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Multiscale methods for levitron problems: Theory and applications

机译：李维创问题的多尺度方法：理论与应用

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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.
机译：描述了基于中性原子或离子阱的静磁阱的多尺度模型。这个想法是让悬浮在空中的磁性陀螺悬浮，并被基础磁铁排斥。现实生活中的应用涉及静磁俘获场，例如[1]，它可以俘获中性原子。在工程中，这种效应用于光谱学和原子钟中，例如[2]。这些问题与结构动力学中的非线性问题有关。这种刚体的动力学建模为具有动力学和潜在零件的机械系统，可以用哈密顿量描述，请参见[3-5]。对于这样的问题，必须处理不同的时空尺度，因此提出了一种新的解决levitron问题的拆分方法，请参见[6]。在本文中，我们主要关注显式和外推时间积分器方法，与Verlet算法有关。由于我们可以将此多尺度问题分解为动力学部分T和潜在部分U，因此，显式方法非常合适。我们试图限制为获得稳定的轨迹（对于给定的精度）所必需的评估次数，并尝试避免隐式方案中涉及的迭代循环，请参见[7]。动力学部分和潜在部分可以看作是流动的生成器，请参见[5]。主要问题是要精确地公式化哈密顿方程，本文提出了一种新的高阶分裂方案来获得相对平衡附近的稳定状态。为了改善分割方案，一种称为MPE（多产品扩展方法）的新方法被应用（参见[8]），其中包括高阶外推方案。使用新型改进的时间积分器，通过数值研究讨论了该相对平衡附近的稳定性。使用外推Verlet方案而不是高阶显式Runge-Kutta方案可获得最佳结果。实验是在轴对称磁场（即levitron）中用磁顶进行的，并将讨论量子计算的未来应用。

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