102,198 research outputs found
Sobre un teorema de G. Contopoulos referente a un cierto tipo de transformaciones canónicas
No full textEn un trabajo publicado por G. Contopoulos en 1963 este autor ha demostrado un teorema según el cual Hamiltonianos de la forma: F=s₁q₁ + s₂q₂ + μ F(q₁,q₂,ρ₁,ρ₂) donde ρ₁ y ρ₂ son funciones periódicas de período 2π, mantienen su forma cuando son sometidos a transformaciones canónicas que satisfagan el teorema de Poincaré.Asociación Argentina de Astronomí
Analytical study of chaos and applications
No full textWe summarize various cases where chaotic orbits can be described analytically. First we consider the case of a magnetic bottle where we have non-resonant and resonant ordered and chaotic orbits. In the sequence we consider the hyperbolic H,non map, where chaos appears mainly around the origin, which is an unstable periodic orbit. In this case the chaotic orbits around the origin are represented by analytic series (Moser series). We find the domain of convergence of these Moser series and of similar series around other unstable periodic orbits. The asymptotic manifolds from the various unstable periodic orbits intersect at homoclinic and heteroclinic orbits that are given analytically. Then we consider some Hamiltonian systems and we find their homoclinic orbits by using a new method of analytic prolongation. An application of astronomical interest is the domain of convergence of the analytical series that determine the spiral structure of barred-spiral galaxies
Origin of chaos near three-dimensional quantum vortices: A general Bohmian theory
No full textWe provide a general theory for the structure of the quantum flow near three-dimensional (3D) nodal lines, i.e., one-dimensional loci where the 3D wave function becomes equal to zero. In suitably defined coordinates (comoving with the nodal line) the generic structure of the flow implies the formation of 3D quantum vortices. We show that such vortices are accompanied by nearby invariant lines of the comoving quantum flow, called X lines, which are normally hyperbolic. Furthermore, the stable and unstable manifolds of the X lines produce chaotic scatterings of nearby quantum (Bohmian) trajectories, thus inducing an intricate form of the quantum current in the neighborhood of each 3D quantum vortex. Generic formulas describing the structure around 3D quantum vortices are provided, applicable to an arbitrary choice of 3D wave function. We also give specific numerical examples as well as a discussion of the physical consequences of chaos near 3D quantum vortices
Resonant normal form and asymptotic normal form behaviour in magnetic bottle Hamiltonians
No full textWe consider normal forms in 'magnetic bottle' type Hamiltonians of the form H = 1/2 (rho(2)(rho) +omega(2)(1)rho(2)) + 1/2 p(z)(2) + hot (second frequency omega 2 equal to zero in the lowest order). Our main results are: (i) a novel method to construct the normal form in cases of resonance, and (ii) a study of the asymptotic behaviour of both the non-resonant and the resonant series. We find that, if we truncate the normal form series at order r, the series remainder in both constructions decreases with increasing r down to a minimum, and then it increases with r. The computed minimum remainder turns to be exponentially small in 1/Delta E, where Delta E is the mirror oscillation energy, while the optimal order scales as an inverse power of Delta E. We estimate numerically the exponents associated with the optimal order and the remainder's exponential asymptotic behaviour. In the resonant case, our novel method allows to compute a 'quasi-integral' (i.e. truncated formal integral) valid both for each particular resonance as well as away from all resonances. We applied these results to a specific magnetic bottle Hamiltonian. The non-resonant normal form yields theoretical invariant curves on a surface of section which fit well the empirical curves away from resonances. On the other hand the resonant normal form fits very well both the invariant curves inside the islands of a particular resonance as well as the non-resonant invariant curves. Finally, we discuss how normal forms allow to compute a critical threshold for the onset of global chaos in the magnetic bottle
Precessing ellipses as the building blocks of spiral arms
