1560354711050029
 Bolsinov A. V.,  Borisov A. V.,  Mamaev I. S. 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
 Borisov A. V.,  Kilin A. A.,  Mamaev I. S. Abstract We consider a novel mechanical system consisting of two spherical bodies rolling over each other, which is a natural extension of the famous Chaplygin problem of rolling motion of a ball on a plane. In contrast to the previously explored non-holonomic systems, this one has a higher dimension and is considerably more complicated. One remarkable property of our system is the existence of "clandestine" linear in momenta first integrals. For a more trivial integrable system, their counterparts were discovered by Chaplygin. We have also found a few cases of integrability. Keywords: nonholonomic constraint, rolling motion, Chaplygin ball, integral, invariant measure Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 465-483 DOI:10.1134/S1560354711050042
 Dragović V.,  Kukić K. Abstract A new examples of integrable dynamical systems are constructed. An integration procedure leading to genus two theta-functions is presented. It is based on a recent notion of discriminantly separable polynomials. They have appeared in a recent reconsideration of the celebrated Kowalevski top, and their role here is analogue to the situation with the classical Kowalevski integration procedure. Keywords: integrable dynamical system, Kowalevski top, discriminantly separable polynomials, systems of the Kowalevski type, invariant measure Citation: Dragović V.,  Kukić K., New Examples of Systems of the Kowalevski Type, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 484-495 DOI:10.1134/S1560354711050054
 Chanu C.,  Degiovanni L.,  Rastelli G. Abstract Families of three-body Hamiltonian systems in one dimension have been recently proved to be maximally superintegrable by interpreting them as one-body systems in the three-dimensional Euclidean space, examples are the Calogero, Wolfes and Tramblay Turbiner Winternitz systems. For some of these systems, we show in a new way how the superintegrability is associated with their dihedral symmetry in the three-dimensional space, the order of the dihedral symmetries being associated with the degree of the polynomial in the momenta first integrals. As a generalization, we introduce the analysis of integrability and superintegrability of four-body systems in one dimension by interpreting them as one-body systems with the symmetries of the Platonic polyhedra in the four-dimensional Euclidean space. The paper is intended as a short review of recent results in the sector, emphasizing the relevance of discrete symmetries for the superintegrability of the systems considered. Keywords: superintegrability, higher-degree first integrals, discrete symmetries, Tremblay-Turbiner–Winterniz system Citation: Chanu C.,  Degiovanni L.,  Rastelli G., Three and Four-body Systems in One Dimension: Integrability, Superintegrability and Discrete Symmetries, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 496-503 DOI:10.1134/S1560354711050066
 Jovanović B. Abstract We prove the integrability of geodesic flows on the Riemannian g.o. spaces of compact Lie groups, as well as on a related class of Riemannian homogeneous spaces having an additional principal bundle structure. Keywords: noncommutative integrability, geodesic orbit spaces, complexity of homogeneous spaces, fiber bundles Citation: Jovanović B., Geodesic Flows on Riemannian g.o. Spaces, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 504-513 DOI:10.1134/S1560354711050078
 Jurdjevic V. Abstract Control theory, initially conceived in the 1950’s as an engineering subject motivated by the needs of automatic control, has undergone an important mathematical transformation since then, in which its basic question, understood in a larger geometric context, led to a theory that provides distinctive and innovative insights, not only to the original problems of engineering, but also to the problems of differential geometry and mechanics. This paper elaborates the contributions of control theory to geometry and mechanics by focusing on the class of problems which have played an important part in the evolution of integrable systems. In particular the paper identifies a large class of Hamiltonians obtained by the Maximum principle that admit isospectral representation on the Lie algebras $\mathfrak{g} = \mathfrak{p} ⊕ \mathfrak{k}$ of the form $$\frac{dL_\lambda}{dt} = [\Omega_\lambda, L_\lambda] L_\lambda = L_{\mathfrak{p}} − \lambda L_{\mathfrak{k}} − (\lambda^2 − s)A, \quad L_{\mathfrak{p}} \in \mathfrak{p}, \quad L_{\mathfrak{k}} \in \mathfrak{k}.$$ The spectral invariants associated with $L_\lambda$ recover the integrability results of C.G.J. Jacobi concerning the geodesics on an ellipsoid as well as the results of C. Newmann for mechanical problem on the sphere with a quadratic potential. More significantly, this study reveals a large class of integrable systems in which these classical examples appear only as very special cases. Keywords: Lie groups, control systems, the Maximum principle, symplectic structure, Hamiltonians, integrable systems Citation: Jurdjevic V., Optimal Control on Lie groups and Integrable Hamiltonian Systems, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 514-535 DOI:10.1134/S156035471105008X
 Kozlov V. V. Abstract The Kac circular model is a discrete dynamical system which has the property of recurrence and reversibility. Within the framework of this model M.Kac formulated necessary conditions for irreversibility over "short" time intervals to take place and demonstrated Boltzmann’s most important exploration methods and ideas, outlining their advantages and limitations. We study the circular model within the realm of the theory of Gibbs ensembles and offer a new approach to a rigorous proof of the "zeroth" law of thermodynamics based on the analysis of weak convergence of probability distributions. Keywords: reversibility, stochastic equilibrium, weak convergence Citation: Kozlov V. V., Statistical Irreversibility of the Kac Reversible Circular Model, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 536-549 DOI:10.1134/S1560354711050091
 Levi-Civita T. Abstract We bring to the reader’s attention a translation of Levi-Civita’s work "Sugli integrali algebrici delle equazioni dinamiche" (1896). In this work, Levi-Civita exposes some of its results concerning the first integrals of the Lagrangian dynamical systems, which are rational in the velocities. Of a particular historical interest is the fact that here he introduces the concept of Killing tensor and the Killing equation. The Editorial Board is grateful to Professor Sergio Benenti for the translation of the original Italian text and valuable comments. Citation: Levi-Civita T., On the Algebraic Integrals of the Dynamical Equations, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 550-554 DOI:10.1134/S1560354711050108

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