Rotation Problem: Multi-Domain Insights

• Rotation problem is a multifaceted challenge that examines mismatches between observed rotational phenomena and baseline models across several disciplines.
• It spans cosmological inertial frame selection, galaxy rotation-curve diversity, optimization on SO(3), imaging geometries, and rotational effects in PDEs.
• The topic features inverse inference, nonconvex optimization, and dynamical forcing that together drive practical innovations and theoretical insights.

The expression rotation problem is used in several technically distinct senses across contemporary research. In cosmology, it denotes the question of why inertial frames appear almost non-rotating relative to the average matter distribution despite the existence of rotating solutions of Einstein’s equations (Jones, 2017, Jones, 8 Jun 2026). In galaxy dynamics, closely related formulations concern the interpretation of galaxy rotation curves, including the dwarf-galaxy rotation-curve diversity problem and the limits of inferring dark-matter profiles from observed H I kinematics (Popolo et al., 2024, Sands et al., 2024). In computer vision and robotics, the term refers to optimization problems such as rotation averaging and robust rotation estimation on $SO(3)$ (Dong et al., 2021, Kapić et al., 2021, Olsson et al., 10 Mar 2025, Liu et al., 13 Jun 2025). Additional domain-specific uses occur in computed tomography, nonlinear dispersive PDEs, quantum mechanics, combinatorics, rigid-body dynamics, and railway operations (Duan et al., 10 Feb 2025, Antonelli et al., 2010, Yan et al., 2020, Cunha et al., 2021, Cleary et al., 2020, Lara, 2013, Prause et al., 2024). Across these settings, “rotation problem” typically names a mismatch between an observed rotational phenomenon and the structure of a baseline model, or an inference task in which rotational degrees of freedom are central and nontrivial.

1. Cosmological relative rotation

In cosmology, the rotation problem is the problem of explaining why the measured relative rotation between matter and inertial frames is extremely small, even though General Relativity admits cosmological solutions with arbitrarily large relative rotation if arbitrary initial conditions are allowed (Jones, 2017, Jones, 8 Jun 2026). The central contrast is between classical permissiveness and observational near-absence of large-scale rotation.

A path-integral resolution is proposed in both "The rotation problem" (Jones, 2017) and "An approximate application of quantum gravity to the rotation problem" (Jones, 8 Jun 2026). In these papers, cosmologies are weighted by phases of the form

$\exp\!\left(\frac{iI}{\hbar}\right),$

and the action is treated as a function of an averaged vorticity or rms relative rotation rate. Because the action is even in vorticity, the zero-vorticity configuration is a saddlepoint, and rotating cosmologies with sufficiently large vorticity contribute destructively by phase interference (Jones, 2017). The later treatment keeps the same mechanism but evaluates the suppression with an explicit dependence on the visible-universe size, Planck time, Hubble parameter, and inflationary e-fold count (Jones, 8 Jun 2026).

The quantitative thresholds are extremely small. A saddlepoint approximation in a perfect-fluid cosmology gives significant contribution only from cosmologies with average present relative rotation rate smaller than about

$T^*H^2 \approx 10^{-71}\ \text{rad yr}^{-1},$

while a refined second-order treatment yields

$$T^*H^2 a_1^{1/2} \approx 10^{-73}\ \text{rad yr}^{-1},$$

with $a_1 \approx 10^{-4}$ the scale factor when matter became more significant than radiation in the cosmological expansion (Jones, 2017). A later approximation, evaluated very early in cosmic history, reports bounds of about $2\times 10^{-51}\ \text{rad yr}^{-1}$ at about a quarter of a second after the initial singularity with 50 e-foldings, even smaller values for 55 or 60 e-foldings, and about $5\times 10^{-33}\ \text{rad yr}^{-1}$ without inflation (Jones, 8 Jun 2026).

This line of work treats the cosmological rotation problem not as a failure of classical field equations, but as a selection effect produced by quantum-gravitational interference. A plausible implication is that “rotation problem” here names a boundary-selection problem in quantum cosmology rather than a local dynamical instability.

