Papers by Elizabeth Mansfield
arXiv (Cornell University), Jul 31, 2012
In this work we prove a weak Noether type theorem for a class of variational problems which inclu... more In this work we prove a weak Noether type theorem for a class of variational problems which include broken extremals. We then use this result to prove discrete Noether type conservation laws for certain classes of finite element discretisation of a model elliptic problem. In addition we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether's 1st Theorem (E. Noether 1918). We summarise extensive numerical tests, illustrating the conservativity of the discrete Noether law using the p-Laplacian as an example.
In the seminal paper “Invariante Variationsprobleme” by Emmy Noether, she showed that for systems... more In the seminal paper “Invariante Variationsprobleme” by Emmy Noether, she showed that for systems derived from a variational principle, the associated conservation laws could be obtained from Lie group actions that left the variational problem unchanged. Recently, we proved that these conservation laws could be rewritten as the divergence of the product of a moving frame and a vector of invariants. The aim of this talk is to illustrate how the knowledge of the conservation laws structure helps reduce the extremising problem, in particular for variational problems that are invariant under the special Euclidean group SE(3).
This volume is a collection of articles based on the plenary talks presented at the 2005 meeting ... more This volume is a collection of articles based on the plenary talks presented at the 2005 meeting in Santander of the Society for the Foundations of Computational Mathematics. The talks were given by some of the foremost world authorities in computational mathematics. The topics covered reflect the breadth of research within the area as well as the richness and fertility of interactions between seemingly unrelated branches of pure and applied mathematics. As a result this volume will be of interest to researchers in the field of computational mathematics and also to non-experts who wish to gain some insight into the state of the art in this active and significant field.
arXiv (Cornell University), Aug 10, 2018
In this second part of the paper, we consider finite difference Lagrangians which are invariant u... more In this second part of the paper, we consider finite difference Lagrangians which are invariant under linear and projective actions of SL(2), and the linear equi-affine action which preserves area in the plane. We first find the generating invariants, and then use the results of the first part of the paper to write the Euler-Lagrange difference equations and Noether's difference conservation laws for any invariant Lagrangian, in terms of the invariants and a difference moving frame. We then give the details of the final integration step, assuming the Euler Lagrange equations have been solved for the invariants. This last step relies on understanding the Adjoint action of the Lie group on its Lie algebra. We also use methods to integrate Lie group invariant difference equations developed in Part I. Effectively, for all three actions, we show that solutions to the Euler-Lagrange equations, in terms of the original dependent variables, share a common structure for the whole set of Lagrangians invariant under each given group action, once the invariants are known as functions on the lattice.
arXiv (Cornell University), Jun 26, 2019
We investigate some infinite dimensional Lie algebras and their associated Poisson structures whi... more We investigate some infinite dimensional Lie algebras and their associated Poisson structures which arise from a Lie group action on a manifold. If G is a Lie group, g its Lie algebra and M is a manifold on which G acts, then the set of smooth maps from M to g has at least two Lie algebra structures, both satisfying the required property to be a Lie algebroid. We may then apply a construction by Marle to obtain a Poisson bracket on the set of smooth real or complex valued functions on M × g *. In this paper, we investigate these Poisson brackets. We show that the set of examples include the standard Darboux symplectic structure and the classical Lie Poisson brackets, but is a strictly larger class of Poisson brackets than these. Our study includes the associated Hamiltonian flows and their invariants, canonical maps induced by the Lie group action, and compatible Poisson structures. Our approach is mainly computational and we detail numerous examples. The Lie brackets from which our results derive, arose from the consideration of connections on bundles with zero curvature and constant torsion. We give an alternate derivation of the Lie bracket which will be suited to applications to Lie group actions for applications not involving a Riemannian metric. We also begin a study of the infinite dimensional Poisson brackets which may be obtained by considering a central extension of the Lie algebras.
Foundations of Computational Mathematics, 2016
In this paper, we develop the theory of the discrete moving frame in two different ways. In the f... more In this paper, we develop the theory of the discrete moving frame in two different ways. In the first half of the paper, we consider a discrete moving frame defined on a lattice variety and the equivalence classes of global syzygies that result from the first fundamental group of the variety. In the second half, we consider the continuum limit of discrete moving frames as a local lattice coalesces to a point. To achieve a well-defined limit of discrete frames, we construct multispace, a generalisation of the jet bundle that also generalises Olver's one-dimensional construction. Using interpolation to provide coordinates, we prove that it is a manifold containing the usual jet bundle as a submanifold. We show that continuity of a multispace moving frame ensures that the discrete moving frame converges to a continuous one as lattices coalesce. The smooth frame is, at the same time, the restriction of the multispace frame to the embedded jet bundle. We prove further that the discrete invariants and syzygies approximate their smooth counterparts. In effect, a frame on multispace allows smooth frames and their discretisations to be studied simultaneously. In our last chapter we discuss two important applications, one to the discrete variational calculus, Communicated by Peter Olver.
