Conference “Geometry of Integrable Systems”, Hanoi, 09-13 April 2007
Titles and Abstracts of the Talks:
Michèle Audin (Universté de Strasbourg), Geometry of quadratic systems
I will discuss various integrability properties of some quadratic differential systems related to the algebra and the geometry of the system.
Alexei Bolsinov (Loughborough), Noncommutative integrability and the Mischenko-Fomenko conjecture.
Holger Dullin (University of Loughborough), Action variables near resonant equilibria
Abstract: Elliptic equilibria of Hamiltonian have only imaginary eigenvalues. Nevertheless they can be non-linearly unstable when the eigenvalues are in resonance, i.e. in two degrees of freedom their ratio is rational. The truncated normal form at such an equilibrium is integrable. We study the actions, periods, and rotation number of this integrable system and derive their leading order singularities. These results are used to study the non-degeneracy conditions of the KAM theorem, and the existence of monodromy near these equilibria.
Gregorio Falqui (Univercità di Milano), Poisson Pencils, Integrability, and Separation of Variables
We will discuss a recently introduced method for solving the Hamilton-Jacobi equations by the method of Separation of Variables. This
method is based on the notion of pencil of Poisson brackets and on the bihamiltonian approach to integrable systems. We will show how separability conditions can be intrinsically characterized within such a geometrical set-up, the definition of the separation coordinates being
encompassed in the bihamiltonian structure itself. We will finally provide examples of these constructions.
Tamara Grava (SISSA), Universality in singular limits of Hamiltonian PDEs
Abstract: We consider Hamiltonian PDEs which depend on a small parameter.
In the limit when the parameter goes to zero the PDEs become an elliptic
or hyperbolic hydrodynamic equations and the corresponding solution
has a singular behaviour. The behaviour of the solution
of the Hamiltonian PDE near the singular points of the corresponding
hydrodynamic equations does not depend on the initial data
and it is described by an ODE.
Heinz Hanssmann (University of Utrecht), On the destruction of resonant Lagrangian tori in Hamiltonian Systems
Poincaré’s fundamental problem of dynamics concerns the behaviour of an integrable Hamiltonian system under a (small) non-integrable
perturbation. Under rather weak conditions K(olmogorov)A(rnol’d)M(oser) theory settles this question for the majority of initial values. The perturbed motion is (again) quasi-periodic, the number of frequencies equals the number of degrees of freedom. KAM theory proves such Lagrangean tori to persist provided that the frequencies are bounded away from resonances by means of Diophantine inequalities.
How do Lagrangean tori with resonant frequencies behave under perturbation ? We concentrate on a single resonance, whence many n-parameter families of n-tori are expected to be generated by the perturbation ; here n+1 is the number of degrees of freedom. For non-degenerate systems we explain the pattern how these families of lower-dimensional tori come into existence, and then discuss what happens in the presence of degeneracies.
Jarmo Hietarinta, Hirota’s bilinear method and soliton solutions
We will give an overview of Hirota’s direct method and its usage in the construction of soliton solutions. An application of Hirota’s method for the search of integrable soliton equations is outlined. The role of Hirota’s method for discrete systems is also discussed.
Thomas Kappeler (University of Zurich), Fermi-Pasta-Ulam lattices: normal form and KAM theorem
Abstract: In the fifties, Fermi, Pasta, and Ulam discovered numerically recurrence phenomena for solutions of 1-dimensional lattices with nearest neighbor interaction. This work had a profound impact on the theory of integrable systems and led to the discovery of integrable Hamiltonian PDE’s. In this talk I will report on recent joint work with A. Henrici leading to an explanation of the recurrence phenomena observed numerically for small energy solutions of a lattice with endpoints kept fixed or of periodic lattices with an odd number of particles. Surprisingly, the case of periodic lattices with an even number of particles is more complicated. It admits only a resonant normal form. We will discuss the underlying geometry of the phase space.
Yuji Kodama, Geometry of the Pfaff lattice
The Pfaff lattice was introduced by Adler and van Moerbeke to describe the partition functions for the random matrix models of GOE and GSE type. The partition functions are given by the $\tau$-functions of the Pfaff lattice. In the talk, I will discuss the finite version of the Pfaff lattice as a Hamiltonian system. In particular, I will show the the complete integrability in the sense of Arnold-Liouville, and with a moment map, I will describe the real isospectral variety for the Pfaff Lattice. The image of the moment map is a convex polytope whose vertices are identified as the fixed points of the Pfaff lattice. This work is under a collaboration with V. Pierce.
Jean-Pierre Marco (Paris VI), Some recent questions and results in hamiltonian perturbation theory
Yoshiaki Maeda, TBA
Juan Morales, Differential Galois Theory and Spectral Theory
Abstract: We will explain some applications of the Picard-Vessiot theory to the integrability of the Schrödinger one-dimensional stationary equation. We consider this equation as a second order linear differential equation with a parameter (the spectral parameter). This is a work in progress with Primitivo B. Acosta-Humanez.
