# Professor Elizabeth Mansfield

Professor of Mathematics

Co-organiser of the Isaac Newton Institute Programme in
Geometry, Compatibility and Structure Preservation in computational
differential equations.https://www.newton.ac.uk/event/gcsCo-organiser of the joint IMA-LMS meeting to celebrate the
100 anniversary of Emmy Noether’s paper on conservation laws, September, 2018.https://ima.org.uk/8936/ima-lms-joint-meeting-noether-celebration/Professor of Mathematics at the University of Kent since
2005; the first ever female professor in mathematics at Kent.PhD University of Sydney, 1992.

## Research interests

Lie groups, their actions and applications. Moving frames.Noether’s Theorem, both smooth and discrete. Poisson
structures, Lie algebroids.One major line of research has been inducing Lie group actions on functional approximation spaces in order to create discrete analogues of Noether’s laws, in order to incorporate, in a bone fide way, the physics into the numerical methods.

## Supervision

Recent students have looked at the application of moving frames in a variety of settings: discrete Noether’s Theorem, discrete integrability, and numerical methods involving Lie groups. New projects involve new classes of Poisson structures arising from a Lie group action.
My main collaborators are Gloria Mari Beffa (Madison Wisconsin) and Peter Hydon (Kent).
Think Kent: Mathematics as a conceptual art form 

## Professional

Vice President of the Institute of Mathematics and its Applications, 2014-2018.Member, LMS Committee for Society Lectures and Meetings, to end 2019.Editorial Board member of the Journal of Foundations of Computational Mathematics.
Member, Scientific Advisory Committee for the Australian Mathematical Sciences Institute.Former member of LMS Council, LMS Programme committee and Editor of the LMS Journal of Computation and Mathematics.Former VP of the ACM SIGSAM. 