2. Galactic rotation curves and the diversity problem

In extragalactic astrophysics, one major usage concerns the diversity of dwarf-galaxy rotation curves. "On the dwarf galaxies rotation curves diversity problem" (Popolo et al., 2024) argues that the problem is not a failure of $\Lambda$CDM itself, but a consequence of baryonic physics reshaping dark-matter haloes in a non-universal way. The paper compares a SPARC subsample with 100 simulated galaxies in the

$V_{\rm 2kpc}-V_{\rm Rlast}$

plane, where $V_{\rm 2kpc}$ measures the inner rise of the rotation curve and $\exp\!\left(\frac{iI}{\hbar}\right),$0 probes the overall halo depth.

The diagnostic is built around the expectation that self-similar haloes would produce a narrow relation between inner and outer velocities. Observationally, galaxies with similar $\exp\!\left(\frac{iI}{\hbar}\right),$1 can have very different $\exp\!\left(\frac{iI}{\hbar}\right),$2, yielding the diversity problem. The paper’s semi-analytic DFBC framework includes adiabatic contraction/compression from baryon infall, dynamical friction between baryonic clumps and dark matter, cooling, star formation, reionization, supernova feedback, and AGN feedback (Popolo et al., 2024). The key mechanism is that baryonic clumps transfer energy and angular momentum to dark matter, heating the central halo and transforming a cusp into a core.

The mass dependence is explicit. Around $\exp\!\left(\frac{iI}{\hbar}\right),$3, galaxies tend to be the most cored, with $\exp\!\left(\frac{iI}{\hbar}\right),$4; below that mass, feedback is weaker and galaxies become cuspier; above that mass, deeper stellar potentials again make the inner profile cuspier (Popolo et al., 2024). The paper states that the scatter in the $\exp\!\left(\frac{iI}{\hbar}\right),$5–$\exp\!\left(\frac{iI}{\hbar}\right),$6 plane is “not possible in the CDM scenario, producing self-similar DM haloes,” but is naturally produced once baryonic effects are included (Popolo et al., 2024).

A particularly important test case is IC2574, described as a galaxy with a very slowly rising rotation curve and a large core extending to $\exp\!\left(\frac{iI}{\hbar}\right),$7 kpc. The paper reports that its baryonic model reproduces the stellar, gas-disk, and total baryonic contributions and that, once circular velocity is evaluated in the galactic plane and observational errors are included, IC2574 and UGC05750 are no longer true outliers (Popolo et al., 2024). Within this literature, the “rotation problem” is therefore a problem of halo self-similarity and baryon–halo coupling.

A distinct but related intervention is made in "Confronting the Diversity Problem: The Limits of Galaxy Rotation Curves as a tool to Understand Dark Matter Profiles" (Sands et al., 2024). That paper argues that measured H I rotation curves do not always equal the true circular velocity. In FIRE simulations, well-ordered gaseous disks show deviations of at most about $\exp\!\left(\frac{iI}{\hbar}\right),$8 within the disk radius, but non-equilibrium behavior, non-circular motions, and non-thermal and non-kinetic stresses can cause discrepancies of $\exp\!\left(\frac{iI}{\hbar}\right),$9 or more (Sands et al., 2024). The momentum equation is written as

$T^*H^2 \approx 10^{-71}\ \text{rad yr}^{-1},$0

with

$T^*H^2 \approx 10^{-71}\ \text{rad yr}^{-1},$1

The paper concludes that some apparent diversity may be artificial, produced by failures of the steady-state, axisymmetric, thin-disk, and purely circular-motion assumptions (Sands et al., 2024).

Taken together, these two papers frame the galactic rotation problem in two different ways. One attributes diversity to genuine baryon-driven, mass-dependent halo restructuring (Popolo et al., 2024). The other emphasizes that the observable itself may fail to trace the relevant dynamical quantity and can therefore produce artificial diversity (Sands et al., 2024). This suggests that the astrophysical rotation problem is partly a model-building problem and partly an inverse-problem problem.

