arXiv subjects

Geoff Prince

Publications and source records attributed to Geoff Prince.

6 recordsLinked to original sources

arXiv2023

Quadratic forms and the expansion and rotations of linear endomorphisms

New expansionary and rotational quadratic forms are constructed for $E^n$-endomorphisms. Relations amongst the various eigenvalues, eigendirections and matrix invariants are established, including propositions on complexity and geometric multiplicity. The underlying construction involves a novel, almost-orthogonal expansion based on two-plane rotations. The development is strongly geometric in flavour and has application to the theory of connections, of which the Frenet case on $E^3$ is given as a model.

math.RA↗

arXiv2020

The inverse problem in the calculus of variations: new developments

We deal with the problem of determining the existence and uniqueness of Lagrangians for systems of $n$ second order ordinary differential equations. A number of recent theorems are presented, using exterior differential systems theory (EDS). In particular, we indicate how to generalise Jesse Douglas's famous solution for $n=2$. We then examine a new class of solutions in arbitrary dimension $n$ and give some non-trivial examples in dimension 3

math.DG↗

arXiv2015

An intrinsic and exterior form of the Bianchi identities

We give an elegant formulation of the structure equations (of Cartan) and the Bianchi identities in terms of exterior calculus without reference to a particular basis and without the exterior covariant derivative. This approach allows both structure equations and the Bianchi identities to be expressed in terms of forms of arbitrary degree. We demonstrate the relationship with both the conventional vector version of the Bianchi identities and to the exterior covariant derivative approach. Contact manifolds, codimension one foliations and the Cartan form of classical mechanics are studied as examples of its flexibility and utility.

math.DG↗

arXiv2014

New progress in the inverse problem in the calculus of variations

We present a new class of solutions for the inverse problem in the calculus of variations in arbitrary dimension $n$. This is the problem of determining the existence and uniqueness of Lagrangians for systems of $n$ second order ordinary differential equations. We also provide a number of new theorems concerning the inverse problem using exterior differential systems theory (EDS). Concentrating on the differential step of the EDS process, our new results provide a significant advance in the understanding of the inverse problem in arbitrary dimension. In particular, we indicate how to generalise Jesse Douglas's famous solution for $n=2$. We give some non-trivial examples in dimensions 2,3 and 4. We finish with a new classification scheme for the inverse problem in arbitrary dimension.

math.DG↗

arXiv2014

Evolution equations: Frobenius integrability, conservation laws and travelling waves

We give new results concerning the Frobenius integrability and solution of evolution equations admitting travelling wave solutions. In particular, we give a powerful result which explains the extraordinary integrability of some of these equations. We also discuss "local" conservations laws for evolution equations in general and demonstrate all the results for the Korteweg de Vries equation.

math.AP↗

arXiv2013

Integration of PDEs by differential geometric means

We use Vessiot theory and exterior calculus to solve partial differential equations(PDEs) of the type uyy = F(x, y,u,ux,uy,uxx,uxy) and associated evolution equations. These equations are represented by the Vessiot distribution of vector fields. We develop and apply an algorithm to find the largest integrable sub-distributions and hence solutions of the PDEs. We then apply the integrating factor technique [19] to integrate this integrable Vessiot sub-distribution. The method is successfully applied to a large class of linear and non-linear PDEs.

math.DG↗