Global singularities propagate in magnetic mechanical systems on Riemannian manifolds.
problem Propagation of singularities in magnetic mechanical systems.
method Combines reduction from magnetic to Riemannian systems, analysis of reparameterized flows, and regularization techniques.
result Invariant singular set under generalized gradient flow dynamics.
New findings on magnetic geodesic flows and periodic motions.
problem Characterizing superintegrable systems in magnetic geodesic flows.
method Analyzing rotationally symmetric magnetic geodesic flows.
result All sufficiently slow motions in a central magnetic field are periodic under specific curvature and homogeneity conditions.
A new Dirac algebroid approach for nonholonomic systems.
problem Nonholonomic constraints in mechanical systems.
method Developed a Dirac algebroid to generate phase equations for systems with linear nonholonomic constraints.
result Unified approach to describe systems with different potentials.
Integrates magnetic geodesic and sub-Riemannian flows on Stefel variety, proving integrability and Lax presentations.
problem Integrability of magnetic geodesic and sub-Riemannian flows on Vn,2. method Proves integrability of magnetic geodesic and sub-Riemannian flows on Vn,2 with respect to magnetic field ηdα. result Integrable cases of a heavy rigid body with a gyrostat are derived.
Extends E. Hopf's theorem to magnetic systems without conjugate points.
problem Proving magnetic curvature non-positive for magnetic systems without conjugate points.
method Using magnetic curvature introduced by the first author, proving magnetic flatness conditions.
result Magnetic flatness is a rigid condition with specific metric and curvature properties.
This paper studies Hamilton-Jacobi equations for magnetic systems with constraints.
problem Understanding dynamics of magnetic systems with geometric constraints.
method Developed Hamilton-Jacobi equations for magnetic systems with nonholonomic constraints.
result Revealed relationships between magnetic structures, constraints, and dynamics.
We apply Arnold's theory of generic smooth plane curves to Stark-Zeeman systems. This is a class of Hamiltonian dynamical systems that describes the dynamics of an electron in an external electric and magnetic field, and includes many systems from celestial mechanics. Based on Arnold's J+-invariant, we introduce inv…
Study preserves symplectic structure in forced discrete mechanical systems.
problem Preserving symplectic structure in forced discrete mechanical systems.
method Analyzes a specific type of forced discrete mechanical system (Q,Ld,fd), preserving a symplectic structure on QimesQ. result The preserved symplectic structure can be seen as Marsden-Weinstein reduction of the canonical symplectic structure.
In this paper we study rigidity aspects of Zoll magnetic systems on closed surfaces. We characterize magnetic systems on surfaces of positive genus given by constant curvature metrics and constant magnetic functions as the only magnetic systems such that the associated Hamiltonian flow is Zoll, i.e. every orbit is clos…
Study shows finiteness of magnetic hypersurfaces on closed manifolds.
problem Understanding the finiteness of magnetic hypersurfaces on closed manifolds.
method Introduced a dynamical version of the second fundamental form to generalize a previous result.
result Real-analytic negatively s-curved magnetic systems on closed real-analytic manifolds have only finitely many closed totally s-magnetic hypersurfaces. The paper explores how magnetic systems' spectra can identify metrics and 1-forms.
problem Can the marked magnetic action spectrum of magnetic systems with Anosov flow determine the metric and 1-form?
method The paper addresses this question in two settings: locally for systems with close metrics and 1-forms, and for metrics in the same conformal class.
result The paper answers the question affirmatively in both settings.
Researchers prove integrability of magnetic systems on spheres up to dimension 6.
problem Integrability of magnetic systems on spheres restricted to their surface.
method Proved complete integrability for n ≤ 6, noncommutative integrability for n ≥ 7, conjectured integrability for all n.
result Complete integrability of magnetic flows on spheres for n ≤ 6, noncommutative integrability for n ≥ 7.
Metrics are isometric for certain Anosov magnetic systems.
problem Isometry of metrics for Anosov magnetic systems.
method Conjugacy isotopic to the identity, volume-preserving conjugacy, cohomology class.
result Isometric metrics for conjugate Anosov magnetic systems.
Study magnetic curvature on Lie groups, extending Milnor's work.
problem Exploring magnetic curvatures on Lie groups.
method Computing magnetic curvatures and analyzing algebraic properties.
result Extending results from Milnor's classic paper on left-invariant metrics.
The paper defines a critical value for a magnetic system and extends solutions beyond blow-up.
problem Analyzing blow-up behavior and extending solutions for a magnetic system.
method Formulated as a magnetic geodesic equation on an infinite-dimensional Lie group, computed Mañé's critical value, established Hopf-Rinow theorem.
result Computed Mañé's critical value for the magnetic two-component Hunter-Saxton system and extended solutions beyond blow-up.
