Andrea Giacobbe

Monodromy of the quantum 1:1:2 resonant swing spring

We describe the qualitative features of the joint spectrum of the quantum 1:1:2 resonant swing spring. The monodromy of the classical analogue of this problem is studied in Dullin et al. [Physica D 190, 15–37 (2004)]. Using symmetry arguments and numerical calculations we compute its three-dimensional (3D) lattice of quantum states and show that it possesses a codimension 2 defect characterized by a nontrivial 3D-monodromy matrix. The form of the monodromy matrix is obtained from the lattice of quantum states and depends on the choice of an elementary cell of the lattice. We compute the quantum monodromy matrix, that is the inverse transpose of the classical monodromy matrix. Finally we show that the lattice of quantum states for the 1:1:2 quantum swing spring can be obtained—preserving the symmetries—from the regular 3D-cubic lattice by means of three “elementary monodromy cuts.”

Gauge conservation laws and the momentum equation in nonholonomic mechanics

The gauge mechanism is a generalization of the momentum map which links conservation laws to symmetry groups of nonholonomic systems. This method has been so far employed to interpret conserved quantities as momenta of vector fields which are sections of the constraint distribution. In order to obtain the largest class of conserved quantities of this type, we extend this method to an over-distribution of the constraint distribution, the so-called reaction-annihilator distribution, which encodes the effects that the nonholonomic reaction force has on the conservation laws. We provide examples showing the effectiveness of this generalization. Furthermore, we discuss the Noetherian properties of these conserved quantities, that is, whether and to which extent they depend only on the group, and not on the system. In this context, we introduce a notion of ‘weak Noetherianity’. Finally, we point out that the gauge mechanism is equivalent to the momentum equation (at least for locally free actions), we generalize the momentum equation to the reaction-annihilator distribution, and we introduce a ‘gauge momentum map’ which embodies both methods. For simplicity, we treat only the case of linear constraints, natural Lagrangians, and lifted actions.

The topology associated with cusp singular points

In this paper we investigate the global geometry associated with cusp singular points of two-degree of freedom completely integrable systems. It typically happens that such singular points appear in couples, connected by a curve of hyperbolic singular points. We show that such a couple gives rise to two possible topological types as base of the integrable torus bundle, that we call pleat and flap. When the topological type is a flap, the system can have non-trivial monodromy, and this is equivalent to the existence in phase space of a lens space compatible with the singular Lagrangian foliation associated to the completely integrable system.

Geometry of Invariant Tori of Certain Integrable Systems with Symmetry and an Application to a Nonholonomic System

Bifibrations, in symplectic geometry called also dual pairs, play a relevant role in the theory of superintegrable Hamiltonian systems. We prove the existence of an analogous bifibrated geometry in dynamical systems with a symmetry group such that the reduced dynamics is periodic. The integrability of such systems has been proven by M. Field and J. Hermans with a reconstruction technique. We apply the result to the nonholonomic system of a ball rolling on a surface of revolution.

Geometric structure of “broadly integrable” Hamiltonian systems

Abstract We study the geometry of the fibration in invariant tori of a Hamiltonian system which is integrable in Bogoyavlenskij’s “broad sense”—a generalization of the standard cases of Liouville and non-commutative integrability. We show that the structure of such a fibration generalizes that of the standard cases. Firstly, the base manifold has a Poisson structure. Secondly, there is a natural way of arranging the invariant tori which generates a second foliation of the phase space; however, such a foliation is not just the polar to the invariant tori. Finally, under suitable conditions, there is a notion of an “action manifold” with an affine structure. We also study the analogous of the problem of the existence of “global action-angle coordinates” for these systems.

Rotation forms and local Hamiltonian monodromy

The monodromy of torus bundles associated with completely integrable systems can be computed using geometric techniques (constructing homology cycles) or analytic arguments (computing discontinuities of abelian integrals). In this article, we give a general approach to the computation of monodromy that resembles the analytical one, reducing the problem to the computation of residues of polar 1-forms. We apply our technique to three celebrated examples of systems with monodromy (the champagne bottle, the spherical pendulum, the hydrogen atom) and to the case of non-degenerate focus-focus singularities, re-obtaining the classical results. An advantage of this approach is that the residue-like formula can be shown to be local in a neighborhood of a singularity, hence allowing the definition of monodromy also in the case of non-compact fibers. This idea has been introduced in the literature under the name of scattering monodromy. We prove the coincidence of the two definitions with the monodromy of an appropr...

Infinitesimally stable and unstable singularities of 2-degrees of freedom completely integrable systems

In this article we give a list of 10 rank zero and 6 rank one singularities of 2-degrees of freedom completely integrable systems. Among such singularities, 14 are the singularities that satisfy a non-vanishing condition on the quadratic part, the remaining 2 are rank 1 singularities that play a role in the geometry of completely integrable systems with fractional monodromy. We describe which of them are stable and which are unstable under infinitesimal completely integrable deformations of the system.

Some remarks on the Gelfand–Cetlin system

In the first section of this paper, we show that the functions in involution of the Gelfand–Cetlin system can be obtained from a λ-parametric Lax equation. In the second section, we observe that the Gelfand–Cetlin system has no obstructions to global action–angle coordinates, and we give an explicit expression of global (action) angle coordinates. In the third section, we remark the fact that the Gelfand–Cetlin system is obtained via a nesting of superintegrable systems, and show they all present a non-vanishing Chern class.

Laminar hydromagnetic flows in an inclined heated layer

In this paper we investigate, analytically, stationary laminar flow solutions of an inclined layer filled with a hydromagnetic fluid heated from below and subject to the gravity field. In particular we describe in a systematic way the many basic solutions associated to the system. This extensive work is the basis to linear instability and nonlinear stability analysis of such motions.

Some results in the nonlinear stability for rotating Bénard problem with rigid boundary condition

The scope of this article is to expose the stabilizing properties of rotation and solute gradient for the Benard problem with (at least one-sided) rigid boundary conditions. We perform a linear investigation of the critical threshold for the rotating Benard problem with a binary fluid, and we also make an investigation with a Lyapunov function for the particular problem of a rotating single fluid. In all the these cases an increase of the Taylor number has stabilizing effects.