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

The multiplier approach to the projective Finsler metrizability problem

This paper is concerned with the problem of determining whether a projective-equivalence class of sprays is the geodesic class of a Finsler function. We address both the local and the global aspects of this problem. We present our results entirely in terms of a multiplier, that is, a type (0,2) tensor field along the tangent bundle projection. In the course of the analysis we consider several related issues of interest including the positivity and strong convexity of positively-homogeneous functions, the relation to the so-called Rapcs\'ak conditions, some peculiarities of the two-dimensional case, and geodesic convexity for sprays.Comment: 25 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 Berwald-type linearisation of generalised connections

We study the existence of a natural `linearisation' process for generalised connections on an affine bundle. It is shown that this leads to an affine generalised connection over a prolonged bundle, which is the analogue of what is called a connection of Berwald type in the standard theory of connections. Various new insights are being obtained in the fine structure of affine bundles over an anchored vector bundle and affineness of generalised connections on such bundles.Comment: 25 page

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

Isometries, submetries and distance coordinates on Finsler manifolds

This paper considers fundamental issues related to Finslerian iso- metries, submetries, distance and geodesics. It is shown that at each point of a Finsler manifold there is a distance coordinate system. Us- ing distance coordinates, a simple proof is given for the Finslerian version of the Myers-Steenrod theorem and for the differentiability of Finslerian submetries

A setting for higher order differential equations fields and higher order Lagrange and Finsler spaces

We use the Fr\"olicher-Nijenhuis formalism to reformulate the inverse problem of the calculus of variations for a system of differential equations of order 2k in terms of a semi-basic 1-form of order k. Within this general context, we use the homogeneity proposed by Crampin and Saunders in [14] to formulate and discuss the projective metrizability problem for higher order differential equation fields. We provide necessary and sufficient conditions for higher order projectivpre-e metrizability in terms of homogeneous semi-basic 1-forms. Such a semi-basic 1-form is the Poincar\'e-Cartan 1-form of a higher order Finsler function, while the potential of such semi-basic 1-form is a higher order Finsler function.Comment: final, pre-published versio

Prospects for K+→π+ννˉK^+ \to \pi^+ \nu \bar{ \nu } at CERN in NA62

The NA62 experiment will begin taking data in 2015. Its primary purpose is a 10% measurement of the branching ratio of the ultrarare kaon decay $K^+ \to \pi^+ \nu \bar{ \nu }$, using the decay in flight of kaons in an unseparated beam with momentum 75 GeV/c.The detector and analysis technique are described here.Comment: 8 pages for proceedings of 50 Years of CP