D. J. Saunders

The Cartan form and its generalizations in the calculus of variations

In this paper, we discuss possible extensions of the concept of the Cartan form of classical mechanics to higher-order mechanics on manifolds, higher-order field theory on jet bundles and to parametric variational problems on slit tangent bundles and on bundles of nondegenerate velocities. We present a generalization of the Cartan form, known as a Lepage form, and basic properties of the Lepage forms. Both earlier and recent examples of differential forms generalizing the Cartan form are reviewed.

Thirty years of the inverse problem in the calculus of variations

We consider several aspects of the inverse problem of the calculus of variations as they have developed since 1979, giving some of the principal results, listing significant primary sources and mentioning review articles for further references. The main topics involve questions of the existence and uniqueness of a Lagrangian giving rise to a given family of differential equations; we also describe some different types of inverse problem which arise in the context of quantum mechanics.

The multiplier approach to the projective Finsler metrizability problem

Abstract 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 Rapcsak conditions, some peculiarities of the two-dimensional case, and geodesic convexity for sprays.

Lie Groupoids and Lie Algebroids

The idea of a groupoid is a generalization of that of a group, where not every pair of elements can be combined. In this chapter we review the concepts of Lie groupoid and Lie algebroid.

Hilbert forms for a Finsler metrizable projective class of sprays

Abstract 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.

AFFINE DUALITY AND LAGRANGIAN AND HAMILTONIAN SYSTEMS

We use affine duals of jet bundles to describe how Legendre maps may be used to provide Hamiltonian representations of variational problems in a single independent variable. Such a problem may be given as a Lagrangian (of first-order or of higher-order), or alternatively as a locally variational form on a jet bundle of arbitrary order with no preferred Lagrangian.

Homotopy operators for the variational bicomplex, representations of the Euler-Lagrange complex, and the Helmholtz-Sonin conditions

We give formulæ for two distinct local homotopy operators for the horizontal differential in the variational bicomplex. We deduce two different representations of the classes of forms in the Euler-Lagrange complex, and hence two different versions of the Helmholtz-Sonin equations for the local variationality of a source form. We give explicit relationships between these two versions of the equations.

Cartan geometries and their symmetries: a Lie algebroid approach.

In this book we first review the ideas of Lie groupoid and Lie algebroid, and the associated concepts of connection. We next consider Lie groupoids of fibre morphisms of a fibre bundle, and the connections on such groupoids together with their symmetries. We also see how the infinitesimal approach, using Lie algebroids rather than Lie groupoids, and in particular using Lie algebroids of vector fields along the projection of the fibre bundle, may be of benefit. We then introduce Cartan geometries, together with a number of tools we shall use to study them. We take, as particular examples, the four classical types of geometry: affine, projective, Riemannian and conformal geometry. We also see how our approach can start to fit into a more general theory. Finally, we specialize to the geometries (affine and projective) associated with path spaces and geodesics, and consider their symmetries and other properties.

Holonomy of a class of bundles with fibre metrics

This paper is concerned with the holonomy of a class of spaces which includes Landsberg spaces of Finsler geometry. The methods used are those of Lie groupoids and algebroids as developed by Mackenzie. We prove a version of the Ambrose{Singer Theorem for such spaces. The paper ends with a discussion of how the results may be extended to Finsler spaces and homogeneous nonlinear connections in general.

Fefferman-type metrics and the projective geometry of sprays in two dimensions

A spray is a second-order differential equation field on the slit tangent bundle of a differentiable manifold, which is homogeneous of degree 1 in the fibre coordinates in an appropriate sense; two sprays which are projectively equivalent have the same base-integral curves up to reparametrization. We show how, when the base manifold is two-dimensional, to construct from any projective equivalence class of sprays a conformal class of metrics on a four-dimensional manifold, of signature (2, 2); the Weyl conformal curvature of these metrics is simply related to the projective curvature of the sprays, and the geodesics of the sprays determine null geodesics of the metrics. The metrics in question have previously been obtained by Nurowski and Sparling (Classical and Quantum Gravity 20 (2003) 4995?5016), by a different method involving the exploitation of a close analogy between the Cartan geometry of second-order ordinary differential equations and of three-dimensional Cauchy?Riemann structures. From this perspective the derived metrics are seen to be analoguous to those defined by Fefferman in the CR theory, and are therefore said to be of Fefferman type. Our version of the construction reveals that the Fefferman-type metrics are derivable from the scalar triple product, both directly in the flat case (which we discuss in some detail) and by a simple extension in general. There is accordingly in our formulation a very simple expression for a representative metric of the class in suitable coordinates.