While studying monodromy of the classical Lamé equation, the simultaneous spectrum of the real and imaginary parts of Lamé operators turns out to be important. We review an example where the simultaneous spectrum is discrete and which undergoes monodromy substitutions if we change the parameter of the Lamé operator.
In this talk we consider different types of non-holonomic systems, connected with rolling motion of rigid bodies. Some new cases of existence of tensor invariants - first integrals, symmetry fields and an invariant measure are indicated. Some obstacles prevent these tensor invariants from existence, and it leads to complex dynamical behavior, typical for celtic stones. Methods for obtaining new cases of integrability and its explicit integration are investigated. We discuss new rubber-rolling model, also described by non-holonomic equation.
We discuss resonance phenomena in various settings, both dissipative and conservative. Resonance roughly speaking means that several oscillatory subcomponents of a system interact, which can give rise to losses of stability or to periodic dynamics of a specific period. The geometry of this often involves `tongues' in a suitable parameter space and the organisation of this is part of bifurcation theory, where often singularity theory can be applied. We discuss several local and global aspects of this.
In 1937, Brauer described the centralizer algebra of a classical group in the n-fold tensor product of its natural representation in terms of diagrams. The diagrams can be viewed as monomials in this algebra, known as the Brauer algebra on n strands. Taking certain basic diagrams as generators, a set of defining relations has been identified as well as a set of rewrite rules that enables us to compute a normal form for elements of the Brauer algebra. We show how the diagram can be described in terms of root system data for the Coxeter group of type A_(n-1), otherwise known as the symmetric group on n letters.
The Kauffman tangle algebra is a quantized version of it, also known as the BMW (for Brauer-Murakami-Wenzl) algebra. It plays a role in knot theory (the Kauffman invariant can be defined by means of a trace function on it) and in representation theory (it is the centralizer algebra of a quantum group of classical type in the n-fold tensor product of the natural representation). Under control of the Brauer algebra rewrite rules, its elements can also be written in normal form.
The above algebras and the normal form results generalize to the types D_n and E_6, E_7, E_8. If time permits, we will show how.
In the Cartesian product of two complex projective lines, any biquadratic curve intersects each horizontal and vertical axis in two points. The self map of the biquadratic curve, which interchanges the two points on the same horizontal and vertical axis, is called the horizontal and vertical switch on the biquadratic curve, respectively. The QRT map is the horizontal switch followed by the vertical switch. If the plane is filled by a pencil of biquadratic curves, then QRT maps of the members of the pencil piece together to a birational transformation of the plane, which may not be well-defined in some of the base points of the pencil. Blowing up the base points one arrives at a compact complex analytic complex two-dimensional surface S which is fibered over the complex projective line, where the fibers correspond to the members of the pencil. On S, the QRT map extends to a complex analytic diffeomorphism of S which leaves each of the fibers invariant. Each smooth fiber is an elliptic curve, on which the QRT map acts as a translation. This implies that if a point of S is k-periodic for the QRT map, then every point of the fiber through it is a k-periodic point. Because the QRT map is an automorphism of the surface S, it has a well-defined action on the homology of S, which can be explicitly computed. Using intersection theory this leads among others to an explicit formula for the number of k-periodic fibers.
The surfaces S which arise in this way are the general so-called rational elliptic surfaces, with the exception of the few rational elliptic surfaces which do not admit any nontrivial automorphisms acting as translations on the smooth fibers. Therefore the study of QRT maps is essentially equivalent to the study of rational elliptic surfaces. On the other hand a large number of explicitly given birational transformations of the plane which appear in the literature turn out to be QRT maps, such as many examples from the mathematical physics literature, the elliptic billiard map, and the Lyness map. About the latter, Beukers and Cushman proved a conjecture of Zeeman, and one of my remaining questions is in how far their methods can be used to obtain a better understanding of arbitrary QRT maps.