No full textStable periodic orbits in spiral galactic models that form families of precessing ellipses can create spiral density waves similar to those that are observed in real grand-design galaxies. We study the range in parameter space for which the amplitude of the spiral perturbation, the pattern speed, and the pitch angle collaborate so as to lead to the creation of density waves that are supported by precessing ellipses and their surrounding matter in ordered motion. Quantitative estimates lead to a correlation between the pitch angle and the amplitude of the spiral perturbation and also between the pitch angle and the pattern speed of the spiral arms. These correlations can be regarded as an orbital analog of a nonlinear dispersion relation in density wave theory
Analytical description of the structure of chaos
No full textWe consider analytical formulae that describe the chaotic regions around the main periodic orbit (x = y = 0) of the Henon map. Following our previous paper (Efthymiopoulos et al 2014 Celest. Mech. Dyn. Astron. 119 331) we introduce new variables (xi, eta) in which the product xi eta = c (constant) gives hyperbolic invariant curves. These hyperbolae are mapped by a canonical transformation Phi to the plane (x, y), giving 'Moser invariant curves'. We find that the series Phi are convergent up to a maximum value of c = c(max). We give estimates of the errors due to the finite truncation of the series and discuss how these errors affect the applicability of analytical computations. For values of the basic parameter kappa of the Henon map smaller than a critical value, there is an island of stability, around a stable periodic orbit S, containing KAM invariant curves. The Moser curves for c <= 0.32 are completely outside the last KAM curve around S, the curves with 0.32 < c < 0.41 intersect the last KAM curve and the curves with 0.41 <= c < c(max) 0.49 are completely inside the last KAM curve. All orbits in the chaotic region around the periodic orbit (x = y = 0), although they seem random, belong to Moser invariant curves, which, therefore define a 'structure of chaos'. Orbits starting close and outside the last KAM curve remain close to it for a stickiness time that is estimated analytically using the series F. We finally calculate the periodic orbits that accumulate close to the homoclinic points, i.e. the points of intersection of the asymptotic curves from x = y = 0, exploiting a method based on the self-intersections of the invariant Moser curves. We find that all the computed periodic orbits are generated from the stable orbit S for smaller values of the Henon parameter., i.e. they are all regular periodic orbits
Bohmian trajectories in an entangled two-qubit system
No full textIn this paper we examine the evolution of Bohmian trajectories in the presence of quantum entanglement (QE). We study a simple two-qubit system composed of two coherent states and investigate the impact of QE on chaotic and ordered trajectories via both numerical and analytical calculation
Partial integrability of 3d Bohmian trajectories
No full textIn this paper we study the integrability of 3d Bohmian trajectories of a system of quantum harmonic oscillators. We show that the initial choice of quantum numbers is responsible for the existence (or not) of an integral of motion which confines the trajectories on certain invariant surfaces. We give a few examples of orbits in cases where there is or there is not an integral and make some comments on the impact of partial integrability in Bohmian Mechanics. Finally, we make a connection between our present results for the integrability in the 3d case and analogous results found in the 2d and 4d cases
Chaos in de Broglie - Bohm quantum mechanics and the dynamics of quantum relaxation
No full textWe discuss the main mechanisms generating chaotic behavior of the quantum trajectories in the de Broglie - Bohm picture of quantum mechanics, in systems of two and three degrees of freedom. In the 2D case, chaos is generated via multiple scatterings of the trajectories with one or more 'nodal point - X-point complexes'. In the 3D case, these complexes form foliations along 'nodal lines' accompanied by 'X-lines'. We also identify cases of integrable or partially integrable quantum trajectories. The role of chaos is important in interpreting the dynamical origin of the 'quantum relaxation' effect, i.e. the dynamical emergence of Born's rule for the quantum probabilities, which has been proposed as an extension of the Bohmian picture of quantum mechanics. In particular, the local scaling laws characterizing the chaotic scattering phenomena near X-points, or X-lines, are related to the global rate at which the quantum relaxation is observed to proceed. Also, the degree of chaos determines the rate at which nearly-coherent initial wavepacket states lose their spatial coherence in the course of time
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