Hamiltonian Monodromy via Picard-Lefschetz Theory

 In this paper, we investigate the ``Hamiltonian' monodromy of the fibration in Liouville tori of certain integrable systems via (real) algebraic geometry. Using Picard-Lefschetz theory in a relative Prym variety, we determine the Hamiltonian monodromy of the ``geodesic flow on SO(4)'. Using a relative generalized Jacobian, we prove that the Hamiltonian monodromy of the spherical pendulum can also be obtained by the Picard-Lefschetz formula. Received: 28 September 2001 / Accepted: 12 April 2002 Published online: 12 August 2002     

Monodromy in non-integrable systems on certain compact classical phase spaces

On the example of bending vibrational polyads of the acetylene molecule (C₂H₂) in the approximation of the $1:1:1:1$ resonant oscillator with axial symmetry, whose geometry is similar to the n-shell approximation of the perturbed hydrogen atom, we show how remaining invariant tori of the underlying classical non-integrable system form a nontrivial continuous family with monodromy. We read this monodromy off the quantum energy spectrum which was observed experimentally by spectroscopists, and we uncover its origins through the particular topology, geometry, and symmetry. We explain how monodromy characterizes the chaotic region surrounded by the tori. We detail the explicit correspondence between the bending polyads of C₂H₂ and the n-shells of the hydrogen atom, and uncover the dynamical SO(3) symmetry of the bending polyads and the corresponding spherically localized vibrational states.     

Dynamical manifestations of Hamiltonian monodromy

Monodromy is the simplest obstruction to the existence of global action–angle variables in integrable Hamiltonian dynamical systems. We consider one of the simplest possible systems with monodromy: a particle in a circular box containing a cylindrically symmetric potential-energy barrier. Systems with monodromy have nontrivial smooth connections between their regular Liouville tori. We consider a dynamical connection produced by an appropriate time-dependent perturbation of our system. This turns studying monodromy into studying a physical process. We explain what aspects of this process are to be looked upon in order to uncover the interesting and somewhat unexpected dynamical behavior resulting from the nontrivial properties of the connection. We compute and analyze this behavior.     

Theory of Tensor Invariants of Integrable Hamiltonian Systems. II. Theorem on Symmetries and Its Applications

The theorem on symmetries is proved that states that a Liouville-integrable Hamiltonian system is non-degene\-rate in Kolmogorov's sense and has compact invariant submanifolds if and only if the corresponding Lie algebra of symmetries is abelian. The theorem on symmetries has applications to the characterization problem, to the integrable hierarchies problem, to the necessary conditions for the strong dynamical compatibility problem, and to the problem on master symmetries. The invariant necessary conditions for the non-degenerate C-integrability in Kolmogorov's sense of a given dynamical system V are derived. It is proved that the C-integrable Hamiltonian system is non-degenerate in the iso-energetic sense if and only if the corresponding Lie algebra of the iso-energetic conformal symmetries is abelian. An extended concept of integrability of Hamiltonian systems on the symplectic manifolds M ⁿ , n= 2k, is introduced. The concept of integrability describes the Hamiltonian systems that have quasi-periodic dynamics on tori or on toroidal cylinders of an arbitrary dimension . This concept includes, as a particular case, all Hamiltonian systems that are integrable in Liouville's classical sense, for which . The A-B-C-cohomologies are introduced for dynamical systems on smooth manifolds. Received: 16 January 1996 / Accepted: 3 July 1996     

Hamiltonian systems with detuned 1:1:2 resonance: Manifestation of bidromy

We consider a generalization of the 1:1:2 resonant swing-spring [see H. Dullin, A. Giacobbe, R.H. Cushman, Physica D 190 (2004) 15] which is suggested both by the symmetries of this system and by its physical and in particular molecular realizations [see R.H. Cushman, H.R. Dullin, A. Giacobbe, D.D. Holm, M. Joyeux, P. Lynch, D.A. Sadovskií, B.I. Zhilinskií, Phys. Rev. Lett. 93 (2004) 024302-1-024302-4]. Our generic integrable system is detuned off the exact Fermi resonance 1:2. The three-dimensional (3D) image of its energy-momentum map EM consists either of two or three qualitatively different non-intersecting 3D regions: a regular region at low vibrational excitation, a region with monodromy similar to that studied for the exact resonance, and in some cases—an intermediate region in which the 3D set of regular values of EM is partially self-overlapping while remaining connected. In the presence of this latter region, the system has an interesting property which we called bidromy. We analyze monodromy and bidromy for a concrete integrable classical Hamiltonian system of three coupled oscillators and for its quantum analog. We also show that the bifurcation involved in the transition from the regular region to the region with monodromy can be regarded as a special resonant equivariant analog of the Hamiltonian Hopf bifurcation.     