3. Rotation averaging and rotation estimation on $T^*H^2 \approx 10^{-71}\ \text{rad yr}^{-1},$2

In computer vision and robotics, the rotation problem commonly refers to recovering absolute rotations from noisy relative data or point correspondences. The canonical formulation of rotation averaging estimates $T^*H^2 \approx 10^{-71}\ \text{rad yr}^{-1},$3 from pairwise relative rotations $T^*H^2 \approx 10^{-71}\ \text{rad yr}^{-1},$4 by minimizing

$T^*H^2 \approx 10^{-71}\ \text{rad yr}^{-1},$5

or equivalently

$T^*H^2 \approx 10^{-71}\ \text{rad yr}^{-1},$6

(Dong et al., 2021). The difficulty arises from the nonconvexity of $T^*H^2 \approx 10^{-71}\ \text{rad yr}^{-1},$7, the coupling across variables, and scalability on large graphs (Dong et al., 2021).

"Efficient Algorithms for Rotation Averaging Problems" (Dong et al., 2021) develops two structure-exploiting solvers. The first is a BCD-based method whose single-block subproblem is a LOSSO problem solved in closed form by SVD; the second is a SUM-based method that constructs a local upper bound and decomposes the update into independent LOSSO problems, making parallel implementation possible (Dong et al., 2021). The paper uses the sufficient optimality condition

$T^*H^2 \approx 10^{-71}\ \text{rad yr}^{-1},$8

to certify global optimality of stationary points (Dong et al., 2021).

"A new dynamical model for solving rotation averaging problem" (Kapić et al., 2021) treats the problem differently: as synchronization on $T^*H^2 \approx 10^{-71}\ \text{rad yr}^{-1},$9. Given $$T^*H^2 a_1^{1/2} \approx 10^{-73}\ \text{rad yr}^{-1},$$0, the paper formulates unweighted and weighted averaging as minimization of sums of squared distances on the manifold. It introduces gradient flows

$$T^*H^2 a_1^{1/2} \approx 10^{-73}\ \text{rad yr}^{-1},$$1

and

$$T^*H^2 a_1^{1/2} \approx 10^{-73}\ \text{rad yr}^{-1},$$2

for the unweighted and weighted cases, respectively (Kapić et al., 2021). The paper explicitly places these dynamics in the family of non-Abelian Kuramoto models on $$T^*H^2 a_1^{1/2} \approx 10^{-73}\ \text{rad yr}^{-1},$$3 and uses the synchronized configuration $$T^*H^2 a_1^{1/2} \approx 10^{-73}\ \text{rad yr}^{-1},$$4 as the global minimizer of the potential (Kapić et al., 2021). The empirical result reported is that the KL average is almost identical to the geometric average and that the projected average deviates slightly more (Kapić et al., 2021).

"Certifiably Optimal Anisotropic Rotation Averaging" (Olsson et al., 10 Mar 2025) addresses a different failure mode: the mismatch between isotropic solvers and anisotropic uncertainty models. The anisotropic objective is derived from local quadratic uncertainty

$$T^*H^2 a_1^{1/2} \approx 10^{-73}\ \text{rad yr}^{-1},$$5

with

$$T^*H^2 a_1^{1/2} \approx 10^{-73}\ \text{rad yr}^{-1},$$6

leading to a matrix-weighted objective

$$T^*H^2 a_1^{1/2} \approx 10^{-73}\ \text{rad yr}^{-1},$$7

(Olsson et al., 10 Mar 2025). Because $$T^*H^2 a_1^{1/2} \approx 10^{-73}\ \text{rad yr}^{-1},$$8 is generally indefinite even though $$T^*H^2 a_1^{1/2} \approx 10^{-73}\ \text{rad yr}^{-1},$$9, the usual $a_1 \approx 10^{-4}$0-based SDP relaxation is too weak. The proposed stronger relaxation constrains pairwise blocks to $a_1 \approx 10^{-4}$1: $a_1 \approx 10^{-4}$2 On 1000 synthetic anisotropic instances, the standard anisotropic relaxation never recovered a rank-3 solution, while the proposed $a_1 \approx 10^{-4}$3 always did; on real datasets, the proposed method returned rank-3 solutions in all tested scenes (Olsson et al., 10 Mar 2025).