In their paper, only the real points are considered. A feature which might make the complex setting interesting to Richard Cushman is that any elliptic surface, which is a real four-dimensional manifold in which the smooth fibers are real two-dimensional tori, can be viewed as a completely integrable Hamiltonian system of two degrees of freedom. In it, singular fibers do occur. The monodromy around each singular fiber is nontrivial, and actually determines the type of the singular fiber. As there already is a great abundance of rational elliptic surfaces, this provides a very rich class of examples of integrable Hamiltonian systems with nontrivial monodromy, existing already in the early 1960's, when Kodaira developed his theory of elliptic fibrations. Warning: I do not claim that every completely integrable Hamiltonian system in two degrees of freedom on a compact manifold arises in this way, because for instance elliptic singular points cannot arise in a complex analytic elliptic fibration.
At the 1:-2 resonance an elliptic equilibrium is generically non-linearly unstable. The unfolding of the resonance shows a collision of a period doubling bifurcation with the equilibrium. The long period Lyapunov orbit thus becomes unstable at resonance. Since the equilibrium is degenerate, the actions, frequencies and rotation number show an unusual singular behaviour near the resonance. In particular there are algebraic singularities instead of the usual logarithmic ones. As a consequence, using a local Poincare mapping, we are able to show the action variables exhibit fractional monodromy at the 1:-2 resonance.
We consider a class of systems on Lie groups and grupoids with left-invariant metric and right-invariant nonholonomic constraints (so called LR systems) which always possess an invariant measure and, under some generic conditions on the constraints, can be regarded as generalized Chaplygin systems. The reduced systems are shown to possess an invariant measure as well. We give sufficient conditions for these systems 1) to be reducible to a Hamiltonian form by a time reparameterization (the Chaplygin method of a reducing multiplier); 2) to be integrable by quadratures.
The approach will be illustrated on several classical and new examples. The talk presents the results of a joint work with B. Jovanovic.
The topology of the Liouville-Arnold's torus bundle associated to a completely integrable system is determined by its singularities. The community has a list of "relevant" singularities, but what does relevant mean? Are there relevant singularities that have not been described in detail?
We will discuss an approach to answer these questions and will study a singularity that appeared in many situations but has escaped a thorough description.
Say k is a field of positive characteristic. Associated to a partition with at most n parts, or a Young diagram with at most n rows, one has a so called Schur module of the general linear linear group GL(n,k). It is a polynomial representation constructed in a characteristic free manner. Examples are symmetric and exterior powers of the defining representation of GL(n,k). These modules have favorable invariant theoretic properties. For instance, let U be the subgroup of upper triangular unipotent matrices in GL(n,k). Then one can use Weyl's character formula to compute the character of a Schur module from the character of its submodule of invariants for U. We say that a polynomial GL(n,k) module has good filtration if it has a filtration whose successive quotients are Schur modules. Modules with good filtration have similar nice invariant theoretic properties. Recall that the first fundamental theorem of invariant theory tells that if SL(n,k) acts polynomially on a finitely generated commutative k algebra A, then the ring of invariants is also a finitely generated k algebra. Now this ring of invariants may be viewed as the degree zero part in a cohomology algebra. Modules with good filtrations help to investigate a conjectural extension of the finite generation result to the full cohomology algebra.
Determination of Hamiltonian Hopf bifurcations in systems with more than two degrees of freedom admitting additional symmetry can be done using geometric arguments. After an introduction reviewing the standard Hamiltonian Hopf bifurcation in two degrees of freedom, especially its geometry, we focus on systems in threefold and fourfold 1:1 resonance with additional rotational symmetries. The latter example includes regularized perturbed Keplerian systems as a special case.
I will present a normalization à la Birkhoff for Hamiltonian systems on Poisson manifolds near a singular point (where the Hamiltonian vector field vanishes, and the Poisson structure is also singular). I will discuss the convergence problem for this normalization, and (formal) integrability. This talk is based on joint work with Philippe Monnier.
Classical holonomic mechanical systems, and discrete analogues of those, and also classical field theories, admit well known variational formulations. Solutions of these systems are critical points of an action functional subject to a fixed-boundary condition.
