Two-dimensional Sphere S2 Research Articles

Integrability and dynamics of a simplified class B laser system.

A simplified class B laser system is a family of differential polynomial systems of degree two depending on the parameters a and b. Its rich dynamics has already been observed in 1980s, see Arecchi et al. [Opt. Commun. 51, 308-314 (1984)] and Politi et al. [Phys. Rev. A 33, 4055 (1986)], and still nowadays, it attracts the interest of the researchers. In this paper, we characterize its dynamics near infinity for all values of the parameters. When a=0, the partial integrability was already proved by Oppo and Politi [Z. Phys. B Con. Mat. 59, 111-115 (1985)]. Here, we prove that for a=0, it is completely integrable with two independent first integrals given by Liouvillian functions, and we present a complete study of its dynamics. When a≠0, we study its dynamics in the Poincaré ball B3, i.e., the interior of this ball is identified with R3 and its boundary the two-dimensional sphere S2 is identified with the infinity of R3.

Convergence of elastic flows of curves into manifolds

For a given p∈[2,+∞), we define the p-elastic energy E of a closed curve γ:S1→M immersed in a complete Riemannian manifold (M,g) as the sum of the length of the curve and the Lp-norm of its curvature (with respect to the length measure). We are interested in the convergence of the (Lp,Lp′)-gradient flow of these energies to critical points. By means of parabolic estimates, it is usually possible to prove sub-convergence of the flow, that is, convergence to critical points up to reparametrizations and, more importantly, up to isometry of the ambient. Assuming that the flow sub-converges, we are interested in proving the smooth convergence of the flow, that is, the existence of the full limit of the evolving flow.We first give an overview of the general strategy one can apply for proving such a statement. The crucial step is the application of a Łojasiewicz–Simon gradient inequality, of which we present a versatile version. Then we apply such strategy to the flow of E of curves into manifolds, proving the desired improvement of sub-convergence to full smooth convergence of the flow to critical points. As corollaries, we obtain the smooth convergence of the flow for p=2 in the Euclidean space Rn, in the hyperbolic plane H2, and in the two-dimensional sphere S2. In particular, the result implies that such flow in Rn or H2 remains in a bounded region of the space for any time.

Dynamical Liouville

The aim of this paper is to analyse an SPDE which arises naturally in the context of Liouville quantum gravity. This SPDE shares some common features with the so-called Sine-Gordon equation and is built to preserve the Liouville measure which has been constructed recently on the two-dimensional sphere S2 and the torus T2 in the work by David-Kupiainen-Rhodes-Vargas [14,16]. The SPDE we shall focus on has the following (simplified) form:∂tX=14πΔX−eγX+ξ, where ξ is a space-time white noise on R+×S2 or R+×T2. The main aspect which distinguishes this singular stochastic SPDE with well-known SPDEs studied recently (KPZ, dynamical Φ34, dynamical Sine-Gordon, generalised KPZ, etc.) is the presence of intermittency. One way of picturing this effect is that a naive rescaling argument suggests the above SPDE is sub-critical for all γ>0, while we don't expect solutions to exist when γ>2. In this work, we initiate the study of this intermittent SPDE by analysing what one might call the “classical” or “Da Prato-Debussche” phase which corresponds here to γ∈[0,γdPD=22−6). By exploiting the positivity of the non-linearity eγX, we can push this classical threshold further and obtain this way a weaker notion of solution when γ∈[γdPD,γpos=22−2). Our proof requires an analysis of the Besov regularity of natural space/time Gaussian multiplicative chaos (GMC) measures. Regularity Structures of arbitrary high degree should potentially give strong solutions all the way to the same threshold γpos and should not push this threshold further unless the notion of regularity is suitably adapted to the present intermittent situation. Of independent interest, we prove along the way (using techniques from [32]) a stronger convergence result for approximate GMC measures μϵ→μ which holds in Besov spaces.

Stochastic Landau–Lifshitz–Gilbert equation with anisotropy energy driven by pure jump noise

In this work we study a stochastic three-dimensional Landau–Lifshitz–Gilbert equation with non-zero anisotropy energy, which is drive by pure jump noise. We show existence of weak martingale solutions taking values in a two-dimensional sphere S2. The construction of the solution is based on the classical Faedo–Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces.

