The well-known invariants of conics are computed for classes of Finsler and Lagrange spaces. For the Finsler case, some (α, β)-metrics namely Randers, Kropina and ”Riemann”-type metrics provides conics as indicatrices and a Randers-Funk metric on the unit disk is treated as example. The relations between algebraic and differential invariants of (α, β)-metrics are pointed out as a method to use ...

We deal with Lagrangian systems that are invariant under the action of a symmetry group. The mechanical connection is a principal connection that is associated to Lagrangians which have a kinetic energy function that is defined by a Riemannian metric. In this paper we extend this notion to arbitrary Lagrangians. We then derive the reduced Lagrange-Poincaré equations in a new fashion and we show...

Abstract. It is well known that Lagrangian dynamical systems naturally arise in describing wave front dynamics in the limit of short waves (which is called pseudoclassical limit or limit of geometrical optics). Wave fronts are the surfaces of constant phase, their points move along lines which are called rays. In non-homogeneous anisotropic media rays are not straight lines. Their shape is dete...

Finsler and Lagrange spaces can be equivalently represented as almost Kähler manifolds endowed with a metric compatible canonical distinguished connection structure generalizing the Levi Civita connection. The goal of this paper is to perform a natural Fedosov– type deformation quantization of such geometries. All constructions are canonically derived for regular Lagrangians and/or fundamental ...

Abstract. Let G be a Banach Lie group modeled on the Banach space, possibly infinite dimensional, E. In this paper first we introduce Euler-Lagrange equations on the Lie group G with potential and right invariant metric. Euler-Lagrange equations are natural extensions of the geodesic equations on manifolds and Lie groups. In the second part, we study the geometry of the mechanical system of a r...

Line triangulation, a classical geometric problem in computer vision, is to determine the 3D coordinates of a line based on its 2D image projections from more than two views of cameras with known projection matrices. Compared to point features, line segments are more robust to matching errors, occlusions, and image uncertainties. In addition to line triangulation, a better metric is needed to e...

We review the current status of Finsler–Lagrange geometry and generalizations. The goal is to aid non–experts on Finsler spaces, but physicists and geometers skilled in general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. We also would like to orient mathematicians working in generalized Finsler and Kähler g...

We consider a class of condensed matter theories in a Newtonian framework with a Lagrange formalism so that a variant of Noether’s theorem gives the classical conservation laws: ∂tρ+ ∂i(ρv ) = 0 ∂t(ρv ) + ∂i(ρv v + p) = 0. We show that for the metric gμν defined by ĝ = g √−g = ρ ĝ = g √−g = ρv ĝ = g √−g = ρvv + p these theories are equivalent to a metric theory of gravity with Lagrangian L = LG...

We consider a classical condensed matter theory in a Newtonian framework where conservation laws ∂tρ+ ∂i(ρv ) = 0 ∂t(ρv ) + ∂i(ρv v + p) = 0 are related with the Lagrange formalism in a natural way. For an “effective Lorentz metric” gμν it is equivalent to a metric theory of gravity close to general relativity with Lagrangian L = LGR − (8πG)(Υg − Ξ(g + g + g)) √−g We consider the differences be...

In this paper, we study under which conditions the Reissner–Nordström anti-de Sitter black hole can be a solution of the vacuum mimetic F(R) gravity with Lagrange multiplier and mimetic scalar potential. As the author demonstrates, the resulting picture in the mimetic F(R) gravity case is a trivial extension of the standard F(R) approach, and in effect, the metric perturbations in the mimetic F...