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Volume 13, Number 6

Volume 13, Number 6, 2008
Jürgen Moser 80th Anniversary. Special memorial issue, part 1

J.K. Moser curriculum vitae and bibliography
Abstract
The year 2008 marks the eightieth anniversary of J. Moser’s birth; the following year it will be the tenth anniversary of his death. This special issue honors his memory as a leading mathematician of the modern era and as a great human being. The articles in this memorial issue are presented in two parts and cover a broad range of topics in which J. Moser worked. The first part contains contributions devoted to integrable systems and some related issues. The second part (RCD, January 2009, vol. 14, no. 1) contains papers on KAM theory and applications, stability problems, problems of celestial mechanics, etc.
Some of the contributions are brief recollections of meetings and working with J. Moser. We also reprint his remarkable and still very relevant lecture (ICM’1998, Berlin) "Dynamical Systems — Past and Present".
The main content is preceded by Moser’s curriculum vitae and bibliography.
Citation: J.K. Moser curriculum vitae and bibliography, Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 493-498
Moser J.
Dynamical systems — Past and present
Abstract
Plenary lecture at the International Congress of Mathematicians (Berlin 1998, August 18–27). Reprinted from Doc. Math. J., DMV, Extra Volume ICM I, 1998, pp. 381–402. ©J.Moser, 1998.
Keywords: KAM theory and applications, integrable Hamiltonian systems, historical review
Citation: Moser J., Dynamical systems — Past and present, Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 499-513
DOI:10.1134/S1560354708060014
Adler M.
Remembering Jürgen Moser
Abstract
Citation: Adler M., Remembering Jürgen Moser, Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 514-514
DOI:10.1134/S1560354708060026
Veselov A. P.
A few things I learnt from Jürgen Moser
Abstract
A few remarks on integrable dynamical systems inspired by discussions with Jürgen Moser and by his work.
Keywords: integrability, adiabatic invariants, geodesics
Citation: Veselov A. P., A few things I learnt from Jürgen Moser, Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 515-524
DOI:10.1134/S1560354708060038
Albouy A.
Projective dynamics and classical gravitation
Abstract
We show that there exists a projective dynamics of a particle. It underlies intrinsically the classical particle dynamics as projective geometry underlies Euclidean geometry. In classical particle dynamics a particle moves in the Euclidean space subjected to a potential. In projective dynamics the position space has only the local structure of the real projective space. The particle is subjected to a field of projective forces. A projective force is not an element of the tangent bundle to the position space, but of some fibre bundle isomorphic to the tangent bundle.
These statements are direct consequences of Appell’s remarks on the homography in mechanics, and are compatible with similar statements due to Tabachnikov concerning projective billiards. When we study Euclidean geometry we meet some particular properties that we recognize as projective properties. The same is true for the dynamics of a particle. We show that two properties in classical particle dynamics are projective properties. The fact that the Keplerian orbits close after one turn is a consequence of a more general projective statement. The fact that the fields of gravitational forces are divergence free is a projective property of these fields.
Keywords: divergence free, Kepler problem
Citation: Albouy A., Projective dynamics and classical gravitation, Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 525-542
DOI:10.1134/S156035470806004X
Bogoyavlenskij O. I.
Integrable Lotka–Volterra systems
Abstract
Infinite- and finite-dimensional lattices of Lotka–Volterra type are derived that possess Lax representations and have large families of first integrals. The obtained systems are Hamiltonian and contain perturbations of Volterra lattice. Examples of Liouville-integrable 4-dimensional Hamiltonian Lotka-Volterra systems are presented. Several 5-dimensional Lotka–Volterra systems are found that have Lax representations and are Liouville-integrable on constant levels of Casimir functions.
Keywords: Lax representation, Hamiltonian structures, Casimir functions, Riemannian surfaces, Lotka–Volterra systems, integrable lattices
Citation: Bogoyavlenskij O. I., Integrable Lotka–Volterra systems, Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 543-556
DOI:10.1134/S1560354708060051
Borisov A. V.,  Fedorov Y. N.,  Mamaev I. S.
Chaplygin ball over a fixed sphere: an explicit integration
Abstract
We consider a nonholonomic system describing the rolling of a dynamically nonsymmetric sphere over a fixed sphere without slipping. The system generalizes the classical nonholonomic Chaplygin sphere problem and it is shown to be integrable for one special ratio of radii of the spheres. After a time reparameterization the system becomes a Hamiltonian one and admits a separation of variables and reduction to Abel–Jacobi quadratures. The separating variables that we found appear to be a non-trivial generalization of ellipsoidal (spheroconic) coordinates on the Poisson sphere, which can be useful in other integrable problems.