## Publications

### Article

• Mansfield, E. and Goncalves, T. (2016). Moving Frames and Noetherâs Conservation Laws â the General Case. Forum of Mathematics, Sigma [Online] 4. Available at: http://dx.doi.org/10.1017/fms.2016.24.
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 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.
• Beffa, G. and Mansfield, E. (2016). Discrete moving frames on lattice varieties and lattice based multispace. Foundations of Computational Mathematics [Online]. Available at: http://dx.doi.org/10.1007/s10208-016-9337-5.
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 generalization of the jet bundle that also generalizes Olverâ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, and the second to discrete integrable systems. Finally, in an appendix, we discuss a more general result concerning equicontinuous families of discretisations of moving frames, which are consistent with a smooth frame.
• Mansfield, E. and Pryer, T. (2014). Noether type discrete conserved quantities arising from a finite element approximation of a variational problem. Foundations of Computational Mathematics [Online]:1-34. Available at: http://dx.doi.org/10.1007/s10208-015-9298-0.
In this work, we prove a weak Noether-type Theorem for a class of variational problems that admit broken extremals. We use this result to prove discrete Noether-type conservation laws for a conforming 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 Noetherâs first theorem (1918). We summarise extensive numerical tests, illustrating the conservation of the discrete Noether law using the p-Laplacian as an example and derive a geometric-based adaptive algorithm where an appropriate Noether quantity is the goal functional.
• Goncalves, T. and Mansfield, E. (2013). Moving Frames and Conservation Laws for Euclidean Invariant Lagrangians. Studies in Applied Mathematics [Online] 130:134-166. Available at: http://dx.doi.org/10.1111/j.1467-9590.2012.00566.x.
In recent work, the authors show the mathematical structure behind both the Eulerâ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, the authors demonstrate that the knowledge of this structure allows to find the first integrals of the EulerâLagrange equations, and subsequently, to solve by quadratures, variational problems that are invariant under the special Euclidean groups SE(2) and SE(3).
• Mansfield, E., MarÃ­ Beffa, G. and Wang, J. (2013). Discrete Moving Frames and Discrete Integrable Systems. Foundations of Computational Mathematics [Online] 13:545-582. Available at: http://dx.doi.org/10.1007/s10208-013-9153-0.
• Hydon, P. and Mansfield, E. (2011). Extensions of Noether's Second Theorem: from continuous to discrete systems. Proceedings of the Royal Society A- Mathematical Physical and Engineering Sciences [Online] 467:3206-3221. Available at: http://dx.doi.org/10.1098/rspa.2011.0158.
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 EulerâLagrange equations of any variational problem whose symmetries depend on 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.
• Mansfield, E. and Goncalves, T. (2011). On Moving Frames and Noetherâs Conservation Laws. Studies in Applied Mathematics [Online] 128:1-29. Available at: http://dx.doi.org/10.1111/j.1467-9590.2011.00522.x.
Noetherâs Theorem yields conservation laws for a Lagrangian 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. The aim of this paper is to explain the mathematical structure of 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. For the examples, we demonstrate, knowledge of this structure allows the Euler-Lagrange equations to be integrated with relative ease. Our methods take advantage of recent advances in the theory of moving frames by Fels and Olver, and in the symbolic invariant calculus by Hubert. The results here generalize those appearing in Kogan and Olver [1] and in Mansfield [2]. In particular, we show results for high-dimensional problems and classify those for the three inequivalent SL(2) actions in the plane.
• Mansfield, E. and Hydon, P. (2008). Difference forms. Foundations of Computational Mathematics [Online] 8:427-467. Available at: http://dx.doi.org/10.1007/s10208-007-9015-8.
Currently, there is much interest in the development of geometric integrators, which retain analogues of geometric properties of an approximated system. This paper provides a means of ensuring that finite difference schemes accurately mirror global properties of approximated systems. To this end, we introduce a cohomology theory for lattice varieties, on which finite difference schemes and other difference equations are defined. We do not assume that there is any continuous space, or that a scheme or difference equation has a continuum limit. This distinguishes our approach from theories of "discrete differential forms" built on simplicial approximations and Whitney forms, and from cohomology theories built on cubical complexes. Indeed, whereas cochains on cubical complexes can be mapped injectively to our difference forms, a bijection may not exist. Thus our approach generalizes what can be achieved with cubical cohomology. The fundamental property that we use to prove our results is the natural ordering on the integers. We show that our cohomology can be calculated from a good cover, just as de Rham cohomology can. We postulate that the dimension of solution space of a globally defined linear recurrence relation equals the analogue of the Euler characteristic for the lattice variety. Most of our exposition deals with forward differences, but little modification is needed to treat other finite difference schemes, including Gauss-Legendre and Preissmann schemes.
• Bila, N., Mansfield, E. and Clarkson, P. (2006). Symmetry group analysis of the shallow water and semi-geostrophic equations. Quarterly Journal of Mechanics and Applied Mathematics [Online] 59:95-123. Available at: http://dx.doi.org/10.1093/qjmam/hbi033.
The two-dimensional shallow water equations and their semi-geostrophic approximation that arise in meteorology and oceanography are analysed from the point of view of symmetry groups theory. A complete classification of their associated classical symmetries, potential symmetries, variational symmetries and conservation laws is found. The semi-geostrophic equations are found to lack conservation of angular momentum. We also show how the particle relabelling symmetry can be used to rewrite the semi-geostrophic equations in such a way that a well-defined formal series solution, smooth only in time, may be carried out. We show that such solutions are in the form of an infinite linear cascade'.
• Mansfield, E. and van der Kamp, P. (2006). Evolution of curvature invariants and lifting integrability. Journal of Geometry and Physics [Online] 56:1294-1325. Available at: http://dx.doi.org/10.1016/j.geomphys.2005.07.002.
