arXiv research

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

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

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

Study shows magnetic trajectories in Berger spheres are homogeneous.

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…

Homogeneous magnetic trajectories in a special linear group proven.

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

Study magnetic geodesics on Heisenberg groups and manifolds.

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

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

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

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.

Extends magnetic flow theory results to higher dimensions.

Let $Q$ be a closed manifold admitting a locally-free action of a compact Lie group $G$. In this paper we study the properties of geodesic flows on $Q$ given by Riemannian metrics which are invariant by such an action. In particular, we will be interested in the existence of geodesics which are closed up to the action …

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…

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…

Smooth orbit equivalence proves metric equivalence for geodesic flows.

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

Study magnetic geodesic flows on spheres, describing their bifurcations.

Study of gyroscopic Chaplygin systems and magnetic flows on spheres.

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…

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…

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

New proofs of Cheeger-like inequalities for coexact 1-forms on hyperbolic manifolds.

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…

Researchers compute trace formula for magnetic Laplacian on hyperbolic 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…

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

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)…

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 …

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…

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…

Study of magnetic geodesics on Heisenberg nilmanifolds.

Study magnetic geodesics on Kähler potentials using variational methods.

Study proves uniqueness for ray transform on surfaces with obstacles.

We prove the existence of Alexandrov embedded closed magnetic geodesics on closed hyperbolic surfaces. Closed magnetic geodesics correspond to closed curves with prescribed geodesic curvature.

Magnetic Brunn-Minkowski inequalities on Riemannian manifolds

Homogeneous magnetic paths found in Heisenberg space.

Study on Killing magnetic curves in Heisenberg group geometry.

We prove that, generically, magnetic geodesics on surfaces will turn away from points with lightlike tangent planes, and we motivate our result with numerical solutions for closed magnetic geodesics.

We study the integrability of the conformal geodesic flow (also known as the conformal circle flow) on the $SO(3)$--invariant gravitational instantons. On a hyper--Kähler four--manifold the conformal geodesic equations reduce to geodesic equations of a charged particle moving in a constant self--dual magnetic field. In…

New tensors help solve magnetic flow integrability.

Anosov magnetic flows on surfaces are characterized.

We apply a local systolic-diastolic inequality for contact forms and odd-symplectic forms on three-manifolds to bound the magnetic length of closed curves with prescribed geodesic curvature (also known as magnetic geodesics) on an oriented closed surface. Our results hold when the prescribed curvature is either close t…

We prove the existence of two Alexandrov embedded closed magnetic geodesics on any two dimensional sphere with nonnegative Gauss curvature.

We give existence results for simple closed curves with prescribed geodesic curvature on $S^{2}$, which correspond to periodic orbits of a charge in a magnetic field.

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…