Multiscale methods for levitron problems: Theory and applications

A multiscale model based on magneto-static traps of neutral atoms or ion traps is described. The idea is to levitate a magnetic spinning top in the air, repelled by a base magnet. Real-life applications are related to magnetostatic trapping fields, e.g., [1], which allows trapping neutral atoms. In engineering, such effects are used in spectroscopy and atomic clocks, e.g., [2]. Such problems are related to nonlinear problems in structural dynamics. The dynamics of such rigid bodies are modeled as a mechanical system with kinetic and potential parts, and can be described by a Hamiltonian, see [3-5]. For such a problem, one must deal with different temporal and spatial scales, and so a novel splitting method for solving the levitron problem is proposed, see [6]. In the present paper, we focus on explicit and extrapolated time-integrator methods, which are related to the Verlet algorithms. Due to the fact that we can decouple this multiscale problem into a kinetic part T and a potential part U, explicit methods are very appropriate. We try to limit the number of evaluations which are necessary (for a given accuracy) to obtain stable trajectories, and try to avoid the iterative cycles which are involved in implicit schemes, see [7]. The kinetic and potential parts can be seen as generators of flows, see [5]. The main problem is that of accurately formulating the Hamiltonian equation and this paper proposes a novel higher order splitting scheme to obtain stable states near the relative equilibrium. To improve the splitting scheme, a novel method, called MPE (multiproduct expansion method), is applied (see [8]), which includes higher order extrapolation schemes. The stability near this relative equilibrium is discussed with numerical studies using novel improved time-integrators. The best results are obtained with extrapolated Verlet schemes rather than higher order explicit Runge-Kutta schemes. Experiments are carried out with a magnetic top in an axisymmetric magnetic field (i.e., the levitron) and future applications to quantum computation will be discussed. (C) 2012 Elsevier Ltd. All rights reserved.