In this work we illustrate the Arnold diffusion in a concrete example-the a priori unstable Hamiltonian system of 2 + 1/2 degrees of freedom H(p, q, I, ϕ, s) = p 2 /2 + cos q − 1 + I 2 /2+h(q, ϕ, s; ε)-proving that for any small periodic perturbation of the form h(q, ϕ, s; ε) = ε cos q (a00 + a10 cos ϕ + a01 cos s) (a10a01 = 0) there is global instability for the action. For the proof we apply a geometrical mechanism based in the so-called Scattering map.This work has the following structure: In a first stage, for a more restricted case (I * ∼ π/2µ, µ = a10/a01), we use only one scattering map, with a special property: the existence of simple paths of diffusion called highways. Later, in the general case we combine a scattering map with the inner map (inner dynamics) to prove the more general result (the existence of the instability for any µ). The bifurcations of the scattering map are also studied as a function of µ. Finally, we give an estimate for the time of diffusion, and we show that this time is primarily the time spent under the scattering map. MSC2010 numbers: 37J40
A major challenge for our understanding of the mathematical basis of particle dynamics is the formulation of N-body and N-vortex dynamics on Riemann surfaces. In this paper, we show how the two problems are, in fact, closely related when considering the role played by the intrinsic geometry of the surface. This enables a straightforward deduction of the dynamics of point masses, using recently derived results for point vortices on general closed differentiable surfaces endowed with a metric. We find, generally, that Kepler's Laws do not hold. What is more, even Newton's First Law (the law of inertia) fails on closed surfaces with variable curvature (e.g. the ellipsoid).
We prove that for any non-trivial perturbation depending on any two independent harmonics of a pendulum and a rotor there is global instability. The proof is based on the geometrical method and relies on the concrete computation of several scattering maps. A complete description of the different kinds of scattering maps taking place as well as the existence of piecewise smooth global scattering maps is also provided. MSC2010 numbers: 37J40
An algorithm for the numerical solution of the Schrödinger equation in the case of a time dependent potential is proposed. Our simple modification upgrades the well known method of Koonin while negligibly increasing the computing time. In the presented test the accuracy is enhanced by up to an order of magnitude. 02.60.+y, 25.85.-w The microscopic description of many-body systems like atoms or nuclei is based on a many-body Hamiltonian. The related wave functions are given by Slater determinants for fermions. In the case of time dependent processes like atomic or nuclear collisions, nuclear fission or fusion, however, the situation is too complex due to the great number of degrees of freedom. Therefore, in most cases, a collective coordinate is introduced according to the essential physical properties of the considered system. This procedure leads to a macroscopic model with one degree of freedom, which is governed by an effective one-body Schrödinger equation including a time dependent potential in the considered examples [1]. The well known coordinate representation readsA momentum dependent potential V , leading to an integro-differential equation, is excluded here. In almost all practical cases eq.(1) has to be evaluated numerically. The standard technique essentially consists of the following two steps: first apply a suitable scheme for the space discretization and then perform the time integration. The algorithm presented here is only related to the second step: The time integration in the case of an explicitly time dependent potential V (r, t). Our proposed convenient modification upgrades the standard method to a much more efficient version while negligibly increasing the computing time.Our experience is due to the description of ternary fission, i.e. a fission process accompanied by the emission of an α-particle, with results to be published elsewhere. However, the presented algorithm might be of broader interest, not necessarily restricted to this nuclear physics theme.
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