Finsler Geodesics Research Articles

Snell’s law revisited and generalized via Finsler geometry

We study the variational problem of finding the fastest path between two points that belong to different anisotropic media, each with a prescribed speed profile and a common interface. The optimal curves are Finsler geodesics that are refracted — broken — as they pass through the interface, due to the discontinuity of their velocities. This “breaking” must satisfy a specific condition in terms of the Finsler metrics defined by the speed profiles, thus establishing the generalized Snell’s law. In the same way, optimal paths bouncing off the interface — without crossing into the second domain — provide the generalized law of reflection. The classical Snell’s and reflection laws are recovered in this setting when the velocities are isotropic. If one considers a wave that propagates in all directions from a given ignition point, the trajectories that globally minimize the traveltime generate the wavefront at each instant of time. We study in detail the global properties of such wavefronts in the Euclidean plane with anisotropic speed profiles. Like the individual rays, they break when they encounter the discontinuity interface. But they are also broken due to the formation of cut loci — stemming from the self-intersection of the wavefronts — which typically appear when they approach a high-speed profile domain from a low-speed profile.

Symmetric periodic orbits and invariant disk-like global surfaces of section on the three-sphere

We study Reeb dynamics on the three-sphere equipped with a tight contact form and an anti-contact involution. We prove the existence of a symmetric periodic orbit and provide necessary and sufficient conditions for it to bound an invariant disk-like global surface of section. We also study the same questions under the presence of additional symmetry and obtain similar results in this case. The proofs make use of pseudoholomorphic curves in symplectizations. As applications, we study Birkhoff’s conjecture on disk-like global surfaces of section in the planar circular restricted three-body problem and the existence of symmetric closed Finsler geodesics on the two-sphere. We also present applications to some classical Hamiltonian systems.

Existence results for closed Finsler geodesics via spherical complexities

We apply topological methods and a Lusternik-Schnirelmann-type approach to prove existence results for closed geodesics of Finsler metrics on spheres and projective spaces. The main tool in the proofs are spherical complexities, which have been introduced in earlier work of the author. Using them, we show how pinching conditions and inequalities between a Finsler metric and a globally symmetric metric yield the existence of multiple closed geodesics as well as upper bounds on their lengths.

Geodesic motion in Bogoslovsky-Finsler spacetimes

We study the free motion of a massive particle moving in the background of a Finslerian deformation of a plane gravitational wave in Einstein's general relativity. The deformation is a curved version of a one-parameter family of relativistic Finsler structures introduced by Bogoslovsky, which are invariant under a certain deformation of Cohen and Glashow's very special relativity group $\mathrm{ISIM}(2)$. The partially broken Carroll symmetry we derive using Baldwin-Jeffery-Rosen coordinates allows us to integrate the geodesics equations. The transverse coordinates of timelike Finsler geodesics are identical to those of the underlying plane gravitational wave for any value of the Bogoslovsky-Finsler parameter $b$. We then replace the underlying plane gravitational wave with a homogeneous $pp$-wave solution of the Einstein-Maxwell equations. We conclude by extending the theory to the Finsler-Friedmann-Lema\^{\i}tre model.

Finsler geodesics, periodic Reeb orbits, and open books

We survey some results on the existence (and non-existence) of periodic Reeb orbits on contact manifolds, both in the open and closed case. We place these statements in the context of Finsler geometry by including a proof of the folklore theorem that the Finsler geodesic flow can be interpreted as a Reeb flow. As a mild extension of previous results we present existence statements on periodic Reeb orbits on contact manifolds with suitable supporting open books.

Differentiable Distance Spaces

The distance function $${\varrho(p, q) ({\rm or} d(p, q))}$$ of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over $${\mathbb{R}^n}$$ , whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are not Finsler spaces (which are special distance spaces). They are situated between general metric spaces (distance spaces) and Finsler spaces. We will investigate such curves of differentiable distance spaces, which possess the same properties as geodesics do in Finsler spaces. So these curves can be considered as forerunners of Finsler geodesics. They are in greater plenitude than Finsler geodesics, but they become geodesics in a Finsler space. We show some properties of these curves, as well as some relations between differentiable distance spaces and Finsler spaces. We arrive to these curves and to our results by using distance spheres, and using no variational calculus. We often apply direct geometric considerations.

Finsler Geodesics of Lagrangian Systems Through Routh Reduction

We make use of a symmetry reduction technique called Routh reduction to show that the solutions of the Euler–Lagrange equations of a strongly convex autonomous Lagrangian which lie on a specific energy level can be thought of as geodesics of an associated Finsler function.

Time-optimal solutions of Parallel navigation and Finsler geodesics

A geometric approach to kinematics in control theory is illustrated. A non-linear control system is derived for the problem and the Pontryagin maximum principle is used to find the time-optimal trajectories of the Parallel navigation. It is proved that the time-optimal relative trajectories of the Parallel navigation are geodesics of a Finsler metric. It is notable that the approach has the advantages using feedback.

Finsler geodesics in the presence of a convex function and their applications

In this paper, we obtain a result about the existence of only a finite number of geodesics between two fixed non-conjugate points in a Finsler manifold endowed with a convex function. We apply it to Randers and Zermelo metrics. As a by-product, we also get a result about the finiteness of the number of lightlike and timelike geodesics connecting an event to a line in a standard stationary spacetime.

Seismic rays as Finsler geodesics

We prove that, in general, for anisotropic nonuniform continua, seismic rays are geodesics in Finsler geometry. In particular, for separable velocity functions, the geometry is Wagnerian. We provide concrete examples with theoretical discussions and introduce the seismic Finsler metric.