Proceedings of the 2001 international symposium on Symbolic and algebraic computation, 2001
Acta Applicandae Mathematicae, 1995
In this paper we study symmetry reductions and exact solutions of the shallow water wave (SWW) eq... more In this paper we study symmetry reductions and exact solutions of the shallow water wave (SWW) equation u xxxt + αu x u xt + βu t u xx − u xt − u xx = 0, (1)
Noether's First Theorem yields conservation laws for Lagrangians with a variational symmetry grou... more Noether's First Theorem yields conservation laws for Lagrangians with a variational symmetry group. The explicit formulae for the laws are well known and the symmetry group is known to act on the linear space generated by the conservation laws. In recent work the authors showed the mathematical structure behind both the Euler-Lagrange system and the set of conservation laws, in terms of the differential invariants of the group action and a moving frame. In this paper we demonstrate that the knowledge of this structure considerably eases finding the extremal curves for variational problems invariant under the special Euclidean groups SE(2) and SE(3).
Foundations of Computational Mathematics, 2013
Group based moving frames have a wide range of applications, from the classical equivalence probl... more Group based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. Here we describe what we call a discrete group based moving frame, which is essentially a sequence of moving frames with overlapping domains. We demonstrate a small set of generators of the algebra of invariants, which we call the discrete Maurer-Cartan invariants, for which there are recursion formulae. We show that this offers significant computational advantages over a single moving frame for our study of discrete integrable systems. We demonstrate that the discrete analogues of some curvature flows lead naturally to Hamiltonian pairs, which generate integrable differential-difference systems. In particular, we show that in the centro-affine plane and the projective space, the Hamiltonian pairs obtained can be transformed into the known Hamiltonian pairs for the Toda and modified Volterra lattices respectively under Miura transformations. We also show that a specified invariant map of polygons in the centro-affine plane can be transformed to the integrable discretization of the Toda Lattice. Moreover, we describe in detail the case of discrete flows in the homogeneous 2-sphere and we obtain realizations of equations of Volterra type as evolutions of polygons on the sphere.
In recent works [1, 2], the authors considered various Lagrangians, which are invariant under a L... more In recent works [1, 2], the authors considered various Lagrangians, which are invariant under a Lie group action, in the case where the independent variables are themselves invariant. Using a moving frame for the Lie group action, they showed how to obtain the invariantized Euler-Lagrange equations and the space of conservation laws in terms of vectors of invariants and the adjoint representation of a moving frame. In this paper, we show how these calculations extend to the general case where the independent variables may participate in the action. We take for our main expository example the standard linear action of SL(2) on the two independent variables. This choice is motivated by applications to variational fluid problems
In recent works, the authors considered various Lagrangians, which are invariant under a Lie grou... more In recent works, the authors considered various Lagrangians, which are invariant under a Lie group action, in the case where the independent variables are themselves invariant. Using a moving frame for the Lie group action, they showed how to obtain the invariantized Euler-Lagrange equations and the space of conservation laws in terms of vectors of invariants and the adjoint representation of a moving frame. In this paper, we show how these calculations extend to the general case where the independent variables may participate in the action. We take for our main expository example the standard linear action of SL(2) on the two independent variables. This choice is motivated by applications to variational fluid problems which conserve potential vorticity. We also give the results for Lagrangians invariant under the standard linear action of SL(3) on the three independent variables.
A simple local proof of Noether's Second Theorem is given. This proof immediately leads to a gene... more A simple local proof of Noether's Second Theorem is given. This proof immediately leads to a generalization of the theorem, yielding conservation laws and/or explicit relationships between the Euler-Lagrange equations of any variational problem whose symmetries depend upon a set of free or partly-constrained functions. Our approach extends further to deal with finite difference systems. The results are easy to apply; several well-known continuous and discrete systems are used as illustrations.