Minimal generating reflexive lattices of projections in finite von Neumann algebras

We show that the reflexive lattice generated by a double triangle lattice of projections in a finite von Neumann algebra is topologically homeomorphic to the two-dimensional sphere S2 (plus two distinct points corresponding to zero and I). Furthermore, such a reflexive lattice is in general minimally generating for the von Neumann algebra it generates. As an application, we show that if a reflexive lattice \({\mathcal F}\) generates the algebra \({M_n(\mathbb C)}\) of all n × n complex matrices, for some n ≥ 3, then \({\mathcal F\setminus\{0,I\}}\) is connected if and only if it is homeomorphic to S2.

Response to “Comment on ‘Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere S2 and the hyperbolic plane H2’ ” [J. Math. Phys. 46, 052702 (2005)

Response to “Comment on ‘Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere S2 and the hyperbolic plane H2’ ” [J. Math. Phys. 46, 052702 (2005)

Comment on “Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere S2 and the hyperbolic plane H2” [J. Math. Phys. 46, 052702 (2005)

Cariñena, Rañada, and Santander consider a classical Kepler problem in constant curvature spaces and related theory of conics in these spaces. Here we point out that earlier results exist in the literature. Some of these results date back to the nineteenth and the beginning of the twentieth century.

Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere S2 and the hyperbolic plane H2

The Kepler problem is a dynamical system that is well defined not only on the Euclidean plane but also on the sphere and on the hyperbolic plane. First, the theory of central potentials on spaces of constant curvature is studied. All the mathematical expressions are presented using the curvature κ as a parameter, in such a way that they reduce to the appropriate property for the system on the sphere S2, or on the hyperbolic plane H2, when particularized for κ>0, or κ<0, respectively; in addition, the Euclidean case arises as the particular case κ=0. In the second part we study the main properties of the Kepler problem on spaces with curvature, we solve the equations and we obtain the explicit expressions of the orbits by using two different methods, first by direct integration and second by obtaining the κ-dependent version of the Binet’s equation. The final part of the paper, that has a more geometric character, is devoted to the study of the theory of conics on spaces of constant curvature.

Separable Potentials and a Triality in Two-Dimensional Spaces of Constant Curvature

We characterize and completely describe some types of separable potentials in two-dimensional spaces, , of any (positive, zero or negative) constant curvature and either definite or indefinite signature type. The results are formulated in a way which applies at once for the two-dimensional sphere S2, hyperbolic plane H2, AntiDeSitter / DeSitter two-dimensional spaces AdS1+1 / dS1+1 as well as for their flat analogues E2 and M1+1. This is achieved through an approach of Cayley-Klein type with two parameters, κ1 and κ2, to encompass all curvatures and signature types. We discuss six coordinate systems allowing separation of the Hamilton-Jacobi equation for natural Hamiltonians in and relate them by a formal triality transformation, which seems to be a clue to introduce general “elliptic coordinates” for any CK space concisely. As an application we give, in any , the explicit expressions for the Fradkin tensor and for the Runge-Lenz vector, i.e., the constants of motion for the harmonic oscillator and Kepler potential on any .

A non-linear oscillator with quasi-harmonic behaviour: two- and n-dimensional oscillators

A non-linear two-dimensional system is studied by making use of both the Lagrangian and the Hamiltonian formalisms. This model is obtained as a two-dimensional version of a one-dimensional oscillator previously studied at the classical and also at the quantum level. First, it is proved that it is a super-integrable system, and then the non-linear equations are solved and the solutions are explicitly obtained. All the bounded motions are quasiperiodic oscillations and the unbounded (scattering) motions are represented by hyperbolic functions. In the second part the system is generalized to the case of n degrees of freedom. Finally, the relation of this non-linear system to the harmonic oscillator on spaces of constant curvature, the two-dimensional sphere S2 and hyperbolic plane H2, is discussed.

On harmonic oscillators on the two-dimensional sphere S2 and the hyperbolic plane H2. II.

The properties of several noncentral n=2 harmonic oscillators are examined on spaces of constant curvature. All the mathematical expressions are presented using the curvature κ as a parameter, in such a way that particularizing for κ>0, κ=0, or κ<0, the corresponding properties are obtained for the system on the sphere S2, the Euclidean plane E2, or the hyperbolic plane H2, respectively. First, the separability in several κ-dependent systems of coordinates, as well as the existence of four families of κ-dependent superintegrable potentials related with the harmonic oscillator, are studied. Then three harmonic oscillators (1:1, 2:1 and 12:1) are studied by using two different methods: superseparability and complex factorization. The second part deals with the problem of the existence of superintegrable but not superseparable systems. Several κ-dependent superintegrable harmonic oscillators with higher-order constants of motion are studied. The constants of motion are obtained by making use of the method of the complex factorization.