A complementary problem is robust rotation estimation from correspondences $a_1 \approx 10^{-4}$4. "Linearly Solving Robust Rotation Estimation" (Liu et al., 13 Jun 2025) reformulates each correspondence as two linear equations in quaternion space: $a_1 \approx 10^{-4}$5 so that stacked data produce $a_1 \approx 10^{-4}$6 (Liu et al., 13 Jun 2025). The paper’s geometric claim is that each correspondence defines a great circle on the unit quaternion sphere $a_1 \approx 10^{-4}$7, or “quaternion circle,” and that robust estimation becomes a voting problem for the point most frequently intersected by these circles (Liu et al., 13 Jun 2025). Using GPU computation, the method is reported to solve large-scale $a_1 \approx 10^{-4}$8 and severely corrupted $a_1 \approx 10^{-4}$9 outlier ratio) problems in under 0.5 seconds (Liu et al., 13 Jun 2025).

Across these papers, the rotation problem is an optimization problem on a nonlinear manifold. Its technical variants differ in whether the main challenge is nonconvexity, anisotropic uncertainty, certification, synchronization dynamics, or extreme robustness.

4. Imaging geometry and multiple centers of rotation in CT

In computed tomography, the rotation problem appears in the design of acquisition geometry. "A CT Geometry With Multiple Centers Of Rotation For Solving Sparse View Problem" (Duan et al., 10 Feb 2025) studies static CNT-based CT, where the finite packaging size of CNT emitters makes densely sampled ring arrays impractical and produces sparse-view sinograms with streak artifacts.

The paper argues that sparse-view degradation is not only a matter of too few rays but also of poor boundary conditions for PDE-based interpolation in projection space (Duan et al., 10 Feb 2025). In conventional CT, projections are acquired around a single rotation center, which yields measured rays that are dense in angle but sparse in detector position, leaving large “black” regions in projection space. The proposed geometry divides a circular ring array into several arcs such that the sources within each arc share one fixed rotation center, while all arc centers are uniformly distributed on a small circle (Duan et al., 10 Feb 2025).

For the example reported, the geometry has $2\times 10^{-51}\ \text{rad yr}^{-1}$0 rotation centers and satisfies

$2\times 10^{-51}\ \text{rad yr}^{-1}$1

The design parameter $2\times 10^{-51}\ \text{rad yr}^{-1}$2 is optimized by binary search to make the angular distribution of projections more dense and uniform and to maximize the overlapped field of view (Duan et al., 10 Feb 2025). Uniformity is quantified by the coefficient of variance

$2\times 10^{-51}\ \text{rad yr}^{-1}$3

and the optimized multi-center geometry reduces the reported $2\times 10^{-51}\ \text{rad yr}^{-1}$4 from $2\times 10^{-51}\ \text{rad yr}^{-1}$5 for the single-center case to $2\times 10^{-51}\ \text{rad yr}^{-1}$6 (Duan et al., 10 Feb 2025).

Interpolation relies on the local correlation equation (LCE), described as a family of PDEs capturing local redundancy of the Radon transform. For circular fan-beam geometry, the first-order cLCE is

$2\times 10^{-51}\ \text{rad yr}^{-1}$7

and the reconstruction task is formulated as

$2\times 10^{-51}\ \text{rad yr}^{-1}$8

(Duan et al., 10 Feb 2025). Measured data serve as boundary conditions, so more uniform distribution of measured projections improves the interpolation.