Study magnetic Hamiltonian systems with constraints, deriving Hamilton-Jacobi equations.
problem Understanding dynamics of controlled magnetic Hamiltonian systems with constraints.
method Defined CMH system, derived Hamilton-Jacobi equations for different constraints.
result Invariant solutions of Hamilton-Jacobi equations under CMH-equivalence.
Study reveals new geometric structures for magnetic field Hamiltonian systems.
problem Understanding Hamiltonian systems in magnetic fields.
method Investigation of symplectic-Haantjes geometry.
result Non-trivial symplectic-Haantjes manifolds found.
Researchers solve boundary and scattering rigidity problems for magnetic systems.
problem Recovering magnetic systems from boundary or scattering data.
method Reduced to magnetic systems and applied results from [DPSU07].
result Recovering MP-system up to a gauge. Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.
problem Deriving new equations for magnetic systems and proving their well-posedness.
method Introducing the magnetic Euler-Arnold equation and proving well-posedness for specific equations.
result Local and global well-posedness results for the magnetic Euler-Arnold equation associated with the global quasi-geostrophic equations.
Study magnetic geodesics on odd spheres, computing critical energy values.
problem Understanding magnetic geodesics on odd-dimensional spheres.
method Explicit computation and analysis of submanifolds and symmetries.
result Energy values determine magnetic geodesic connectivity on spheres.
The article finds non-trivial Zoll magnetic systems for surfaces of any genus.
problem Finding non-trivial Zoll magnetic systems for surfaces of any genus.
method Twistor theoretic approach, constructing holomorphic blow-down maps into ruled surfaces.
result Construction of non-trivial Zoll magnetic systems for surfaces of any genus.
Study of gyroscopic Chaplygin systems and magnetic flows on spheres.
problem Integrability and Hamiltonization of magnetic geodesic flows on spheres.
method Analysis of gyroscopic Chaplygin systems with magnetic forces, Hamiltonization, invariant measure existence.
result Integrable magnetic geodesic flows on spheres Sn−1 for n>3. Challenge to separate Earth's magnetic field from vehicle's magnetic field for accurate navigation.
problem Separate Earth's magnetic field from vehicle's magnetic field for accurate magnetic navigation.
method Use machine learning (ML) and integrate physics of magnetic navigation (SciML) to remove aircraft magnetic field from total magnetic field.
result A model can be constructed to effectively remove aircraft magnetic field from the dataset.
A controlled magnetic Hamiltonian (CMH) system is a regular controlled Hamiltonian (RCH) system with magnetic symplectic form, it is an important special case of RCH system. Note that there is a magnetic term on the cotangent bundle of the Heisenberg group, such that we can define a CMH system with symmetry of the Heis…
Constructs smooth integrable magnetic systems on a two-torus.
problem Creating smooth magnetic systems on a two-torus with specific properties.
method Uses Nash-Moser implicit function theorem to find zeros of an action functional.
result Characterizes Zoll magnetic systems and proves their existence.
Hamiltonian Monte Carlo (HMC) exploits Hamiltonian dynamics to construct efficient proposals for Markov chain Monte Carlo (MCMC). In this paper, we present a generalization of HMC which exploits \textit{non-canonical} Hamiltonian dynamics. We refer to this algorithm as magnetic HMC, since in 3 dimensions a subset of th…
We propose a general framework for solving statistical mechanics of systems with finite size. The approach extends the celebrated variational mean-field approaches using autoregressive neural networks, which support direct sampling and exact calculation of normalized probability of configurations. It computes variation…
Study rigidity of Hamiltonians near a minimum in symplectic and magnetic settings.
problem Rigidity of Hamiltonians near a minimum in symplectic and magnetic settings.
method Analyzing Hamiltonian systems near a compact symplectic Morse-Bott minimum, focusing on Zoll flows and magnetic forms.
result A constant curvature quantity characterizes complex space forms among Kähler manifolds.
Normal forms prove dynamical results for magnetic fields on surfaces.
problem Existence and rigidity of Zoll flows on surfaces.
method Proved normal forms for strong magnetic fields and used them to derive dynamical results.
result Flow cannot be Zoll unless specific conditions hold.
Examples are presented of how the geometric notion of the mean curvature is used for general magnetic field configurations and magnetic surfaces. It is shown that the mean magnetic curvature is related to the variation of the absolute value of the magnetic field along its lines. Magnetic surfaces of constant mean curva…
We describe three perspectives on higher quantization, using the example of magnetic Poisson structures which embody recent discussions of nonassociativity in quantum mechanics with magnetic monopoles and string theory with non-geometric fluxes. We survey approaches based on deformation quantization of twisted Poisson …
MagNet uses neural networks to predict multi-agent dynamics from observations.
problem Predicting the evolution of complex multi-agent systems.
method Formulated a coupled non-linear network with ODE-based state evolution, trained a neural network to discover dynamics from observations.
result Orders of magnitude improvement in prediction accuracy over traditional models.
