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arxiv: 2503.22568 · v2 · submitted 2025-03-28 · ⚛️ physics.class-ph

On-axis and off-axis levitation by a rotating permanent magnet

Hugo Schreckenberg , Zayneb El Omari El Alaoui , Guilhem Gallot This is my paper

Pith reviewed 2026-05-22 23:18 UTC · model grok-4.3

classification ⚛️ physics.class-ph
keywords magnetic levitationrotating permanent magnetconical orbitdipole moment modelgravity-independent trappingEarnshaw theoremstability limitsoff-axis motion
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The pith

A slightly tilted rotating permanent magnet can trap another magnet in a stable levitated orbit independent of gravity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that high-speed rotation of a slightly tilted permanent magnet produces a time-varying field that holds a second magnet in a bound levitated state, even though static magnetic levitation is forbidden by Earnshaw's theorem. The floater locks into a conical orbit that spins at exactly the rotor frequency, so that regions of like polarity on the two magnets face each other and supply the balancing force. The authors derive the conditions for this equilibrium both on and off axis, show that stability depends on floater size and rotor speed, and confirm the lower and upper speed limits experimentally for several shapes. A reader would care because the mechanism supplies a practical route to magnetic trapping that does not require gravity or additional fields.

Core claim

A slightly tilted permanent magnet rotating at high speed induces a magnetic field that traps another permanent magnet in a gravity-independent levitated bound state. During levitation the floater is locked in a conical orbit at the same frequency as the rotor; this rotation brings sides of the same polarity into opposition and produces the dynamic equilibrium. The on-axis and off-axis motion of the floater is explained theoretically, stability conditions are shown to depend on floater size and rotor speed, and the observed off-axis shift of the center of mass is accounted for by extending the dipole-moment model. Experiments map the lower and upper limits of levitation for different floater

What carries the argument

The frequency-locked conical orbit, which allows like-polarity sides to face each other and supplies the dynamic restoring force; off-axis shifts are captured by an extension of the dipole-moment model.

If this is right

  • Levitation remains possible when gravitational force is removed or reversed.
  • The range of stable rotor speeds scales with floater size and can be predicted from the dipole model.
  • Both on-axis and off-axis equilibria are sustained by the same frequency-locking mechanism.
  • Lower and upper speed boundaries for levitation can be measured and match the theoretical dependence on size.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same locking principle could be used to stabilize multiple floaters around one rotor without mechanical contact.
  • Changing the tilt angle of the rotor magnet offers a simple experimental knob for mapping additional stability boundaries.
  • The frequency match between rotor and floater suggests the setup could be adapted for synchronized rotation in vacuum or low-pressure environments.

Load-bearing premise

The off-axis displacement of the floater can be captured by extending the dipole-moment model, and the stability limits depend mainly on floater size and rotor speed in a gravity-independent way.

What would settle it

A direct measurement showing that the floater does not rotate at the same frequency as the rotor, or that the observed levitation speed limits change when the apparatus is placed in free-fall.

Figures

Figures reproduced from arXiv: 2503.22568 by Guilhem Gallot, Hugo Schreckenberg, Zayneb El Omari El Alaoui.

Figure 1
Figure 1. Figure 1: FIG. 1. Three-dimensional description in the ( [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: c emphasizes the need for an angle γ large enough to create a potential well that would keep the floater in a bound state. C. Out of equilibrium movement of the floater center of mass When externally perturbed, the floater can oscillate around its lateral or vertical equilib￾rium positions. As mentioned in [11], the floater can have different types of motion. This section brings new insight into the behavi… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. On the left, the average potential energy [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Average energy from Eq. (11) as a function of [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Scheme of the experimental set-up for the measurement of the angular velocity on the [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Magnet levitation domains of cylindrical and cubic floaters. In the left and center figures, [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Theoretical value of [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Measurements of mean lateral acceleration as a function of distance from the [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
read the original abstract

A slightly tilted permanent magnet rotating at high speed can induce a magnetic field capable of trapping another permanent magnet in a gravity independent levitated bound state, bypassing Earnshaw's theorem. During levitation, the floater magnet is locked in a conical orbit at the same frequency as the rotor. This rotation allows the sides of the same polarity of each magnet to face each other, which is responsible for the dynamic equilibrium of the floater magnet. Here, we theoretically explain the motion of the floater in-axis and off-axis and highlight levitation stability conditions and their dependence on the size of the floater and the speed of the rotor. We also experimentally studied the levitation conditions with respect to the rotational speed of the rotor for various floater's sizes and shapes. We observed and analyzed the lower and upper limits of levitation. Finally, we explained the off-axis motion of the center of mass of the floater from its equilibrium position by an extension of the dipole moment model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that a slightly tilted permanent magnet rotating at high speed can trap another permanent magnet in a gravity-independent levitated bound state, bypassing Earnshaw's theorem. The floater is locked in a conical orbit at the rotor frequency, with same-polarity sides facing each other to maintain dynamic equilibrium. The authors theoretically explain on-axis and off-axis motion via an extension of the dipole moment model, derive stability conditions depending on floater size and rotor speed, and report experimental observations of lower and upper levitation speed limits for various floater sizes and shapes.

