DOCSLIB.ORG

Equivalence Transformations for Classes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Equivalence Transformations for Classes EQUIVALENCE TRANSFORMATIONS FOR CLASSES OF DIFFERENTIAL EQUATIONS by Ian Lisle B.Sc. (Australian Environmental Studies) Griffith University M.Sc.St. (Mathematics) University of Queensland A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS INSTITUTE OF APPLIED MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February, 1992 © Ian Lisle, 1992 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. ! further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allow/ed without my written permission. Department of Maihemaiics The University of British Columbia Vancouver, Canada Date 15^^ Apr,'!, mi DE-6 (2/88) Abstract We consider classes C of differential equations characterized by the presence of arbitrary ele• ments, that is, arbitrary functions or constants. Based on an idea of Ovsiannikov, we develop a systematic theory of equivalence transformations, that is, point changes of variables which map every equation in C to another equation in C. Examples of nontrivial groups of equivalence transformations are found for some linear wave and nonlinear diffusion convection systems, and used to clarify some previously known results. We show how equivalence transformations may be inherited as symmetries of equations in C, leading to a partial symmetry classification for the class C. New symmetry results for a potential system form of the nonlinear diffusion convection equation are derived by this procedure. Finally we show how to use equivalence group information to facilitate complete symmetry classification for a class of differential equations. The method relies on the geometric concept of a moving frame, that is, an arbitrary (possibly noncommuting) basis for differential operators on the space of independent and dependent variables. We show how to choose a frame which is invariant under the action of the equivalence group, and how to rewrite the determining equations for symmetries in terms of this frame. A symmetry classification algorithm due to Reid is modified to deal with the case of noncommuting operators. The result is an algorithm which combines features of Reid's classification algorithm and Cartan's equivalence method. The method is applied to the potential diffusion convection example, and yields a complete symmetry classification in a particularly elegant form. Table of Contents Abstract ii List of Tables vii List of Figures viii Acknowledgments ix 1 Introduction 1 1.1 Differential equations and their transformation 1 1.2 Equivalence of differential equations: Examples 4 1.3 Symmetries and differential equations 12 1.4 Equivalence transformations 15 1.5 Symmetry classification problem 17 2 Transformation Groups and Differential Equations 22 2.1 Transformation groups 22 2.1.1 Transformations, Lie groups 22 2.1.2 Infinitesimal operators 26 2.1.3 Invariant surface 32 2.2 Extension 33 2.2.1 Notation for derivatives 33 2.2.2 Extension of transformation 34 2.2.3 Extension of group operator 38 2.3 Differential equations eind symmetry 39 2.3.1 Differential equations 39 2.3.2 Symmetries of differential equations 41 2.3.3 Algorithmic construction of symmetries 45 3 The Equivalence Group 48 3.1 Class of differential equations 48 3.1.1 Decoupled systems of d.e.'s 48 3.1.2 Class of d.e.'s 49 3.2 Equivalence transformations 52 3.3 Infinitesimal augmented transformations 60 3.3.1 Infinitesimal augmented transformations 60 3.3.2 Algebra of equivalence operators 64 3.3.3 Algorithm for construction of equivalence group 70 3.3.4 Proposition on form of infinitesimals 72 3.3.5 Structure of the equivalence group 74 3.4 Examples of equivalence groups 80 3.4.1 Boltzmann's similarity solution for nonlinear diffusion 80 3.4.2 Nonlinear diffusion-convection equations 88 3.4.3 Wave equations 98 3.4.4 Hamilton's equations 103 4 Symmetry Group Classification 108 4.1 Symmetry classification problem 108 4.1.1 Example: scalar diffusion convection 109 4.2 Partial symmetry classification 110 4.2.1 Symmetry inherited from equivalence group 110 4.2.2 Optimal system of subalgebras 115 4.2.3 Partial symmetry classification for nonlinear diffusion convection 117 4.3 Modification of Reid algorithm 129 4.3.1 Moving frame and determining equations 131 4.3.2 Frame Reid method 139 4.4 Invariant frame 150 4.4.1 Augmented frame 150 4.4.2 Invariant frame 155 4.4.3 Differential invariants 156 4.4.4 Tresse basis 159 4.5 Symmetry classification 162 4.5.1 Invariant form of group classification 165 4.5.2 Potential diffusion convection system 175 5 Conclusion 196 5.1 Further work 196 5.1.1 Isovector method for frame determining system 197 5.2 Conclusions 201 A Algorithms for Frame Systems 203 A.l Reduction to frame involutive form 203 A. 1.1 Orthonomic form 203 A. 1.2 Reduced orthonomic form 205 A. 1.3 Involutive form 206 A.2 Group classification 207 B Structure Constants 210 C Similarity Solution for Nonlinear Diffusion 215 C.l Power law diffusivity 216 C.1.1 Phase reduction 217 C.1.2 Exact shooting 218 C.l.3 Series solution 219 C.2 Modified power law diffusivity 220 C.2.1 Phase reduction 221 C.2.2 Exact shooting 221 C.2.3 Series solution 222 C.3 Discussion 223 Bibliography 225 List of Tables 3.1 Commutator table of equivalence algebra of nonlinear diffusion potential system. 