But classical holonomic mechanical systems also have a well known symplectic formulation. This can be derived from the corresponding variational principle using a simple procedure, which only refers to generic concepts, such as "solution", and "boundary". The same procedure can be applied in a variety of contexts as a way or recognizing or identifying analogues of symplectic structures. This idea is useful and compelling in discrete analogues of these systems, and can be used to derive symplectic integration algorithms.
So what happens when the procedure is applied, for the purpose of recognizing analogues of the symplectic structure, to nonholonomic systems, which are known not to be, in general, symplectic?
Reference: Variational development of the semi-symplectic geometry of nonholonomic mechanics. Accepted Rep. Math. Phys., 42pp. http://math.usask.ca/~patrick/PatrickGW-2005-3.pdf
The study of perturbations of the Kepler system constitutes a fundamental part of mechanics with a long list of famous names attached. The hydrogen atom is the quantum Kepler system and its perturbations by external electric and magnetic fields is an important fundamental quantum problem with its own long list of more contemporary names: Pieter Zeeman, NP 1902, Johannes Stark, NP 1919, Wolfgang Pauli, NP 1945, ...
Following Richard Cushman's emphatic introduction of monodromy to atomic and molecular physicists around 1996-98, this system became one of the first known fundamental physical sytems with monodromy. We begin with our original result of 1999-2000 on the monodromy of the hydrogen atom in orthogonal fields obtained in collaboration with Richard Cushman, and we place this study within a framework of analysing a more general class of systems with sufficiently small homogeneous static fields of all possible mutual orientations. This analysis is inspired by the recent quantum calculations of John Delos and his PhD student Chris Schleif. Our results are submitted jointly with Konstantinos Efstathiou and Boris Zhilinskií to Proc. Roy. Soc. A in 2006. 80 years after Pauli's first attempt at classifying these systems our work completes the task.
Normalising with regard to the Keplerian symmetry, we uncover resonances and conjecture that the parameter space of this family of dynamical systems is stratified into zones centred on the resonances. The 1:1 resonance corresponds to the orthogonal field limit. Finding deformations of the energy-momentum map of the orthogonal system as the fields get skewed, we describe the structure of the 1:1 zone, where the system may have monodromy of different kinds. Subsequently, we consider briefly the 1:2 zone, where the system exhibits bidromy and fractional monodromy.
I this talk I will discuss the general problem of classifying integrable Hamiltonian systems with singularities. When one restricts to non-degenerate singularities (à la Morse-Bott), the topology of the Liouville foliation is known, mainly thanks to works by Nguyen Tien Zung. A first approach for a symplectic classification is to understand the divergence of the dynamics near hyperbolic singular sets. This was used by Toulet in the 1D case. The 2D case has been recently attacked. In a joint work with Dullin, we have been able to calculate symplectic invariants for two coupled hyperbolic singularities. A more general approach is to consider the behaviour of action integrals. In a joint work with Bolsinov, we have shown that, for analytic systems, the action integrals contain all the symplectic invariants.
I will discuss the qualitative theory of the energy spectra of simple molecules which are finite-particle quantum systems. In many concrete cases stationary molecular states can be described with a physically reasonable accuracy in a classical limit by an integrable approximation. Hamiltonian monodromy naturally appears on this way as a simplest obstruction to the existence of global action-angle variables. Intensive study of the manifestations of classical Hamiltonian monodromy in associated quantum problems was successfully realized during last ten years. The main result is the better understanding of different patterns formed by the joint spectra of mutually commuting quantum operators. At the same time many molecular examples inspire more complicated but nevertheless generic behavior of such patterns. Fractional monodromy [1] and bidromy [2] are two examples of new notions introduced recently and motivated by the qualitative description of quantum molecular systems. In what direction should we go further and what kind of mathematical tools is appropriate to characterize realistic physical systems? I'll try to formulate my physico-chemical point of view on this problem by looking on several concrete examples and making some speculative suggestions.