# Symmetric variational methods

Lead Research Organisation:
University of Kent

Department Name: Sch of Maths Statistics & Actuarial Scie

### 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.

## People |
## ORCID iD |

Elizabeth Louise Mansfield (Principal Investigator) |

### Publications

Goncalves TMN

*Moving Frames and Conservation Laws -- the general case*
Goncalves TMN
(2010)

*Noether's Conservation Laws: smooth and discrete cases*
GonÃ§alves T
(2012)

*On Moving Frames and Noether's Conservation Laws Moving Frames and Noether's Conservation Laws*in Studies in Applied Mathematics
GonÃ§alves T
(2013)

*Moving Frames and Conservation Laws for Euclidean Invariant Lagrangians T. M. N. Gonçalves and E. L. Mansfield*in Studies in Applied 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
Mansfield E
(2013)

*Discrete Moving Frames and Discrete Integrable Systems*in Foundations of Computational Mathematics
Mansfield EL

*Discrete Moving Frames on Lattices*
Mansfield EL
(2010)

*Practical Guide to the Invariant Calculus*
Zhao J
(2013)

*Discrete Variational Calculus for B-spline Approximated Curves*in Mathematics in Computer Science
Zhao Jun
(2011)

*Discrete invariants of curves and surfaces*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) |