1,720,978 research outputs found
Some remarks on the Gelfand-Cetlin system
No full textIn 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
Convexity of multi-valued momentum maps
No full textA famous theorem of Atiyah, Guillemin and Sternberg states that, given a Hamiltonian torus action, the image of the momentum map is a convex polytope. We prove that this result can be extended to the case in which the action is non-Hamiltonian. Our generalization of the theorem states that, given a symplectic torus action, the momentum map can be defined on an appropriate covering of the manifold and its image is the product of a convex polytope along a rational subspace times the orthogonal vector space. We also prove that this decomposition in direct product is stable under small equivariant perturbations of the symplectic structure; this, in particular, means that the property of being Hamiltonian is locally stable. The technique developed allows us to extend the result to any compact group action and also to deduce that any symplectic n-torus action, with fixed points, on a compact 2n-dimensional manifold, is Hamiltonian
Fractional monodromy: parallel transport of homology cycles
Full text linkA 2n-dimensional completely integrable system gives rise to a singular fibration whose generic fiber is the n-torus. In the classical setting, it is possible to define a parallel transport of elements of the first homotopy group of a fiber along a path, when the path describes a loop around a singular fiber, it defines an automorphism of the fundamental group of the torus called monodromy transformation [J.J. Duistermaat, On global action-angle coordinates, Communications on Pure and Applied Mathematics 33 (6) (1980) 687– 706]. Some systems give rise to a non-classical setting, in which the path can wind around a singular fiber only by crossing a codimension 1 submanifold of special singular fibers (a wall), in this case a non-classical parallel transport can be defined on a subgroup of the fundamental group. This gives rise to what is known as monodromy with fractional coefficients [N. Nekhoroshev, D. Sadovskiì, B. Zhilinskiì, Fractional monodromy of resonant classical and quantum oscillators, Comptes Rendus Mathematique 335 (11) (2002) 985–988]. In this article, we give a precise meaning to the non-classical parallel transport. In particular we show that it is a homologic process and not a homotopic one. We justify this statement by describing the type of singular fibers that generate a wall that can be crossed, by describing the parallel transport in a semi-local neighbourhood of the wall of singularities, and by producing a family of 4-dimensional examples
Infinitesimally Stable and Unstable Singularities of 2-Degrees of Freedom Completely Integrable Systems
Full text linkIn 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
Marginal regions for the solute Bénard problem with many types of boundary conditions
Full text linkA large number of variants of the Bénard problem (with a solute, rotating, subject to mag- netic field, etc.) have been extensively studied. Despite this, new interesting results can be obtained imposing very general yet physically relevant boundary conditions. In this frame- work, we develop a technique to analytically compute the marginal region in parameter space.
We investigate the thermal stability of a fluid layer salted from below, subject to finite slip on velocity and Robin conditions on temperature and solute concentration. We write analytical conditions for the onset of stationary convection, obtain simplified formulas for particularly symmetric cases, and draw the associated (convective) marginal regions in some significant cases. Moreover, we describe the analytical conditions for the onset of overstability, and use such equations to numerically draw the associated (overstable) marginal region. We finally perform an asymptotic analysis for small wave numbers
The topology associated to cusp singular points
Full text linkIn 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 nontrivial 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
Double diffusion in rotating porous media under general boundary conditions
Full text linkIn this article we study a binary fluid saturating a rotating porous medium; the fluid is modeled according to Darcy–Brinkman law and the boundary conditions are rigid or stress-free on the velocity field and of Robin type on temperature and solute concentration.
We determine the threshold of linear instability and its dependence on Taylor and Darcy numbers. Using a Lyapunov function we prove analytically, under certain assumptions, the coincidence of linear and nonlinear thresholds. A second Lyapunov function allows us to prove numerically the coincidence of the two thresholds with weaker assumptions on the parameters.
We show that in the particular limit case of fixed heat and solute fluxes this system has a remarkable feature: the wave number of critical cells goes to zero when the Taylor number is below a threshold. Above such threshold, the wave number is non-zero when the Darcy number belongs to a finite interval. These phenomena could perhaps be tested experimentally
Monodromy in the resonant swing spring
Full text linkIn this paper, it is shown that an integrable approximation of the spring pendulum, when tuned to be in 1:1:2 resonance, has monodromy. The stepwise precession angle of the swing plane of the resonant spring pendulum is shown to be a rotation number of the integrable approximation. Due to the monodromy, this rotation number is not a globally defined function of the integrals. In fact at lowest order it is given by arg(χ + iλ), where χ and λ are functions of the integrals. The resonant swing spring is therefore a system where monodromy has easily observed physical consequences
- …