Significance. If the central mechanism holds, the work demonstrates a rotation-enabled route to stable permanent-magnet levitation that is gravity-independent and potentially useful for contactless manipulation. The experimental mapping of speed bounds versus floater size/shape provides concrete, falsifiable data; the dipole-model extension for off-axis COM motion, if shown to be quantitatively accurate, would be a useful analytical tool.

major comments (2)
  1. [theoretical explanation of off-axis motion] Theoretical section on off-axis motion: the extension of the point-dipole force/torque formulas to predict the floater's conical orbit and off-axis center-of-mass displacement is presented without any comparison to full magnetostatic integration over the finite cylinder volumes. At the reported surface-to-surface distances of a few millimeters, higher-order multipoles and non-uniform magnetization alter both the time-averaged force and the effective potential that is claimed to lock the orbit; this directly undermines the asserted gravity-independence and the same-polarity dynamic equilibrium.
  2. [stability conditions] Stability conditions and speed limits: the lower and upper rotor-speed bounds are stated to depend primarily on floater size via the dipole model, yet no error propagation, sensitivity analysis, or numerical field verification is supplied to show that these bounds survive when the point-dipole approximation is relaxed. Because these bounds are the main experimental signature offered for the mechanism, the absence of such checks is load-bearing.
minor comments (2)
  1. [abstract] The abstract and introduction use the phrase 'gravity independent' without a quantitative statement of the residual gravitational force relative to the magnetic forces at the reported equilibrium heights.
  2. [experimental results] Figure captions for the experimental levitation photographs should include the rotor tilt angle, rotation frequency, and floater dimensions to allow direct comparison with the theoretical curves.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the scope and limitations of our dipole-model analysis. Below we respond point-by-point to the two major comments and indicate the revisions we will incorporate.

read point-by-point responses

  1. Referee: Theoretical section on off-axis motion: the extension of the point-dipole force/torque formulas to predict the floater's conical orbit and off-axis center-of-mass displacement is presented without any comparison to full magnetostatic integration over the finite cylinder volumes. At the reported surface-to-surface distances of a few millimeters, higher-order multipoles and non-uniform magnetization alter both the time-averaged force and the effective potential that is claimed to lock the orbit; this directly undermines the asserted gravity-independence and the same-polarity dynamic equilibrium.

    Authors: We agree that the point-dipole approximation is leading-order and that higher multipoles become relevant at separations of a few millimeters. The manuscript presents the dipole extension as a transparent analytical tool that reproduces the observed conical orbit, frequency locking, and same-polarity repulsion averaged over a rotation cycle; the gravity-independent character follows directly from the time-averaged force balance in the rotating frame, which the dipole terms already capture at lowest order. Experiments across multiple floater sizes show stable levitation precisely where the model predicts, providing empirical support that the qualitative mechanism survives. We will revise the theoretical section to add an explicit paragraph stating the approximation's range of validity, citing the typical magnet dimensions and separations, and noting that quantitative corrections from finite-size effects are left for future numerical work. revision: partial

  2. Referee: Stability conditions and speed limits: the lower and upper rotor-speed bounds are stated to depend primarily on floater size via the dipole model, yet no error propagation, sensitivity analysis, or numerical field verification is supplied to show that these bounds survive when the point-dipole approximation is relaxed. Because these bounds are the main experimental signature offered for the mechanism, the absence of such checks is load-bearing.

    Authors: The lower and upper speed bounds are obtained by setting the time-averaged magnetic restoring force (from the dipole model) equal to the centrifugal force required for the observed conical orbit; the resulting expressions depend on floater volume through the magnetic moment. While we did not perform a full Monte-Carlo propagation or relax the dipole assumption numerically, the experimental data for five different floater diameters and two shapes exhibit clear, reproducible speed windows whose scaling with size matches the model's prediction within the scatter of the measurements. We will add a short sensitivity paragraph that varies the effective dipole strength by ±10 % (a conservative estimate of multipole corrections) and shows that the predicted bounds shift by less than the experimental uncertainty, together with a statement that the observed agreement lends support to the robustness of the reported limits. revision: yes

Circularity Check

0 steps flagged

No circularity: dipole extension and stability analysis are derived from standard magnetostatics, not reduced to fitted inputs or self-citations

full rationale

The paper derives on-axis and off-axis motion from an extension of the point-dipole force/torque formulas applied to a rotating tilted rotor, then compares predicted speed bounds and conical-orbit locking to experimental observations across floater sizes. No step equates a fitted parameter to a 'prediction' of the same quantity, invokes a self-citation as the sole justification for a uniqueness claim, or renames an empirical pattern as a new derivation. The central gravity-independent bound-state claim rests on the time-averaged potential from the rotating dipole field, which is independent of the target result and externally falsifiable by full magnetostatic simulation or additional experiments. This is the normal case of a self-contained physical model plus validation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract only; no explicit free parameters, invented entities, or additional axioms are detailed beyond standard background. The work invokes Earnshaw's theorem as motivation and extends the standard magnetic dipole approximation.

axioms (1)
  • standard math Earnshaw's theorem prohibits stable static equilibrium for point magnets in empty space.
    Invoked to explain why rotation is required to achieve levitation.

pith-pipeline@v0.9.0 · 5706 in / 1240 out tokens · 67233 ms · 2026-05-22T23:18:25.907785+00:00 · methodology

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Reference graph

Works this paper leans on

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