Given a geometry defined by the action of a Lie-group on a flat manifold, the Fels-Olver moving frame method yields a complete set of invariants, invariant differential operators, and the differential relations, or syzygies, they satisfy. We give a method that determines, from minimal data, the differential equations the frame must satisfy, in terms of the curvature and evolution invariants that are associated to curves in the given geometry. The syzygy between the curvature and evolution invariants is obtained as a zero curvature relation in the relevant Lie-algebra. An invariant motion of the curve is uniquely associated with a constraint specifying the evolution invariants as a function of the curvature invariants. The zero curvature relation and this constraint together determine the evolution of curvature invariants.
Invariantizing the formal symmetry condition for curve evolutions yield a syzygy between different evolution invariants. We prove that the condition for two curvature evolutions to commute appears as a differential consequence of this syzygy. This implies that integrability of the curvature evolution lifts to integrability of the curve evolution, whenever the kernel of a particular differential operator is empty. We exhibit various examples to illustrate the theorem, the calculations involved in verifying the result are substantial. (c) 2005 Elsevier B.V. All rights reserved.
• Mansfield, E. (2006). Noether's Theorem for Smooth, Difference and Finite Element Schemes. Foundations of Computational Mathematics, Santander 2005 [Online] London:230-254. Available at: http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521681612.
• Mansfield, E. and Quispel, R. (2005). Towards a variational complex for the finite element method. Group Theory and Numerical Analysis 39:207-232.
Variational and symplectic integrators are now popular for mechanical systems, both because of their good long term stability and qualitative fit. Sucy integrators mimic or inherit the Lagrangian, respectively Hamiltonian, structure of the continuous model. A variational complex is a theoretical tool for the rigorous study of Lagrangian systems and their conservation laws. This article examines whether a formulation of a variational calculus for finite element methods, for an arbitrary finite element approximation scheme, is possible. The motivation is that this would allow a variational scheme to be written down for a given approximation model. Moreover, the stability and the conservation laws of such integrators could be studied without any need for individual, ad hoc arguments. A number of examples are considered, mainly one-dimensional, and conditions for a suitable complex derived.
• Hydon, P. and Mansfield, E. (2004). A variational complex for difference equations. Foundations of Computational Mathematics [Online] 4:187-217. Available at: https://doi.org/DOI not available.
An analogue of the Poincare lemma for exact forms on a lattice is stated and proved. Using this result as a starting-point, a variational complex for difference equations is constructed and is proved to be locally exact. The proof uses homotopy maps, which enable one to calculate Lagrangians for discrete Euter-Lagrange systems. Furthermore, such maps lead to a systematic technique for deriving conservation laws of a given system of difference equations (whether or not it is an Euler-Lagrange system).
• Clarkson, P. and Mansfield, E. (2003). The second Painleve equation, its hierarchy and associated special polynomials. Nonlinearity [Online] 16:R1-R26. Available at: http://dx.doi.org/10.1088/0951-7715/16/3/201.
In this paper we are concerned with hierarchies of rational solutions and associated polynomials for the second Painleve equation (P-II) and the equations in the P-II hierarchy which is derived from the modified Korteweg-de Vries hierarchy. These rational solutions of P-II are expressible as the logarithmic derivative of special polynomials, the Yablonskii-Vorob'ev polynomials. The structure of the roots of these Yablonskii-Vorob'ev polynomials is studied and it is shown that these have a highly regular triangular structure. Further, the properties of the Yablonskii-Vorob'ev polynomials are compared and contrasted with those of classical orthogonal polynomials. We derive the special polynomials for the second and third equations of the P-II hierarchy and give a representation of the associated rational solutions in the form of determinants through Schur functions. Additionally the analogous special polynomials associated with rational solutions and representation in the form of determinants are conjectured for higher equations in the P-II hierarchy. The roots of these special polynomials associated with rational solutions for the equations of the P-II hierarchy also have a highly regular structure.
• Mansfield, E. and Szanto, A. (2003). Elimination Theory for differential difference polynomials. Proceedings of the 2003 International Symposium for Symbolic Algebra and Computation [Online]:191-198. Available at: http://doi.acm.org/10.1145/860854.860897.
In this paper we give an elimination algorithm for differential difference polynomial systems. We use the framework of a generalization of Ore algebras, where the independent variables are non-commutative. We prove that for certain term orderings, Buchberger's algorithm applied to differential difference systems terminates and produces a GrÃ¶bner basis. Therefore, differential-difference algebras provide a new instance of non-commutative graded rings which are effective GrÃ¶bner structures.
• Mansfield, E. and Hydon, P. (2001). On a variational complex for difference equations. Contemporary Mathematics 285:195-205.
• Mansfield, E. (2001). Algorithms for symmetric differential systems. Foundations of Computational Mathematics 1:335-383.
Over-determined systems of partial differential equations may be studied using differential-elimination algorithms, as a great deal of information about the solution set of the system may be obtained from the output. Unfortunately, many systems are effectively intractable by these methods due to the expression swell incurred in the intermediate stages of the calculations. This can happen when, for example, the input system depends on many variables and is invariant under a large rotation group, so that there is no natural choice of term ordering in the elimination and reduction processes. This paper describes how systems written in terms of the differential invariants of a Lie group action may be processed in a manner analogous to differential-elimination algorithms. The algorithm described terminates and yields, in a sense which we make precise, a complete set of representative invariant integrability conditions which may be calculated in a "critical pair" completion procedure. Further, we discuss some of the profound differences between algebras of differential invariants and standard differential algebras. We use the new, regularized moving frame method of Fels and Olver [11], [12] to write a differential system in terns of the invariants of a symmetry group. The methods described have been implemented as a package in MAPLE. The main example discussed is the analysis of the (2 + 1)-d'Alembert-Hamilton system
u(xx) + u(yy) - u(zz) = f(u),

