Hilbert forms for a Finsler metrizable projective class of sprays

The projective Finsler metrizability problem deals with the question whether a projective-equivalence class of sprays is the geodesic class of a (locally or globally defined) Finsler function. In this paper we use Hilbert-type forms to state a number of different ways of specifying necessary and sufficient conditions for this to be the case, and we show that they are equivalent. We also address several related issues of interest including path spaces, Jacobi fields, totally-geodesic submanifolds of a spray space, and the equivalence of path geometries and projective-equivalence classes of sprays.Comment: 23 page

On Landsberg spaces and the Landsberg-Berwald problem

This paper is concerned with the geometry of a class of Finsler spaces called Landsberg spaces. A Landsberg space may be characterized by the fact that its fundamental tensor is covariant constant along horizontal curves with respect to its Berwald connection. A Finsler space whose Berwald connection is affine is called a Berwald space. Berwald spaces are necessarily Landsbergian, but whether there are y-global Landsberg spaces which are not of Berwald type is not known. Resolving this question is the Landsberg-Berwald problem of the title. The paper deals with several topics in Landsberg geometry which are related mainly by the possibility that the results obtained may throw light on the Landsberg-Berwald problem. It is assumed throughout that the dimension of the base manifold is at least 3. It is shown that a Landsberg space over a compact base, which is R-quadratic, is necessarily Berwaldian. A model for the holonomy algebra of a Landsberg space is proposed. Finally, the technique of averaging the fundamental tensor over the indicatrix is discussed, and it is shown that for a Landsberg space, with the correct interpretations, the averaged Berwald connection is the Levi-Civita connection of the averaged metric

Some remarks on the Finslerian version of Hilbert's fourth problem

The Finslerian version of Hilbert's fourth problem is the problem of finding projective Finsler functions. Alvarez Paiva (J. Diff. Geom. 69 (2005) 353-378) has shown that projective absolutely homogeneous Finsler functions correspond to symplectic structures on the space of oriented lines in R. with certain properties. I give new and direct proofs of his main results, and show how they are related to the more classical formulations of the problem due to Hamel and Rapcsak

Finsler functions for two-dimensional sprays

I derive the general formula for a local Finsler function for any spray over a two-dimensional manifold specified by its geodesic curvature function relative to a given background Riemannian metric

The inverse problem for Lagrangian systems with certain non-conservative forces

We discuss two generalizations of the inverse problem of the calculus of variations, one in which a given mechanical system can be brought into the form of Lagrangian equations with non-conservative forces of a generalized Rayleigh dissipation type, the other leading to Lagrangian equations with so-called gyroscopic forces. Our approach focusses primarily on obtaining coordinate-free conditions for the existence of a suitable non-singular multiplier matrix, which will lead to an equivalent representation of a given system of second-order equations as one of these Lagrangian systems with non-conservative forces.Comment: 28 page

Invariant Lagrangians, mechanical connections and the Lagrange-Poincare equations

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-Poincare equations in a new fashion and we show how solutions of the Euler-Lagrange equations can be reconstructed with the help of the mechanical connection. Illustrative examples confirm the theory.Comment: 22 pages, to appear in J. Phys. A: Math. Theor., D2HFest special issu

The Cartan form for constrained Lagrangian systems and the nonholonomic Noether theorem

This paper deals with conservation laws for mechanical systems with nonholonomic constraints. It uses a Lagrangian formulation of nonholonomic systems and a Cartan form approach. We present what we believe to be the most general relations between symmetries and first integrals. We discuss the so-called nonholonomic Noether theorem in terms of our formalism, and we give applications to Riemannian submanifolds, to Lagrangians of mechanical type, and to the determination of quadratic first integrals.Comment: 25 page

Homogeneous variational problems: a minicourse

A Finsler geometry may be understood as a homogeneous variational problem, where the Finsler function is the Lagrangian. The extremals in Finsler geometry are curves, but in more general variational problems we might consider extremal submanifolds of dimension $m$. In this minicourse we discuss these problems from a geometric point of view.Comment: This paper is a written-up version of the major part of a minicourse given at the sixth Bilateral Workshop on Differential Geometry and its Applications, held in Ostrava in May 201

Linearization of nonlinear connections on vector and affine bundles, and some applications

A linear connection is associated to a nonlinear connection on a vector bundle by a linearization procedure. Our definition is intrinsic in terms of vector fields on the bundle. For a connection on an affine bundle our procedure can be applied after homogenization and restriction. Several applications in Classical Mechanics are provided