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

Adler M.
Remembering Jürgen Moser
Abstract

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.

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. 
Bogoyavlenskij O. I.
Integrable Lotka–Volterra systems
Abstract
Infinite and finitedimensional 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 Liouvilleintegrable 4dimensional Hamiltonian LotkaVolterra systems are presented. Several 5dimensional Lotka–Volterra systems are found that have Lax representations and are Liouvilleintegrable on constant levels of Casimir functions.

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 nontrivial 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 thetafunctions of the new time. 
Damianou P. A.
Reduction and realization in Toda and Volterra
Abstract
We construct a new symplectic, biHamiltonian realization of the KMsystem by reducing the corresponding one for the Toda lattice. The biHamiltonian 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 KMsystems.

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 secondorder 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.

Kostov N. A., Tsiganov A. V.
New Lax pair for restricted multiple three wave interaction system, quasiperiodic solutions and biHamiltonian 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.

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.

Radnović M., RomKedar 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 nondegenerate integrable two degrees of freedom systems.