The paper reports experiments on a Forbild phantom with sparsity $2\times 10^{-51}\ \text{rad yr}^{-1}$9 and a Mayo Clinic abdomen dataset with sparsity $5\times 10^{-33}\ \text{rad yr}^{-1}$0. Even without interpolation, the multi-center geometry reduces streak artifacts relative to the single-center geometry; with cLCE interpolation and TV regularization, image quality improves further, and fine details are better preserved, especially in low-contrast structures such as the lung region (Duan et al., 10 Feb 2025). In this domain, the rotation problem is therefore a geometric design problem: how to distribute centers of rotation so that sparse-view inversion becomes better conditioned.

5. Rotation as a dynamical or spectral term in PDEs and quantum systems

Another large class of rotation problems concerns evolution equations modified by explicit rotational terms. In "On the Cauchy Problem for nonlinear Schrödinger equations with rotation" (Antonelli et al., 2010), the equation is

$5\times 10^{-33}\ \text{rad yr}^{-1}$1

This is presented as a model for superfluid quantum gases in rotating traps (Antonelli et al., 2010). The main theorem gives global existence in the energy space

$5\times 10^{-33}\ \text{rad yr}^{-1}$2

for defocusing nonlinearities without restriction on $5\times 10^{-33}\ \text{rad yr}^{-1}$3, while focusing nonlinearities admit finite-time blow-up under conditions that depend on axial symmetry and the trap frequencies (Antonelli et al., 2010). The proof removes the explicit rotation by a time-dependent change of coordinates $5\times 10^{-33}\ \text{rad yr}^{-1}$4, converting the problem to an NLS with a time-dependent potential $5\times 10^{-33}\ \text{rad yr}^{-1}$5 (Antonelli et al., 2010). Rotation is benign for defocusing global existence but complicates blow-up analysis in the focusing case.

"Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces" (Yan et al., 2020) studies the RMKP equation

$5\times 10^{-33}\ \text{rad yr}^{-1}$6

The rotation parameter is $5\times 10^{-33}\ \text{rad yr}^{-1}$7, proportional to the Coriolis force (Yan et al., 2020). After normalization $5\times 10^{-33}\ \text{rad yr}^{-1}$8, the linear phase becomes

$5\times 10^{-33}\ \text{rad yr}^{-1}$9

and the extra $\Lambda$0 singularity is the main difficulty. The paper proves local well-posedness in $\Lambda$1 for $\Lambda$2, $\Lambda$3, an endpoint result in $\Lambda$4 via $\Lambda$5 and $\Lambda$6 spaces, and ill-posedness for $\Lambda$7 in the sense that the flow map is not $\Lambda$8 at the origin (Yan et al., 2020). The decisive technique is a decomposition into regular and singular frequency regions, reflecting the rotational singularity (Yan et al., 2020).

"A singular limit problem for rotating capillary fluids with variable rotation axis" (Fanelli, 2015) examines simultaneous incompressible and fast-rotation asymptotics for a Navier–Stokes–Korteweg system with Coriolis force. With constant capillarity $\Lambda$9, the variable-axis case uses

$V_{\rm 2kpc}-V_{\rm Rlast}$0

where $V_{\rm 2kpc}-V_{\rm Rlast}$1 and $V_{\rm 2kpc}-V_{\rm Rlast}$2 (Fanelli, 2015). The main result is convergence to a linear parabolic-type equation with variable coefficients

$V_{\rm 2kpc}-V_{\rm Rlast}$3

(Fanelli, 2015). A key structural consequence is that limit fields satisfy $V_{\rm 2kpc}-V_{\rm Rlast}$4, $V_{\rm 2kpc}-V_{\rm Rlast}$5, $V_{\rm 2kpc}-V_{\rm Rlast}$6, and $V_{\rm 2kpc}-V_{\rm Rlast}$7 (Fanelli, 2015).