Completeness of Determinantal Hamiltonian Flows on the Matrix Affine Poisson Space

The matrix affine Poisson space (M _m,n , π _m,n ) is the space of complex rectangular matrices equipped with a canonical quadratic Poisson structure which in the square case m = n reduces to the standard Poisson structure on ${{\rm GL}_n(\mathbb{C})}$ . We prove that the Hamiltonian flows of all minors are complete. As a corollary we obtain that all Kogan–Zelevinsky integrable systems on M _n,n are complete and thus induce (analytic) Hamiltonian actions of ${\mathbb{C}^{n(n-1)/2}}$ on (M _n,n , π _n,n ) (as well as on ${{\rm GL}_n(\mathbb{C})}$ and on ${{\rm SL}_n(\mathbb{C})}$ ). We define Gelfand–Zeitlin integrable systems on (M _n,n , π _n,n ) from chains of Poisson projections and prove that their flows are also complete. This is an analog for the quadratic Poisson structure π _n,n of the recent result of Kostant and Wallach (Studies in Lie Theory. Progress in Mathematics, vol 243, pp 319–364. Birkhäuser, Boston, 2006) that the flows of the complexified classical Gelfand–Zeitlin integrable systems are complete.     

Canonical Structure and Symmetries of the Schlesinger Equations

The Schlesinger equations S _(n,m) describe monodromy preserving deformations of order m Fuchsian systems with n + 1 poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of n copies of m × m matrix algebras equipped with the standard linear Poisson bracket. In this paper we present a new canonical Hamiltonian formulation of the general Schlesinger equations S _(n,m) for all n, m and we compute the action of the symmetries of the Schlesinger equations in these coordinates.     

Completely integrable systems: Jacobi's heritage

During the last few decades, algebraic geometry has become a tool for solving differential equations and spectral questions of mechanics and mathematical physics. This paper deals with the study of the integrable systems from the point of view of algebraic geometry, inverse spectral problems and mechanics from the point of view of Lie groups. Section 1 is preliminary giving a little background. In Section 2, we study a Lie algebra theoretical method leading to completely integrable systems, based on the Kostant-Kirillov coadjoint action. Section 3 is devoted to illustrate how to decide about the algebraic complete integrability (a.c.i.) of Hamiltonian systems. Algebraic integrability means that the system is completely integrable in the sense of the phase space being foliated by tori, which in addition are real parts of a complex algebraic tori (abelian varieties). Adler-van Moerbeke's method is a very useful tool not only to discover among families of Hamiltonian systems those which are a.c.i., but also to characterize and describe the algebraic nature of the invariant tori (periods, etc.) for the a.c.i. systems. Some integrable systems, such as Kortewege—de Vries equation, Toda lattice, Euler rigid body motion, Kowalewski's top, Manakov's geodesic flow on S O (4), etc. are treated.     

The Geometry of the Semiclassical Wave Front Set for Schrödinger Eigenfunctions on the Torus

This paper deals with the phase space analysis for a family of Schrödinger eigenfunctions ψ _? on the flat torus ?? ⁿ = (?/2π?) ⁿ by the semiclassical Wave Front Set. We study those ψ _? such that WF_?(ψ _?) is contained in the graph of the gradient of some viscosity solutions of the Hamilton-Jacobi equation. It turns out that the semiclassical Wave Front Set of such Schrödinger eigenfunctions is stable under viscous perturbations of Mean Field Game kind. These results provide a further viewpoint, and in a wider setting, of the link between the smooth invariant tori of Liouville integrable Hamiltonian systems and the semiclassical localization of Schrödinger eigenfunctions on the torus.     

On integrable systems close to the toda lattice

We study the generalized discrete self-trapping (DST) system formulated in terms of the u(n) Lie-Poisson algebra as well as its noncompact analog given on the gl(n) algebra. The Hamiltonian is a quadratic-linear function of the algebra generators where the quadratic part consists of the squared generators of the Cartan subalgebra only: $$H = \sum\limits_{i = 1}^n {\frac{{\gamma _i }}{2}A_{ii}^2 + } \sum\limits_{i,j = 1}^n {m_{ij} } A_{ij} $$ Two integrable cases are discovered: one for the u(n) case and the other for the gl(n) case. The correspondingL-operators (2 × 2 andn ×n) are found which give the Lax representation for these systems. The integrable model on the gl(n) algebra looks like the Toda lattice because in this case,m _ij=c _iδ_ij-1. The corresponding 2 × 2L-operator satisfies the Sklyanin algebra.     

A New Model in the Calogero–Ruijsenaars Family

Hamiltonian reduction is used to project a trivially integrable system on the Heisenberg double of SU(n, n), to obtain a system of Ruijsenaars type on a suitable quotient space. This system possesses BC _n symmetry and is shown to be equivalent to the standard three-parameter BC _n hyperbolic Sutherland model in the cotangent bundle limit.     

