Symmetric variational methods
Lead Research Organisation:
University of Kent
Department Name: Sch of Maths Statistics & Actuarial Sci
Abstract
Suppose you are given a dirty photocopy and you need to touch it up.The human eye is very good at guessing how missing pieces of curvesshould be joined up. Now suppose you have hundreds of such copies.Can a computer do it? The problem is that we can do these thingswithout knowing how we do it. A computer has to be told exactlywhat to do, in the computer's own language. This means we needto solve the problem of curve completion using mathematics.This project is a contribution to this and similar problemswhich can be described in the same mathematical terms.Basically, we try lots of different completions and ask,which looks best? Formulating what it means to ``look best is one problem, finding a way to describe the solutionis another. The key to success is to make the solutionmethod resemble mathematics we already know and love.
Organisations
People |
ORCID iD |
| Elizabeth Mansfield (Principal Investigator) |
Publications
Zhao Jun
(2011)
Discrete invariants of curves and surfaces
Zhao J
(2013)
Discrete Variational Calculus for B-Spline Curves
in Mathematics in Computer Science
Mansfield EL
Discrete Moving Frames on Lattices
Mansfield EL
(2010)
Practical Guide to the Invariant Calculus
Mansfield E
(2013)
Discrete Moving Frames and Discrete Integrable Systems
in Foundations of Computational Mathematics
Hydon P
(2011)
Extensions of Noether's Second Theorem: from continuous to discrete systems
in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Gonçalves T
(2011)
On Moving Frames and Noether's Conservation Laws
in Studies in Applied Mathematics
Gonçalves T
(2012)
Moving Frames and Conservation Laws for Euclidean Invariant Lagrangians
in Studies in Applied Mathematics
Goncalves TMN
(2010)
Noether's Conservation Laws: smooth and discrete cases
| Description | The idea that a physical system will be governed by a principle such as least action or least time or least distance is one of the major governing principles in the mathematical modelling of all kinds of physical systems. In mathematics, such models are called variational problems, for reasons that become clear when you look at the calculations involved. One of the most important properties of variational models is that symmetries give rise to conservation laws, such as for energy, linear momentum and angular momentum. The project concerned itself with the study of these conservation laws for both smooth and discrete variational problems. We discovered that for one dimensional smooth systems, the use of a moving frame gave enormous information about the structure of the laws as well as helping with the integration problems. For discrete systems, we looked at the B-spine approximations of variational problems invariant under affine transformations, that is, translation and rotation. |
| Exploitation Route | This work has been picked up by plasma physicists. |
| Sectors | Aerospace, Defence and Marine,Construction,Digital/Communication/Information Technologies (including Software) |