At the quantum-mechanical level, "Effects of rotation and Coulomb type potential on the spin-1/2 Aharonov-Bohm problem" (Cunha et al., 2021) studies a spin-$V_{\rm 2kpc}-V_{\rm Rlast}$8 particle in an AB flux, an attractive Coulomb-type potential $V_{\rm 2kpc}-V_{\rm Rlast}$9, and a rotating frame with $V_{\rm 2kpc}$0. The Pauli–Schrödinger equation is

$V_{\rm 2kpc}$1

Rotation enters the radial equation through

$V_{\rm 2kpc}$2

shifting the spectrum linearly in $V_{\rm 2kpc}$3, modifying degeneracies, and allowing positive energies for some states (Cunha et al., 2021). Because of the singularity at the origin, self-adjoint extension theory is required when $V_{\rm 2kpc}$4 (Cunha et al., 2021).

In these PDE and quantum settings, the rotation problem is not primarily about inference. It is about how explicit rotational operators alter well-posedness thresholds, asymptotic limits, blow-up arguments, spectra, and admissible boundary conditions.

6. Discrete, geometric, and operational formulations

Several additional papers use “rotation problem” in discrete mathematics, dynamical systems, rigid-body mechanics, and transportation planning.

In combinatorics, "Counting difficult tree pairs with respect to the rotation distance problem" (Cleary et al., 2020) studies the minimum number of simple rotations needed to transform one rooted binary tree into another. The rotation distance is

$V_{\rm 2kpc}$5

The hardest instances are difficult tree pairs, defined by the absence of both common edges and one-off edges (Cleary et al., 2020). The total number of instances of size $V_{\rm 2kpc}$6 is $V_{\rm 2kpc}$7, where

$V_{\rm 2kpc}$8

and the paper reports that the fraction of difficult pairs decays exponentially, approximately as

$V_{\rm 2kpc}$9

from exact-data fitting, with sampling estimates

$\exp\!\left(\frac{iI}{\hbar}\right),$00

(Cleary et al., 2020). Here the rotation problem is a shortest-path problem on tree space, and the main phenomenon is the exponential rarity of irreducibly hard instances.

In quasiperiodic dynamics, "Solving the Babylonian Problem of quasiperiodic rotation rates" (Das et al., 2017) defines the problem of recovering a meaningful rotation rate from projected observations of a torus orbit. For

$\exp\!\left(\frac{iI}{\hbar}\right),$01

one does not observe $\exp\!\left(\frac{iI}{\hbar}\right),$02 directly, only $\exp\!\left(\frac{iI}{\hbar}\right),$03. The goal is to compute an observable rotation rate $\exp\!\left(\frac{iI}{\hbar}\right),$04, which for circle-valued projections satisfies

$\exp\!\left(\frac{iI}{\hbar}\right),$05

after choosing a lift (Das et al., 2017). The paper introduces the Embedding Continuation Method, based on Takens embedding and the Birkhoff ergodic theorem, to reconstruct the correct lift from trajectory data (Das et al., 2017). This is a rotation problem in the sense of inverse recovery from projected quasiperiodic motion.

In rigid-body mechanics, "Short-axis-mode rotation of a free rigid body by perturbation series" (Lara, 2013) considers torque-free motion near the principal axis of maximum inertia. The torque-free Hamiltonian

$\exp\!\left(\frac{iI}{\hbar}\right),$06

is rearranged as

$\exp\!\left(\frac{iI}{\hbar}\right),$07

where the perturbation is controlled by

$\exp\!\left(\frac{iI}{\hbar}\right),$08

rather than small triaxiality (Lara, 2013). The paper derives action-angle variables through Hamilton–Jacobi reduction and constructs a Lie-transform perturbation series, recovering Kinoshita’s low-order expansions (Lara, 2013). Here the rotation problem is a perturbative reformulation of rigid-body attitude dynamics near stable short-axis-mode rotation.

In transport operations, "An Iterative Refinement Approach for the Rolling Stock Rotation Problem with Predictive Maintenance" (Prause et al., 2024) uses “rotation” in the railway-planning sense: a vehicle circulation or assignment of trips to rolling stock. The paper defines RSRP-PdM through a state-expanded event-graph with discretized health states $\exp\!\left(\frac{iI}{\hbar}\right),$09 and degradation functions $\exp\!\left(\frac{iI}{\hbar}\right),$10 (Prause et al., 2024). The induced ILP is

$\exp\!\left(\frac{iI}{\hbar}\right),$11

subject to trip coverage, flow conservation, balancedness, and integrality constraints (Prause et al., 2024). A rounding function is designed so that health-state parameters are consistently underestimated, yielding lower bounds that converge from below under iterative refinement (Prause et al., 2024). In this domain, the rotation problem is a fleet-scheduling problem with maintenance and stochastic health evolution.

