David Hartley

A constructive implementation of the Cartan-Kähler theory of exterior differential systems

An efficient algorithm for the construction of a regular chain of involutive integral elements for a general exterior differential system is presented. It is based upon the existence theorems of the Cartan-Kahler theory, and may be used to analyse partial differential equations by formulating them as exterior differential systems.

Energetic electron precipitation characteristics observed from Antarctica during a flux dropout event

Data from two autonomous VLF radio receiver systems installed in a remote region of the Antarctic in 2012 is used to take advantage of the juxtaposition of the L=4.6 contour, and the Hawaii-Halley, Antarctica, great circle path as it passes over thick Antarctic ice shelf. The ice sheet conductivity leads to high sensitivity to changing D-region conditions, and the quasi-constant L-shell highlights outer radiation belt processes. The ground-based instruments observed several energetic electron precipitation events over a moderately active 24-hour period, during which the outer radiation belt electron flux declined at most energies and subsequently recovered. Combining the ground-based data with low- and geosynchronous-orbiting satellite observations on 27 February 2012, different driving mechanisms were observed for three precipitation events with clear signatures in phase space density and electron anisotropy. Comparison between flux measurements made by Polar-orbiting Operational Environmental Satellites (POES) in low Earth orbit and by the Antarctic instrumentation provides evidence of different cases of weak and strong diffusion into the bounce-loss-cone, helping to understand the physical mechanisms controlling the precipitation of energetic electrons into the atmosphere. Strong diffusion events occurred as the 30 keV flux than was reported by POES, more consistent with strong diffusion conditions.

Tensor Distributions on Signature-changing Space-times

Irregularities in the metric tensor of a signature-changing space-time suggest that field equations on such space-times might be regarded as distributional. We review the formalism of tensor distributions on differentiable manifolds, and examine to what extent rigorous meaning can be given to field equations in the presence of signature-change, in particular those involving covariant derivatives. We find that, for both continuous and discontinuous signature-change, covariant differentiation can be defined on a class of tensor distributions wide enough to be physically interesting.

Involution analysis for nonlinear exterior differential systems

Analysing auxiliary systems for integrability conditions is an indispensable part of many indirect studies of partial differential equations, such as symmetry analysis. An invariant differential geometric approach to integrability analysis is described, using the concept of an involutive exterior differential system. The essential theory is first presented, paying particular attention to the nonlinear case, and then algorithms implementing the central techniques are discussed.

Darboux Integrability in More Than Two Dimensions

Abstract. Darbouxs method reduces the problem of integrating some partial differential equations in two independent variables to the easier problem of solving systems of ordinary differential equations. Those equations to which it applies share many of the properties of so-called integrable systems. In the 1950s, Parsons extended Darbouxs method to a restricted class of equations in three or more independent variables. This paper reviews Parsons approach within the geometrical framework of exterior differential systems, which should facilitate the search for more comprehensive generalisations.

On Waylen's Regular Axisymmetric Similarity Solutions

We review the similarity solutions proposed by Waylen for a regular time–dependent axisymmetric vacuum space–time, and show that the key equation introduced to solve the invariant surface conditions is related by a Bäcklund transform to a restriction on the similarity variables. We further show that the vacuum space–times produced via this path automatically possess a (possibly homothetic) Killing vector, which may be time–like.

Dynamical systems and anholonomic constraints

The Dirac analysis of constrained Hamiltonian mechanics is one of the conventional precursors to the quantisation of classical systems. We reformulate this analysis in the language of exterior differential systems, starting from the Lagrangian, moving through the generation of primary and secondary constraints and leading to the construction of symmetry generators for gauge symmetries. This reformulation extends the procedure to systems involving anholonomic elements.

On the motion of charged relativistic particles with spin

We derive an involutive system of equations describing the classical relativistic motion of a charged particle with spin in an external electromagnetic field. The method is based on geometirc variational principle for an integral manifold of an exterior differential system on an extended frame brundle and the dynamics is motivated by the theory of an idealised radiating gyroscope.