Study on magnetic field and potential systems to prove rigidity results.
problem Proving rigidity for magnetic field and potential systems.
method Explicit relation between ray transform and magnetic one, applying results from [DPSU07].
result Existence of a generic set of simple MP-systems with the same boundary action function must be k-gauge equivalent.
Paper studies non-associativity in quantum systems with magnetic fields.
problem Non-associativity of magnetic translations in quantum systems.
method Quantum field theory approach with n-component fermions. result Non-associativity described by a 3-cocycle of Rn with values in S1. We solved the Schr{ö}dinger equation for a particle in a uniform magnetic field in the n-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus T^n = R^n / Λ is defined as a quotient of the Euclidean space R^n by an arbitrary n-dimensional lattice Λ. The lattice is not neces…
Machine learning models simulate molecular spectra and reactions in solvents.
problem Accurate simulation of molecular spectra and reactions in solvent environments.
method Introduced FieldSchNet, a deep neural network for modeling molecular interactions with external fields.
result Demonstrated significant lowering of Claisen rearrangement reaction activation barrier using FieldSchNet.
The paper studies magnetic geodesic flows on 2-surfaces with integrable structures.
problem Analyzing magnetic geodesic flows on 2-surfaces with additional integrals.
method Constructing exact solutions to semi-Hamiltonian systems of PDEs using generalized hodograph method and Legendre transformation.
result Exact solutions constructed for semi-Hamiltonian systems of PDEs.
Study magnetic geodesics on half-Lie groups, proving Hopf-Rinow theorem for energies above critical value.
problem Investigate magnetic geodesics on half-Lie groups using Riemannian and two-form structures.
method Define Mañé's critical value, prove Finsler geodesic flow equivalence, and apply Hopf-Rinow theorem.
result Hopf-Rinow theorem holds for energies above Mañé's critical value on magnetic geodesics.
Symmetry-electronic fingerprints reveal competing magnetic phases in two-dimensional materials.
problem Predicting magnetic ground states, moments, and anisotropy in two-dimensional magnets.
method Introduce the symmetry-electronic fingerprint (SEF), a physically interpretable representation that encodes crystallographic symmetry operations, Wyckoff-site geometry, and site-resolved electronic structure.
result SEF-trained models accurately classify magnetic ordering and regress moments alongside anisotropy energies.
Study magnetic field evolution in inhomogeneous axion stars.
problem Magnetic field evolution in axion stars with spatial inhomogeneity.
method Derived new induction equation for magnetic field, analyzed CS waves interactions, and considered compact domain effects.
result Spatial inhomogeneity of pseudoscalar field significantly affects magnetic field evolution.
Study isotropic embeddings of Lie group orbits and their application to magnetic geodesic flows.
problem Embedding coadjoint orbits and their equivalence to magnetic geodesic flows.
method Review and initiate study of isotropic and Lagrangian embeddings for SO and Sp cases, then apply to magnetic geodesic flows. result Equivalence between magnetic geodesic flows and certain spin chains.
This article provides some estimates for the relative sizes of the electric and magnetic contributions to the energy functional for the minimum energy configuration of an SU(2) gauge field on R^3 in the presence of an source in a fixed ball. The surprising fact is that the contribution to both energies from the free fi…
We consider magnetic geodesic flows of the normal metrics on a class of homogeneous spaces, in particular (co)adjoint orbits of compact Lie groups. We give the proof of the non-commutative integrability of flows and show, in addition, for the case of (co)adjoint orbits, the usual Liouville integrability by means of ana…
For a two-dimensional simple magnetic system, we study the attenuated magnetic ray transform IA,Φ, with attenuation given by a unitary connection A and a skew-Hermitian Higgs field Φ. We give a description for the range of IA,Φ acting on Cn-valued tensor fields.
In this paper, we consider a compact Riemannian manifold with boundary, endowed with a magnetic potential α and a potential U. For brevity, this type of systems are called $\MP$-systems. On simple $\MP$-systems, we consider both the boundary rigidity problem and scattering rigidity problem, see the introduction for…
This study assesses the reproducibility of 1H-MRS scans across different vendors and sessions.
problem Lack of harmonization in magnetic resonance spectroscopy protocols among vendors.
method Analysis of CV and ICC for within- and between-sessions, and correlation coefficients for across machines.
result Metabolite concentrations are highly reproducible across different vendors and sessions.
We consider a periodic problem for the motion of a charged particle in a magnetic field. Introducing a notion of Ricci curvature for such Lagrangian systems and using the methods of the calculus of variations in the large, we prove the existence of periodic motions for such particles under a condition of positivity of …
We propose a twistor construction of surfaces in Lie sphere geometry based on the linear system which copies equations of Wilczynski's projective frame. In the particular case of Lie-applicable surfaces this linear system describes joint eigenfunctions of a pair of commuting Schrödinger operators with magnetic fields.