These examples show that “rotation problem” often names a structural obstacle associated with state spaces that are cyclic, manifold-valued, or combinatorially generated by local rotations. The shared feature is not a single mathematics, but the nontrivial role of rotational structure in inference, optimization, or dynamics.

7. Common themes and cross-domain distinctions

Despite the heterogeneity of the literature, several recurring patterns are visible. First, many rotation problems arise from a mismatch between an observed rotational phenomenon and a simpler baseline model. In cosmology, classical GR admits rotating universes, yet the observed universe is nearly nonrotating relative to matter (Jones, 2017, Jones, 8 Jun 2026). In dwarf-galaxy dynamics, self-similar halo expectations or naive circular-velocity reconstructions fail to capture the observed spread of rotation-curve shapes (Popolo et al., 2024, Sands et al., 2024). In CT, single-center acquisition yields poor projection-space boundary conditions under sparse sampling (Duan et al., 10 Feb 2025).

Second, many formulations are inverse problems on nonlinear or constrained spaces. Rotation averaging and robust rotation estimation operate on $\exp\!\left(\frac{iI}{\hbar}\right),$12 or $\exp\!\left(\frac{iI}{\hbar}\right),$13, and their difficulty stems from nonconvexity, anisotropy, certification, and outlier structure (Dong et al., 2021, Kapić et al., 2021, Olsson et al., 10 Mar 2025, Liu et al., 13 Jun 2025). The Babylonian problem of quasiperiodic rotation rates similarly requires reconstructing a lifted quantity from a projection of torus dynamics (Das et al., 2017).

Third, several papers show that “rotation” is not merely an observable but an operator or singular perturbation that changes the qualitative form of the governing equation. This is explicit in NLS with angular momentum terms, RMKP with a Coriolis parameter, rotating capillary fluids with variable axis, and the rotating-frame AB problem (Antonelli et al., 2010, Yan et al., 2020, Fanelli, 2015, Cunha et al., 2021).

A concise classification is therefore possible.

Usage of “rotation problem”	Representative content	Representative papers
Cosmological relative rotation	Why matter and inertial frames are nearly nonrotating	(Jones, 2017, Jones, 8 Jun 2026)
Galactic rotation curves	Diversity, core formation, and inference limits	(Popolo et al., 2024, Sands et al., 2024)
$\exp\!\left(\frac{iI}{\hbar}\right),$14 estimation and averaging	Synchronization, SDP certification, robust estimation	(Dong et al., 2021, Kapić et al., 2021, Olsson et al., 10 Mar 2025, Liu et al., 13 Jun 2025)
Imaging acquisition geometry	Multiple centers of rotation in sparse-view CT	(Duan et al., 10 Feb 2025)
Rotation-modified evolution equations	NLS, RMKP, rotating capillary fluids, AB spectra	(Antonelli et al., 2010, Yan et al., 2020, Fanelli, 2015, Cunha et al., 2021)
Discrete and operational problems	Tree rotation distance, quasiperiodic rates, rolling-stock rotations	(Cleary et al., 2020, Das et al., 2017, Prause et al., 2024)

This range of meanings makes the phrase inherently context-dependent. In astrophysics, it usually refers to rotation curves or cosmic vorticity; in vision and robotics, to optimization over rotations; in PDEs, to rotational forcing or Coriolis modification; and in combinatorics or scheduling, to local restructuring moves or vehicle circulations. The unifying idea is that rotation introduces either a nontrivial geometry or a nontrivial observable whose interpretation cannot be reduced to a naive Euclidean or static model.