Maria Przybylska, Finiteness of integrable $n$-dimensional homogeneous polynomial potentials
We consider natural Hamiltonian systems of $n>1$ degrees of freedom with polynomial homogeneous potentials of degree $k$. We show that under a genericity assumption, for a fixed $k$, at most only a finite number of such systems is integrable. We also explain how to find explicit forms of these integrable potentials for small $k$. Some exaples will be presented.
Jean-Pierre Ramis (Toulouse), Swinging Atwood’s machine: experimental and theoretical studies.
Abstract: Swinging Atwood’s Machine (SAM) is a system with two degrees of freedom derived from the classical Atwwod’s Machine (1784). There were a lot of studies of this system, using numerical models, but not so many rigourous theoretical results, and no serious physical experiments.
We will study the integrability problem of ideal SAM (when the pulleys are negligible), using numerical simulations and Morales-Ramis theory. The natural parameter is the ratio of the two masses $\mu =M/m$. The final answer is certainly that ideal SAM is integrable only if $\mu =3$. It is a work in progress and not yet a theorem (a rigorous proof needs the higher variational version of M-R theory, due to Morales, Simo and the lecturer, but we got some evidence of the result by delicate numerical simulations).
We have also tested integrability with a true model. In this model the pulleys are no longer negligible, therefore it is necessary to investigate theoretically SAM with pulleys. Some methods used for the ideal cases can be extended, others no. The final answer is certainly that SAM with pulleys is never integrable. The discrete list of exceptional cases of the ideal case is “deformed” in another discrete list in the case with pulleys, and to skip this list, it is, also in this case, necessary to use higher variational equations.
It is a joint work of the lecturer with: O. Pujol, J.P. Pérez, S. Simon, J. Morales-Ruiz, J.A. Weil and C. Simo.
Jonathan Robbins (UK), Maslov indices and singularities of integrable systems
The Maslov index is an integer-valued invariant of closed curves on a Lagrangian submanifold of a cotangent bundle. It appears, eg, in the next-to-leading-order semiclassical asymptotics for quantum mechanics.
I’ll present some results relating Maslov indices to the singularities of integrable systems. Here, the Lagrangian submanifolds are regular level sets of the energy-momentum map, and the singularities are its critical points. The first is the observation that Maslov indices are invariant under, and therefore constrain (provided they don’t vanish), the monodromy matrix associated with codimension-two critical values of the energy-momentum map.
The rest will concern an index formula for the Maslov index in terms of codimension-two critical points; this may be seen as a generalisation of the formula for the Poincare’ index for Hamiltonian vector fields in the plane. I’ll discuss some examples, including SO(n)-invariant systems and the periodic Toda chain. For the Toda chain, there is a general characterisation of codimension-k critical points in terms of eigenvalue degeneracies of the Lax matrices, and the Maslov index is related to the holonomy of the associated eigenvector bundles.
Tudor Ratiu (EPFL), TBA
M. Senthivelan, TBA
Jacek Tafel (Institute of Theoretical Physics, Univ. of Warsaw): Symmetries and completely integrable reductions of the Einstein equations
The crucial role in the Einstein theory of gravity is played by a metric tensor field on a 4-dimensional differential manifold modelling spacetime. Since the Einstein equations form a highly nonlinear system of 10 equations solving them without assuming symmetries of the metric is extremely difficult. Cosmological solutions, plane waves, spherically symmetric solutions and stationary axiallly symmetric solutions (including the black hole solutions of Schwarzschild and Kerr) represent the most important classes of gravitational fields with symmetries.
In the case of stationary axially symmetric metrics the vacuum Einstein equations reduce to the Ernst equation equivalent to a pair of real equations for two real functions. This system is completely integrable and admits solution generating techniques. Moreover, its solutions correspond to particular 2-dimensional surfaces in 3-dimensional flat space. One can generalize this geometrical approach in order to obtain solutions of the Einstein equation with matter fields.
Another example of (almost) completely integrable reduction of the Einstein equations is the Plebanski equation describing metrics with the self-dual Riemann tensor. The Plebanski equation is an equation for one function (the Kahler potential) of the Monge-Ampere type. It has several equivalent forms. One of them, due to Husain, is similar to the completely integrable sigma model. It is believed that self-dual Einstein equations play an important role in classication of other completely integrable equations, a role similar to that played by the self-dual Yang-Mills equations.
Suprisingly, the general Einstein equations with a cosmological constant are integrability conditions of a linear differential system (the Rarita-Schwinger equation) for a spinor field. This fact was a basis of a speculation that these general equations could be also completely integrable. It seems that this expectation is not realistic since for stationary axially symmetric metrics the Rarita-Schwinger equation is useless from the point of view of completely integrable systems.
Tadashi Tokieda, TBA
San Vu-Ngoc (Institut Fourier), Classical and quantum invariant for integrable Hamiltonian systems