Noether’s seminal paper in 1918 has an enduring appeal: variational problems and their conservati... more Noether’s seminal paper in 1918 has an enduring appeal: variational problems and their conservation laws continue to be central in physical applications. Their Lie symmetries arise naturally from the physics, and it appears her theorems are valid far more broadly, but just as powerfully, in all the new kinds of variational problems that are arising. Noether’s First Theorem guarantees conservation laws for smooth variational problems which are invariant under a Lie group action, and typically these laws are of serious physical interest in applications, for example, conservation of linear momentum, angular momentum, energy, potential voticity and so on. The conservation laws are written as divergence expressions which are zero on solutions of the system. Since the formulae for the laws are eye-glazingly recursive and complicated (see Olver [1] for the formulae and historical notes), Noether’s laws are eminently suited to implementation in the symbolic computation domain. However, the ...
Transactions of Mathematics and Its Applications
In this second part of the paper, we consider finite difference Lagrangians that are invariant un... more In this second part of the paper, we consider finite difference Lagrangians that are invariant under linear and projective actions of $SL(2)$, and the linear equi-affine action that preserves area in the plane. We first find the generating invariants, and then use the results of the first part of the paper to write the Euler–Lagrange difference equations and Noether’s difference conservation laws for any invariant Lagrangian, in terms of the invariants and a difference moving frame. We then give the details of the final integration step, assuming the Euler Lagrange equations have been solved for the invariants. This last step relies on understanding the adjoint action of the Lie group on its Lie algebra. We also use methods to integrate Lie group invariant difference equations developed in Part I. Effectively, for all three actions, we show that solutions to the Euler–Lagrange equations, in terms of the original dependent variables, share a common structure for the whole set of Lagr...
Forum of Mathematics, Sigma
In recent works [Gonçalves and Mansfield, Stud. Appl. Math., 128 (2012), 1–29; Mansfield, A Pract... more In recent works [Gonçalves and Mansfield, Stud. Appl. Math., 128 (2012), 1–29; Mansfield, A Practical Guide to the Invariant Calculus (Cambridge University Press, Cambridge, 2010)], the authors considered various Lagrangians, which are invariant under a Lie group action, in the case where the independent variables are themselves invariant. Using a moving frame for the Lie group action, they showed how to obtain the invariantized Euler–Lagrange equations and the space of conservation laws in terms of vectors of invariants and the Adjoint representation of a moving frame. In this paper, we show how these calculations extend to the general case where the independent variables may participate in the action. We take for our main expository example the standard linear action of SL(2) on the two independent variables. This choice is motivated by applications to variational fluid problems which conserve potential vorticity. We also give the results for Lagrangians invariant under the standa...
Transactions of Mathematics and Its Applications
We consider the calculation of Euler–Lagrange systems of ordinary difference equations, including... more We consider the calculation of Euler–Lagrange systems of ordinary difference equations, including the difference Noether’s theorem, in the light of the recently-developed calculus of difference invariants and discrete moving frames. We introduce the difference moving frame, a natural discrete moving frame that is adapted to difference equations by prolongation conditions. For any Lagrangian that is invariant under a Lie group action on the space of dependent variables, we show that the Euler–Lagrange equations can be calculated directly in terms of the invariants of the group action. Furthermore, Noether’s conservation laws can be written in terms of a difference moving frame and the invariants. We show that this form of the laws can significantly ease the problem of solving the Euler–Lagrange equations, and we also show how to use a difference frame to integrate Lie group invariant difference equations. In this Part I, we illustrate the theory by applications to Lagrangians invaria...
CONTEMPORARY MATHEMATICS, 2001
The version in the Kent Academic Repository may differ from the final published version. Users ar... more The version in the Kent Academic Repository may differ from the final published version. Users are advised to check http://kar.kent.ac.uk for the status of the paper. Users should always cite the published version of record.
Arxiv preprint arXiv:1106.3964, 2011
Abstract: Noether's First Theorem yields conservation laws for Lagrangians with a variationa... more Abstract: Noether's First Theorem yields conservation laws for Lagrangians with a variational symmetry group. The explicit formulae for the laws are well known and the symmetry group is known to act on the linear space generated by the conservation laws. In recent work the ...
Journal of Computational Dynamics
The version in the Kent Academic Repository may differ from the final published version. Users ar... more The version in the Kent Academic Repository may differ from the final published version. Users are advised to check http://kar.kent.ac.uk for the status of the paper. Users should always cite the published version of record.
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Papers by Elizabeth Mansfield