Classical Gauge Field Theory

After a sketch of Lagrangian field theory on jet bundles, the notion of a gauge field is introduced as a section of an affine bundle which is naturally constructed without any involvement with structure groups. An original approach to gauge field theory in terms of covariant differentials (alternative to the jet bundle approach) is then developed, and the adaptations needed in order to deal with general theories are laid out. A careful exposition of the replacement principle allows comparisons with approaches commonly found in the literature.

Autonomous Geometrical Mechanics

This chapter examines the structure of the phase space of an integrable system as being constructed from invariant tori using the Arnold–Liouville integrability theorem, and periodic flow and ergodic flow are investigated using action-angle theory. Time-dependent mechanics is formulated by extending the symplectic structure to a contact structure in an extended phase space before it is shown that mechanics has a natural setting on a jet bundle. The chapter then describes phase space of integrable systems and how tori behave when time-dependent dynamics occurs. Adiabatic invariance is discussed, as well as slow and fast Hamiltonian systems, the Hannay angle and counter adiabatic terms. In addition, the chapter discusses foliation, resonant tori, non-resonant tori, contact structures, Pfaffian forms, jet manifolds and Stokes’s theorem.

Covariant-differential formulation of Lagrangian field theory

Building on the Utiyama principle we formulate an approach to Lagrangian field theory in which exterior covariant differentials of vector-valued forms replace partial derivatives, in the sense that they take up the role played by the latter in the usual jet bundle formulation. Actually a natural Lagrangian can be written as a density on a suitable “covariant prolongation bundle”; the related momenta turn out to be natural vector-valued forms, and the field equations can be expressed in terms of covariant exterior differentials of the momenta. Currents and energy-tensors naturally also fit into this formalism. The examples of bosonic fields and spin one-half fields, interacting with non-Abelian gauge fields, are worked out. The “metric-affine” description of the gravitational field is naturally included, too.

Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations

We construct metric connection associated with a first-order differential equation by means of the generator set of a Pfaffian system on a submanifold of an appropriate first-order jet bundle. We firstly show that the inviscid and viscous Burgers’ equations describe surfaces attached to an ODE of the form dx/dt=u(t,x) with certain Gaussian curvatures. In the case of PDEs, we show that the scalar curvature of a three-dimensional manifold encoding a system of first-order PDEs is determined in terms of the integrability condition and the Gaussian curvatures of the surfaces corresponding to the integral curves of the vector fields which are annihilated by the contact form. We see that an integral manifold of any PDE defines intrinsically flat and totally geodesic submanifold.