The geometry of jet bundles, with applications to the calculus of variations

Saunders, David John (1987). The geometry of jet bundles, with applications to the calculus of variations. PhD thesis The Open University.

DOI: https://doi.org/10.21954/ou.ro.000100d7

Abstract

This dissertation presents an exposition of the geometric theory of jet bundles, and describes some applications of this theory to the study of certain types of differential equation, principally those associated with the calculus of variations.

The detailed structure of this dissertation is as follows. Chapter 1 contains a review of the properties of bundles and their vector fields and differential forms, and also describe the theory of derivations along bundle maps. Chapter 2 contains a coherent account of the theory of jet bundles. It includes a definition of derivations of type h* and v* and the vertical bracket of vector-valued forms, and also describes infinite jets in a way which clarifies the manifold structure of J°° and the properties of smooth functions on this space. Chapter 3 contains a description of the prolongations of bundle maps and of vector fields, and also explains some properties of repeated jets. Chapter 4 contains a definition of jet fields and their associated connections, and a summary of their properties. It also relates the construction of Backlund transformations to jet fields along bundle maps. Chapter 5 contains a brief history of the almost tangent structure on a tangent manifold and a generalisation of this structure to vertical endomorphisms on an arbitrary jet manifold and a vertical vector-valued m-form on a first-order jet manifold. Chapter 6 contains a short summary of the modern approach to Lagrangian and Hamiltonian systems in mechanics. It also includes a construction for a C art an form in higher-order field theories and an explanation of the relationship of integral sections of jet fields to extremals of variational problems. Finally, Chapter 7 contains a discussion of completely integrable evolution equations and their Hamiltonian structure. There is also an explanation of the Kac-Moody algebra interpretation of the Inverse Scattering Transform in terms of higher-order tangent manifolds.