u(x)(2) + u(y)(2) - u(z)(2) = 1.

We demonstrate the classification of solutions due to Collins [7] for f not equal 0 using the new methods.
• Clarkson, P., Mansfield, E. and Webster, H. (2000). On the relation between the continuous and discrete Painleve equations. Theoretical and Mathematical Physics [Online] 122:1-16. Available at: http://dx.doi.org/10.1007/BF02551165.
A method for deriving difference equations (the discrete Painleve' equations in particular) from the Backlund transformations of the continuous Painleve' equations is discussed. This technique can be used to derive several of the known discrete Painleve' equations (in particular, the first and second discrete Painleve equations and some of their alternative versions). The Painleve' equations possess hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions for special values of the parameters. Hence, the aforementioned relations can be used to generate hierarchies of exact solutions for the associated discrete Painleve' equations. Exact solutions of the Painleve equations simultaneously satisfy both a differential equation and a difference equation, analogously to the special functions.
• Mansfield, E. (1999). The nonclassical group analysis of the heat equation. Journal of Mathematical Analysis and Applications [Online] 231:526-542. Available at: http://dx.doi.org/10.1006/jmaa.1998.6250.
The nonclassical method of reduction was devised originally by Bluman and Cole in 1969, to find new exact solutions of the heat equation. Much success has been had by many authors using the method to find new exact solutions of nonlinear equations of mathematical and physical significance. However, the defining equations for the nonclassical reductions of the heat equation itself have remained unsolved, although particular solutions have been given. Recently, Arrigo, Goard, and Broadbridge showed that there are no nonclassical reduction solutions of constant coefficient linear equations that are not already classical Lie symmetry reduction solutions. Their arguments leave open the problem of what is the general nonclassical group action, and its effect on the relevant solution of the heat equation. In this article, both these problems are solved. In the final section we use the methods developed to solve the remaining outstanding case of nonclassical reductions of Burgers' equation. (C) 1999 Academic Press.
• Mansfield, E. and Webster, H. (1998). On one-parameter families of Painleve III. Studies in Applied Mathematics [Online] 101:321-341. Available at: http://dx.doi.org/10.1111/1467-9590.00096.
Albrecht, Mans field, arid Milne developed a direct method with which one can calculate special integrals of polynomial type (also known as one parameter family conditions, Darboux polynomials, eigenpolynomials, or algebraic invariant curves) for nonlinear ordinary differential equations of polynomial type. We apply this method to the third Painleve equation and prove that for the generic case, the set of known one-parameter family conditions is complete.
• Mansfield, E., Reid, G. and Clarkson, P. (1998). Nonclassical reductions of a 3+1-cubic nonlinear Schrodinger system. Computer Physics Communications [Online] 115:460-488. Available at: http://dx.doi.org/10.1016/S0010-4655(98)00136-2.
An analytical study, strongly aided by computer algebra packages diffgrob2 by Mansfield and rif by Reid, is made of the 3 + 1-coupled nonlinear Schrodinger (CNLS) system i Psi(t) + del(2)Psi + (\Psi\(2) + \Phi\(2)) Psi = 0, i Phi(t) + del(2)Phi + (\Psi\(2) + \Phi\(2)) Phi = 0. This system describes transverse effects in nonlinear optical systems. It also arises in the study of the transmission of coupled wave packets and "optical solitons", in nonlinear optical fibres. First we apply Lie's method for calculating the classical Lie algebra of vector fields generating symmetries that leave invariant the set of solutions of the CNLS system. The large linear classical determining system of PDE for the Lie algebra is automatically generated and reduced to a standard form by the rif algorithm, then solved, yielding a 15-dimensional classical Lie invariance algebra. A generalization of Lie's classical method, called the nonclassical method of Bluman and Cole, is applied to the CNLS system. This method involves identifying nonclassical vector fields which leave invariant the joint solution set of the CNLS system and a certain additional system, called the invariant surface condition. In the generic case the system of determining equations has 856 PDE, is nonlinear and considerably more complicated than the linear classical system of determining equations whose solutions it possesses as a subset. Very few calculations of this magnitude have been attempted due to the necessity to treat cases, expression explosion and until recent times the dearth of mathematically rigorous algorithms for nonlinear systems. The application of packages diffgrob2 and rif leads to the explicit solution of the nonclassical determining system in eleven cases. Action of the classical group on the nonclassical vector fields considerably simplifies one of these cases. We identify the reduced form of the CNLS system in each case. Many of the cases yield new results which apply equally to a generalized coupled nonlinear Schrodinger system in which \Psi\(2) + \Phi\(2) may be replaced by an arbitrary function of \Psi\(2) + \Phi\(2). Coupling matrices in sl(2, C) feature prominently in this family of reductions. (C) 1998 Elsevier Science B.V.
• Clarkson, P., Mansfield, E. and Priestley, T. (1997). Symmetries of a class of nonlinear third-order partial differential equations. Mathematical and Computer Modelling 25:195-212.