On harmonic oscillators on the two-dimensional sphere S2 and the hyperbolic plane H2

Two Harmonic Oscillators (isotropic and nonisotropic 2:1) are studied on the two-dimensional sphere S2 and the hyperbolic plane H2. Both systems are integrable and super-integrable with constants of motion quadratic in the momenta. These properties are shown to derive from a complex factorization for the constants of motion, which holds for arbitrary values of the curvature κ, and the dynamics of the Euclidean harmonic 1:1 and 2:1 oscillators is directly recovered for κ=0. The harmonic oscillators on either the standard unit sphere (radius R=1) or the unit Lobachewski plane (“radius” R=1) appear as the particular values of the κ-dependent potentials for the values κ=1 and κ=−1. Finally a particular potential is proposed for representing the general spherical (hyperbolic) n:1 anisotropic harmonic oscillator on a two-dimensional manifold of constant curvature.

Topological conjugacy of cascades generated by gradient flows on the two-dimensional sphere

This article presents a theorem about the topological conjugacy of a gradient dynamical system with a constant time step and the cascade generated by its Euler method. It is shown that on the two-dimensional sphere S² the gradient dynamical flow is, under

Superintegrable systems on the two-dimensional sphere S2 and the hyperbolic plane H2

The existence of superintegrable systems with n=2 degrees of freedom possessing three independent globally defined constants of motion which are quadratic in the velocities is studied on the two-dimensional sphere S2 and on the hyperbolic plane H2. The approach used is based on enforcing the conditions for the existence of two independent integrals (further than the energy). This is done in a way which allows us to discuss at once the cases of the sphere S2 and the hyperbolical plane H2, by considering the curvature κ as a parameter. Different superintegrable potentials are obtained as the solutions of certain systems of two κ-dependent second order partial differential equations. The Euclidean results are directly recovered for κ=0, and the superintegrable potentials on either the standard unit sphere (radius R=1) or the unit Lobachewski plane (“radius” R=1) appear as the particular values of the κ-dependent superintegrable potentials for the values κ=1 and κ=−1. Some new superintegrable potentials are found, both on S2 and H2. The correspondence between superintegrable systems in spaces of zero and nonzero curvature is discussed.

VISUALIZING RELATIVE EQUILIBRIA AND BIFURCATIONS BY PAINTING HAMILTONIANS ON PERSONAL COMPUTERS

In this paper, we present a graphical tool running on a PC for studies of integrable systems of one degree of freedom with several parameters, in particular for perturbed Hamiltonian systems with symmetries whose orbit space is the two-dimensional sphere S2 after they are reduced by normalization. The pioneering work was performed by Coffey et al. [1990], who developed a graphical method on a Connection Machine in order to paint the Hamiltonian function ℋ* over phase space, instead of numerical integrations of the corresponding differential equations. We have pursued their idea and the procedure proposed here renders pictures at the rate of one picture per minute. For research or teaching activities, the above mentioned method may be used in two directions: (a) in the stability analysis of equilibria and their bifurcations in parameter space; (b) for the specification of initial conditions and the corresponding qualitative long term global dynamics, in order to make proper choices for numerical studies of the full problem. As examples of how to interact with the program, and of the different role played by physical and dynamical parameters, we consider the dynamics of the Laplace–Runge–Lenz vector in two perturbed Keplerian systems of recent interest: the generalized Van der Waals effect in the hydrogen atom, and Harrington’s model for triple stellar systems.

Dirac and reduced quantization: A Lagrangian approach and application to coset spaces

A Lagrangian treatment of the quantization of first class Hamiltonian systems with constraints and Hamiltonian linear and quadratic in the momenta, respectively, is performed. The ‘‘first reduce and then quantize’’ and the ‘‘first quantize and then reduce’’ (Dirac’s) methods are compared. A source of ambiguities in this latter approach is pointed out and its relevance on issues concerning self-consistency and equivalence with the ‘‘first reduce’’ method is emphasized. One of the main results is the relation between the propagator obtained à la Dirac and the propagator in the full space. As an application of the formalism developed, quantization on coset spaces of compact Lie groups is presented. In this case it is shown that a natural selection of a Dirac quantization allows for full self-consistency and equivalence. Finally, the specific case of the propagator on a two-dimensional sphere S2 viewed as the coset space SU(2)/U(1) is worked out.