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., , which allows trapping neutral atoms. In engineering, such effects are used in spectroscopy and atomic clocks, e.g., .
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 .
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 .
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 .
The kinetic and potential parts can be seen as generators of flows, see .
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 ), 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.