In this paper, we study symmetry reductions of a class of nonlinear third-order partial differential equations (1) U-t - epsilon u(xxt) + 2 kappa u(x) = uu(xxx) + alpha uu(x) + beta u(x)u(xx), where epsilon, kappa, alpha, and beta are arbitrary constants. Three special cases of equation (1) have appeared in the literature, up to some rescalings. In each case, the equation has admitted unusual travelling wave solutions: the Fornberg-Whitham equation, for the parameters epsilon = 1, alpha = -1, beta = 3, and kappa = 1/2, admits a wave of greatest height, as a peaked limiting form of the travelling wave solution; the Rosenau-Hyman equation, for the parameters epsilon = 0, alpha = 1, beta = 3, and kappa = 0, admits a ''compacton'' solitary wave solution; and the Fuchssteiner-Fokas-Camassa-Holm equation,for the parameters epsilon = 1, alpha = -3, and beta = 2, has a ''peakon'' solitary wave solution. A catalogue of symmetry reductions for equation (1) is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole.
• Mansfield, E. and Clarkson, P. (1997). Applications of the differential algebra package diffgrob2 to classical symmetries of differential equations. Journal of Symbolic Computation [Online] 23:517-533. Available at: http://dx.doi.org/10.1006/jsco.1996.0105.
We show how the MAPLE package diffgrob2 can be used to analyse overdetermined systems of PDE. The particular application discussed here is to find classical symmetries of differential equations of mathematical and physical interest. Symmetries of differential equations underly most of the methods of exact integration known; the use and calculation of such symmetries is often introduced at advanced undergraduate level. Examples include cases where heuristics give incomplete information or fail in the integration of the determining equations for the group infinitesimals. The ideas presented here are thus an alternative method of attacking this important problem. The discussion is at a ''hands on'' level suitable as resource material for undergraduate instruction.
• Mansfield, E. and Clarkson, P. (1997). Symmetries and exact solutions for a 2+1-dimensional shallow water wave equation. Mathematics and Computers in Simulation [Online] 43:39-55. Available at: http://dx.doi.org/10.1016/S0378-4754(96)00054-7.
Classical and nonclassical reductions of a 2 + 1-dimensional shallow water wave equation are classified. Using these reductions, we derive some exact solutions, including solutions expressed as the nonlinear superposition of solutions of a generalised variable-coefficient Korteweg-de Vries equation. Many of the reductions obtained involve arbitrary functions and so the associated families of solutions have a rich variety of qualitative behaviours. This suggests that solving the initial value problem for the 2 + 1-dimensional shallow water equation under discussion could pose some fundamental difficulties. The nonlinear overdetermined systems of partial differential equations whose solutions yield the reductions were analysed and solved using the MAPLE package diffgrob2, which we describe briefly.
• Clarkson, P., Mansfield, E. and Milne, A. (1996). Symmetries and exact solutions of a (2+1)-dimensional sine-Gordon system. Philosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences. [Online] 354:1807-1835. Available at: http://dx.doi.org/10.1098/rsta.1996.0079.
We investigate the classical and non-classical reductions of the (2 + 1)-dimensional sine-Gordon system of Konopelchenko and Rogers, which is a strong generalization of the sine-Gordon equation. A family of solutions obtained as a non-classical reduction involves a decoupled sum of solutions of a generalized, real, pumped Maxwell-Bloch system. This implies the existence of families of solutions, all occurring as a decoupled sum, expressible in terms of the second, third and fifth Painleve transcendents, and the sine-Gordon equation. Indeed, hierarchies of such solutions are found, and explicit transformations connecting members of each hierarchy are given. By applying a known Backlund transformation for the system to the new solutions found, rye obtain further families of exact solutions, including some which are expressed as the argument and modulus of sums of products of Bessel functions with arbitrary coefficients. Finally, we show that the sine-Gordon system satisfies the necessary conditions of the Painleve PDE test due to Weiss et al., which requires the usual test to be modified, and derive a non-isospectral Lax pair for the generalized, real, pumped Maxwell-Bloch system.
• Clarkson, P., Mansfield, E. and Milne, A. (1996). Symmetries and exact solutions of a 2+1 dimensional sine-Gordon equation. Philosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences. 354:1807-1835.
• Albrecht, D., Mansfield, E. and Milne, A. (1996). Algorithms for special integrals of ordinary differential equations. Journal of Physics A: Mathematical and General [Online] 29:973-991. Available at: http://dx.doi.org/10.1088/0305-4470/29/5/013.
We give new, conceptually simple procedures for calculating special integrals of polynomial type (also known as Darboux polynomials, algebraic invariant curves, or eigenpolynomials), for ordinary differential equations. In principle, the method requires only that the given ordinary differential equation be itself of polynomial type of degree one and any order. The method is algorithmic, is suited to the use of computer algebra, and does not involve solving large nonlinear algebraic systems. To illustrate the method, special integrals of the second, fourth and sixth Painleve equations, and a third-order ordinary differential equation of Painleve type are investigated. We prove that for the second Painleve equation, the known special integrals are the only ones possible.
• Mansfield, E. (1996). A simple criterion for involutivity. Journal of the London Mathematical Society [Online] 54:323-345. Available at: http://dx.doi.org/10.1112/jlms/54.2.323.
A simple criterion for the involutivity of a system of partial differential equations of polynomial type is proved. The criterion involves the equations themselves and does not require the system to be in orthonomic form. It is proved that a system of partial differential equations is involutive if it is a differential Grobner basis with respect to a total degree ordering, and if the compatibility conditions of the symbol equations of the system consist of equations of degree one. An algorithm for calculating these compatibility conditions is given.
• Mansfield, E. (1996). The differential algebra package diffgrob2. Mapletech 3:33-37.