Finiteness of integrable n-dimensional homogeneous polynomial potentials

We consider natural Hamiltonian systems of n>1$n>1$ 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.     

Integrable Hamiltonians with exponential potential

We construct a large class of integrable Hamiltonian systems with n degrees of freedom. This class naturally extends the nonperiodic Hamiltonians of Toda lattice type.     

Another Note on Focus-Focus Singularities

We show a natural relation between the monodromy formula for focus-focus singularities of integrable Hamiltonian systems and a formula of Duistermaat–Heckman, and extend the main results of our previous note ( ¹-action, monodromy, and topological classification) to the case of degenerate focus-focus singularities. We also consider the non-Hamiltonian case, local normal forms, etc.     

Integrable Hamiltonian systems and interactions through quadratic constraints

O _n -invariant classical relativistic field theories in one time and one space dimension with interactions that are entirely due to quadratic constraints are shown to be closely related to integrable Hamiltonian systems.     

Topological monodromy as an obstruction to Hamiltonization of nonholonomic systems: Pro or contra?

The phenomenon of a topological monodromy in integrable Hamiltonian and nonholonomic systems is discussed. An efficient method for computing and visualizing the monodromy is developed. The comparative analysis of the topological monodromy is given for the rolling ellipsoid of revolution problem in two cases, namely, on a smooth and on a rough plane. The first of these systems is Hamiltonian, the second is nonholonomic. We show that, from the viewpoint of monodromy, there is no difference between the two systems, and thus disprove the conjecture by Cushman and Duistermaat stating that the topological monodromy gives a topological obstruction for Hamiltonization of the rolling ellipsoid of revolution on a rough plane.     

The su(n) Hubbard model

The one-dimensional Hubbard model is known to possess an extended su(2) symmetry and to be integrable. I introduce an integrable model with an extended su(n) symmetry. This model contains the usual su(2) Hubbard model and has a set of features that makes it the natural su(n) generalization of the Hubbard model. Complete integrability is shown by introducing the L-matrix and showing that the transfer matrix commutes with the Hamiltonian. While the model is integrable in one dimension, it provides a generalization of the Hubbard Hamiltonian in any dimension.     

Theory of tensor invariants of integrable Hamiltonian systems. I. Incompatible Poisson structures

This paper develops a new theory of tensor invariants of a completely integrable non-degenerate Hamiltonian system on a smooth manifoldM ⁿ. The central objects in this theory are supplementary invariant Poisson structuresP _c which are incompatable with the original Poisson structureP ₁ for this Hamiltonian system. A complete classification of invariant Poisson structures is derived in a neighbourhood of an invariant toroidal domain. This classification resolves the well-known Inverse Problem that was brought into prominence by Magri's 1978 paper deveoted to the theory of compatible Poisson structures. Applications connected with the KAM theory, with the Kepler problem, with the basic integrable problem of celestial mechanics, and with the harmonic oscillator are pointed out. A cohomology is defined for dynamical systems on smooth manifolds. The physically motivated concepts of dynamical compatibility and strong dynamical compatibility of pairs of Poisson structures are introduced to study the diversity of pairs of Poisson structures incompatible in Magri's sense. It is proved that if a dynamical systemV preserves two strongly dynamically compatible Poisson structuresP ₁ andP ₂ in a general position then this system is completely integrable. Such a systemV generates a hierarchy of integrable dynamical systems which in general are not Hamiltonian neither with respect toP ₁ nor with respect toP ₂. Necessary conditions for dynamical compatibility and for strong dynamical compatibility are derived which connect these global properties with new local invariants of an arbitrary pair of incompatible Poisson structures.Supported by NSERC grant OGPIN 337.     

Vlasov moments, integrable systems and singular solutions

The Vlasov equation governs the evolution of the single-particle probability distribution function (PDF) for a system of particles interacting without dissipation. Its singular solutions correspond to the individual particle motions. The operation of taking the moments of the Vlasov equation is a Poisson map. The resulting Lie-Poisson Hamiltonian dynamics of the Vlasov moments is found to be integrable is several cases. For example, the dynamics for coasting beams in particle accelerators is associated by a hodograph transformation to the known integrable Benney shallow-water equation. After setting the context, the Letter focuses on geodesic Vlasov moment equations. Continuum closures of these equations at two different orders are found to be integrable systems whose singular solutions characterize the geodesic motion of the individual particles.     

Obstructions to the continuation of analytic integrals of Hamiltonian systems under non-Hamiltonian perturbations

In this Letter we consider n degrees-of-freedom integrable Hamiltonian systems subjected to a non-Hamiltonian perturbation controlled by a small parameter ε. An obstruction to the analytic continuation of the integrals of motion of the unperturbed system with respect to ε is developed for sufficiently small perturbations. The theory is applied to a perturbed system of Morse oscillators.