arXiv research

New tensors help solve magnetic flow integrability.

Anosov magnetic flows on surfaces are characterized.

Study magnetic flows on 3D contact sub-Riemannian manifolds using Rumin complex.

Extends magnetic flow theory results to higher dimensions.

A magnetic field is defined by the property that its divergence is zero in a three dimensional oriented Riemannian manifold. Each magnetic field generates a magnetic flow whose trajectories are curves called as magnetic curves. In this paper, we give a new variational approach to studies the magnetic flow asociated wit…

Let $M$ be a closed oriented surface endowed with a Riemannian metric $g$ and let $Ω$ be a 2-form. We show that the magnetic flow of the pair $(g,Ω)$ has zero asymptotic Maslov index and zero Liouville action if and only $g$ has constant Gaussian curvature, $Ω$ is a constant multiple of the area form of $g$ and the mag…

Proves magnetic geodesic flow on sphere is integrable with constant 2-form.

Guillarmou extends X-ray transform to magnetic and thermostat flows.

Study shows magnetic trajectories in Berger spheres are homogeneous.

Homogeneous magnetic trajectories in a special linear group proven.

Normal forms prove dynamical results for magnetic fields on surfaces.

Magnetic geodesics describe the trajectory of a particle in a Riemannian manifold under the influence of an external magnetic field. In this article, we use the heat flow method to derive existence results for such curves. We first establish subconvergence of this flow to a magnetic geodesic under certain boundedness a…

Study magnetic geodesics on half-Lie groups, proving Hopf-Rinow theorem for energies above critical value.

Assume (M,g,Ω) is a closed, oriented Riemannian surface equipped with an Anosov magnetic flow. We establish certain results on the surjectivity of the adjoint of the magnetic ray transform, and use these to prove the injectivity of the magnetic ray transform on sums of tensors of degree at most two. In the final sectio…

Study magnetic geodesics on odd spheres, computing critical energy values.

We consider magnetic flows on 2-step nilmanifolds $M = Γ\backslash G$, where the Riemannian metric $g$ and the magnetic field $σ$ are left-invariant. Our first result is that when $σ$ represents a rational cohomology class and its restriction to $\mathfrak{g} = T_eG$ vanishes on the derived algebra, then the associated…

Researchers prove integrability of magnetic systems on spheres up to dimension 6.

The paper explores how magnetic systems' spectra can identify metrics and 1-forms.

New magnetic flow rigidity theorem for negative curvatures.

Study isotropic embeddings of Lie group orbits and their application to magnetic geodesic flows.

Integrates magnetic geodesic and sub-Riemannian flows on Stefel variety, proving integrability and Lax presentations.

Study of gyroscopic Chaplygin systems and magnetic flows on spheres.

The paper studies magnetic geodesic flows on 2-surfaces with integrable structures.

Researchers study injectivity of magnetic and thermostatic nonabelian ray transforms on compact surfaces.

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…

Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.

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…

The problem of description of superintegrable systems (i.e., systems with closed trajectories in a certain domain) in the class of rotationally symmetric natural mechanical systems goes back to Bertrand and Darboux. We describe all superintegrable (in a domain of slow motions) systems in the class of rotationally symme…

Researchers compute trace formula for magnetic Laplacian on hyperbolic surfaces.

Extends E. Hopf's theorem to magnetic systems without conjugate points.

We interpret magnetic billiards as Finsler ones and describe an analog of the string construction for magnetic billiards. Finsler billiards for which the law "angle of incidence equals angle of reflection" are described. We characterize the Finsler metrics in the plane whose geodesics are circles of a fixed radius. Thi…

Global singularities propagate in magnetic mechanical systems on Riemannian manifolds.

We study the dynamics of magnetic flows on Heisenberg groups. Let $H$ denote the three-dimensional simply connected Heisenberg Lie group endowed with a left-invariant Riemannian metric and an exact, left-invariant magnetic field. Let $Γ$ be a lattice subgroup of $H,$ so that $Γ\backslash H$ is a closed nilmanifold. We …

The problem of the existence of an additional (independent on the energy) first integral, of a geodesic (or magnetic geodesic) flow, which is polynomial in momenta is studied. The relation of this problem to the existence of nontrivial solutions of stationary dispersionless limits of two-dimensional soliton equations i…

The paper finds infinitely many magnetic geodesics on non-compact manifolds.

Consider a compact Riemannian manifold with boundary endowed with a magnetic field. A path taken by a particle of unit charge, mass, and energy is called a magnetic geodesic. It is shown that if everything is real-analytic, the topology, metric, and magnetic field are uniquely determined by the scattering relation of t…

In this paper we study the local magnetic ray transform of symmetric tensor fields up to rank two on a Riemannian manifold of dimension $\geq 3$ with boundary. In particular, we consider the magnetic ray transform of the combinations of tensors of different orders due to the nature of magnetic flows. We show that such …

Study magnetic geodesic flows on spheres, describing their bifurcations.

The Guillemin-Uribe trace formula is a semiclassical version of the Selberg trace formula and more general Duistermaat-Guillemin formula for elliptic operators on compact manifolds, which reflects the dynamics of magnetic geodesic flows in terms of eigenvalues of a natural differential operator (the magnetic Laplacian)…

We consider a magnetic flow without conjugate points on a closed manifold $M$ with generating vector field $\G$. Let $h\in C^{\infty}(M)$ and let $θ$ be a smooth 1-form on $M$. We show that the cohomological equation \[\G(u)=h\circ π+θ\] has a solution $u\in C^{\infty}(SM)$ only if $h=0$ and $θ$ is closed. This result …

We consider billiard ball motion in a convex domain of the Euclidean plane bounded by a piece-wise smooth curve influenced by the constant magnetic field. We show that if there exists a polynomial in velocities integral of the magnetic billiard flow then every smooth piece $γ$ of the boundary must be algebraic and eith…

Let $M$ be a closed oriented surface of negative Gaussian curvature and let $Ω$ be a non-exact 2-form. Let $λ$ be a small positive real number. We show that the longitudinal KAM-cocycle of the magnetic flow given by $\la Ω$ is a coboundary if and only if the Gaussian curvature is constant and $Ω$ is a constant multiple…

Smooth orbit equivalence proves metric equivalence for geodesic flows.

It is well-known that the LIE(Locally Induction Equation) admit soliton-type solutions and same soliton solutions arise from different and apparently irrelevant physical models. By comparing the solitons of LIE and Killing magnetic geodesics, we observe that these solitons are essentially decided by two families of iso…

The main result presented here is that the flow associated with a riemannian metric and a non zero magnetic field on a compact oriented surface without boundary, under assumptions of hyperbolic type, cannot have the same length spectrum of topologically corresponding periodic orbits as the geodesic flow associated with…

Study magnetic Laplacians on hyperbolic surfaces, revealing three regimes of eigenfunction behavior.

Study rigidity of Hamiltonians near a minimum in symplectic and magnetic settings.

This work uses Sylvester normalizing flows for more accurate metabolite quantification in MRS.