### Book

• Mansfield, E. (2010). A Practical Guide to the Invariant Calculus. Cambridge: Cambridge University Press.

### Book section

• Mansfield, E. and Zhao, J. (2011). On the modern notion of a moving frame. in: Dorst, L. and Lasenby, J. eds. Guide to Geometric Algebra in Practice. London: Springer, pp. 411-434. Available at: http://dx.doi.org/10.1007/978-0-85729-811-9_20.
A tutorial on the modern definition and application of moving frames, with a variety
of examples and exercises, is given. We show three types of invariants; differential, joint,
and integral, and the running example is the linear action of $SL(2)$ on smooth surfaces, on
sets of points in the plane, and path integrals over curves in the plane.
We also give details of moving frames for the group of rotations and translations acting
on smooth curves, and on discrete sets of points, in the conformal geometric algebra.
• Mansfield, E. (2002). Moving frames and differential algebra. in: Guo, L. et al. eds. Differential Algebra and Related Topics. Singapore: World Scientific Press, pp. 257-279. Available at: http://www.worldscibooks.com/mathematics/4768.html.
• Clarkson, P., Mansfield, E. and Webster, H. (2002). On Discrete Painleve Equations as Backlund Transformations. in: Coley, A. et al. eds. Backlund and Darboux Transformations: The Geometry of Solitons. United States: American Mathematical Society, pp. 129-139.
• Clarkson, P. and Mansfield, E. (2002). Open problems in symmetry analysis. in: Leslie, J. ed. The Geometrical Study of Differential Equations. United Kingdom: American Mathematical Society, pp. 195-205.