Using the quadratures we also perform an explicit integration of the problem in theta-functions of the new time.
Keywords: Chaplygin ball, explicit integration, nonholonomic mechanics
Citation: Borisov A. V.,  Fedorov Y. N.,  Mamaev I. S., Chaplygin ball over a fixed sphere: an explicit integration, Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 557-571
DOI:10.1134/S1560354708060063
Damianou P. A.
Reduction and realization in Toda and Volterra
Abstract
We construct a new symplectic, bi-Hamiltonian realization of the KM-system by reducing the corresponding one for the Toda lattice. The bi-Hamiltonian pair is constructed using a reduction theorem of Fernandes and Vanhaecke. In this paper we also review the important work of Moser on the Toda and KM-systems.
Keywords: Toda lattice, bi-Hamiltonian systems
Citation: Damianou P. A., Reduction and realization in Toda and Volterra, Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 572-587
DOI:10.1134/S1560354708060075
Inozemtsev V. I.
Equilibrium points of classical integrable particle systems, factorization of wave functions of their quantum analogs and polynomial solutions of the Hill equation
Abstract
The relation between the characteristics of the equilibrium configurations of the classical Calogero–Moser integrable systems and properties of the ground state of their quantum analogs is found. It is shown that under the condition of factorization of the wave function of these systems the coordinates of classical particles at equilibrium are zeroes of the polynomial solutions of the second-order linear differential equation. It turns out that, under these conditions, the dependence of classical and quantum minimal energies on the parameters of the interaction potential is the same.
Keywords: Calogero–Moser systems, equilibrium points, Hill equation
Citation: Inozemtsev V. I., Equilibrium points of classical integrable particle systems, factorization of wave functions of their quantum analogs and polynomial solutions of the Hill equation, Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 588-592
DOI:10.1134/S1560354708060087
Kostov N. A.,  Tsiganov A. V.
New Lax pair for restricted multiple three wave interaction system, quasiperiodic solutions and bi-Hamiltonian structure
Abstract
We study restricted multiple three wave interaction system by the inverse scattering method. We develop the algebraic approach in terms of classical $r$-matrix and give an interpretation of the Poisson brackets as linear $r$-matrix algebra. The solutions are expressed in terms of polynomials of theta functions. In particular case for $n = 1$ in terms of Weierstrass functions.
Keywords: Lax pair, bi-Hamiltonian structure, three wave interaction system
Citation: Kostov N. A.,  Tsiganov A. V., New Lax pair for restricted multiple three wave interaction system, quasiperiodic solutions and bi-Hamiltonian structure, Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 593-601
DOI:10.1134/S1560354708060099
Lukina O. V.,  Takens F.,  Broer H. W.
Global properties of integrable Hamiltonian systems
Abstract
This paper deals with Lagrangian bundles which are symplectic torus bundles that occur in integrable Hamiltonian systems. We review the theory of obstructions to triviality, in particular monodromy, as well as the ensuing classification problems which involve the Chern and Lagrange class. Our approach, which uses simple ideas from differential geometry and algebraic topology, reveals the fundamental role of the integer affine structure on the base space of these bundles. We provide a geometric proof of the classification of Lagrangian bundles with fixed integer affine structure by their Lagrange class.
Keywords: integrable Hamiltonian system, global action-angle coordinates, symplectic topology, monodromy, Lagrange class, classification of integrable systems
Citation: Lukina O. V.,  Takens F.,  Broer H. W., Global properties of integrable Hamiltonian systems, Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 602-644
DOI:10.1134/S1560354708060105
Radnović M.,  Rom-Kedar V.
Foliations of isonergy surfaces and singularities of curves
Abstract
It is well known that changes in the Liouville foliations of the isoenergy surfaces of an integrable system imply that the bifurcation set has singularities at the corresponding energy level.We formulate certain genericity assumptions for two degrees of freedom integrable systems and we prove the opposite statement: the essential critical points of the bifurcation set appear only if the Liouville foliations of the isoenergy surfaces change at the corresponding energy levels. Along the proof, we give full classification of the structure of the isoenergy surfaces near the critical set under our genericity assumptions and we give their complete list using Fomenko graphs. This may be viewed as a step towards completing the Smale program for relating the energy surfaces foliation structure to singularities of the momentum mappings for non-degenerate integrable two degrees of freedom systems.
Keywords: Hamiltonian system, integrable system, singularity, Liouville foliation, isoenergy manifold, bifurcation set, Liouville equivalence
Citation: Radnović M.,  Rom-Kedar V., Foliations of isonergy surfaces and singularities of curves, Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 645-668
DOI:10.1134/S1560354708060117

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