### Conference or workshop item

• Shemyakova, E. and Mansfield, E. (2008). Moving Frames for Laplace Invariants. in: Jeffrey, D. ed. International Symposium in Symbolic and Algebraic Manipulation 2008. New York: Association for Computing Machinery, pp. 291-298. Available at: http://portal.acm.org/toc.cfm?id=1390768&coll=ACM&dl=ACM&type=proceeding&idx=SERIES418&part=series&WantType=Proceedings&title=ISSAC&CFID=://www.google.co.uk/search?q=ISSAC%2001%20Proceedings%20&CFTOKEN=www.google.co.uk/search?q=ISSAC%2001%20Proceedings.
• Mansfield, E. and Hydon, P. (2001). Towards approximations which preserve integrals. in: Mourrain, B. ed. International Symposium in Symbolic and Algebraic Manipulation. New York: Association for Computing Machinery, pp. 217-222. Available at: http://portal.acm.org/toc.cfm?id=384101&coll=ACM&dl=ACM&type=proceeding&idx=SERIES418&part=series&WantType=Proceedings&title=International+Conference+on+Symbolic+and+Algebraic+Computation.

### Forthcoming

• Mansfield, E. et al. (2019). Moving Frames and Noetherâs Finite Difference Conservation Laws I. Transactions of Mathematics and its Applications.
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 invariant under various solvable Lie groups. The theory is also generalized to deal with variational symmetries that do not leave the Lagrangian invariant.

Apart from the study of systems that are inherently discrete, one significant application is to obtain geometric (variational) integrators that have finite difference approximations of the continuous conservation laws embedded \textit{a priori}. This is achieved by taking an invariant finite difference Lagrangian in which the discrete invariants have the correct continuum limit to their smooth counterparts.
We show the calculations for a discretization of the Lagrangian for Euler's elastica, and compare our discrete solution to that of its smooth continuum limit.
• Mansfield, E. and Rojo-Echeburua, A. (2019). On the use of the Rotation Minimizing Frame for Variational Systems with Euclidean Symmetry. Studies in Applied Mathematics [Online]. Available at: https://doi.org/10.1111/sapm.12275.
We study variational systems for space curves, for which the Lagrangian or action principle has a Euclidean symmetry, using the Rotation Minimizing frame, also known as the Normal, Parallel or Bishop frame (see [1], [36]).

Such systems have previously been studied using the FrenetâSerret frame. However, the Rotation Minimizing frame has many advantages, and can be used to study a wider class of examples.

We achieve our results by extending the powerful symbolic invariant cal- culus for Lie group based moving frames, to the Rotation Minimizing frame case. To date, the invariant calculus has been developed for frames defined by algebraic equations. By contrast, the Rotation Minimizing frame is defined by a differential equation.

In this paper, we derive the recurrence formulae for the symbolic invariant differentiation of the symbolic invariants. We then derive the syzygy operator needed to obtain Noetherâs conservation laws as well as the EulerâLagrange equations directly in terms of the invariants, for variational problems with a Euclidean symmetry. We show how to use the six Noether laws to ease the integration problem for the minimizing curve, once the EulerâLagrange equations have been solved for the generating differential invariants. Our applications include variational problems used in the study of strands of pro- teins, nucleid acids and polymers.
• Mansfield, E. and Rojo-Echeburua, A. (2019). Moving Frames and Noetherâs Finite Difference Conservation Laws II. Transactions of Mathematics and its applications.
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 EulerâLagrange difference equations and Noetherâ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 Eulerâ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.
• Zadra, M. and Mansfield, E. (2019). USING LIE GROUP INTEGRATORS TO SOLVE TWO DIMENSIONAL VARIATIONAL PROBLEMS WITH SYMMETRY. American Institute of Mathematical Sciences.
The theory of moving frames has been used successfully to solve one dimensional (1D) variational problems invariant under a Lie group symmetry. Unlike in the 1D case, where Noetherâs laws give first integrals of the EulerâLagrange equations, in higher dimensional problems the conservation laws do not enable the exact integration of the EulerâLagrange system. In this paper we use a moving frame to solve, numerically, a two dimensional (2D)
variational problem, invariant under a projective action of SL(2). In order to find a solution to the variational problem, we may solve a related 2D system of linear, first order, coupled ODEs for the moving frame, evolving on SL(2). We demonstrate that Lie group integrators [12] may be used in this context, by
showing that such systems are also numerically compatible, up to order 5, that is, the result is independent of the order of integration. This compatibility is a testament to the level of geometry built into the Lie group integrators.
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