70 3.2 Commutator table for equivalence operators of scalar diffusion convection equation 91 3.3 Commutation relations of equivalence algebra of scalar wave equation 100 4.1 Symmetry classification for scalar nonlinear diffusion convection equation Ill 4.2 Optimal system of subalgebras for nonlinear diffusion convection potential system. 121 4.3 Partial symmetry classification for diffusion convection potential system: Case K{u) = 0 (diffusion equations) 123 4.4 Partial symmetry classification for diffusion convection potential system: Case with nonlinear convection 124 4.5 Nonlocal symmetries inherited from equivalence group of diffusion convection potential system: Case K{u) = 0 (diffusion equations) 125 4.6 Nonlocal symmetries inherited from equivalence group of diffusion convection potential system: Case K{u) ^0 126 4.7 Commutator table of equivalence algebra of nonlinear diffusion equation 168 4.8 Commutation relations for equivalence algebra of diffusion convection potential system 176 vn List of Figures 3.1 Relationship between linearizable diffusion convection potential systems 98 4.1 Classification tree for symmetries of nonlinear diffusion equation 173 4.2 Preliminary classification tree for potential diffusion convection system 189 4.3 Complete symmetry classification tree for diffusion convection potential system. 195 C.l Relation between concentration, flux, and spatial coordinate for Boltzmann's similarity solution with power law diffusivity 217 C.2 Phase portrait for Boltzmann similarity solution with power law diffusivity . 218 vm Acknowledgments I wish to thank my supervisor Dr. G.W. Bluman for originally suggesting the study of equiva• lence transformations, and for his guidance, insight and unfailing sense of direction. Particular thanks are due for his careful and speedy reading of the drafts of my thesis. My wholehearted appreciation also to Dr. Greg Reid for his constant encouragement and support, for his patient explanations of his symmetry classification algorithm, and for making available the MAPLE implementation of this algorithm. Alan Boulton was a constant source of enthusiasm over the last two years, participating in many lively discussions. The insightful lec• tures of Dr. K.Y. Lam on differential geometry provided me with the tools which made §4.3-§4.5 possible. The problem described in Appendix C resulted from discussions with Dr. J.Y. Par- lange. My fellow graduate students have maintained me with their friendship over the years, and have also enriched my knowledge of mathematics, computing, French, and softball. My most heartfelt appreciation to Maria ChiaroUa for helping maintain my sanity when things did not go well. Honourable mention to the Stanley Cup playoffs, the few sunny days in winter, and Piper's Ale. Funding for much of my stay in Canada was provided by the Canadian Commonwealth Scholarship and Fellowship Administration, and their support is gratefully acknowledged. This thesis was typeset using M?gX. Chapter 1 Introduction 1.1 Differential equations and their transformation In dealing with differential equations, a common situation is that one wishes to analyze simul• taneously a whole class of equations of some given type. It is natural to consider 'the class of second order ordinary differential equations' or 'the nonlinear diffusion equation' ut = [D{u)u^]^ Here w, D are arbitrary (smooth) functions of their arguments, at least in some suitable domain of definition. Thus the entire class of equations under consideration is specified by allowing these arbitrary elements to range over all possible functional forms. In this dissertation, I will be concerned with the transformation properties of a given class C of differential equations. Attention will be restricted to invertible 'point' transformations, which act on a coordinate space of the independent and dependent variables. These are the usual 'changes of variables' in differential equations. For (1.1) for example, the most general such change of variables is x' = Fix,y) y' = Gix,y) (1.2) (subject to the Jacobian FxGy — G^Fy being nonzero). Any transformation applied to the variables in a differentiïJ equation (d.e.) yields another differential equation.
Load more

Recommended publications
  • A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets Khalil Ghorbal∗ Andrew Sogokon∗ Andre´ Platzer∗ November 2014 CMU-CS-14-140
    A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets Khalil Ghorbal∗ Andrew Sogokon∗ Andre´ Platzer∗ November 2014 CMU-CS-14-140 Publisher: School of Computer Science Carnegie Mellon University Pittsburgh, PA, 15213 To appear [14] in the Proceedings of the 16th International Conference on Verification, Model Checking, and Abstract Interpretation (VMCAI 2015), 12-14 January 2015, Mumbai, India. This material is based upon work supported by the National Science Foundation (NSF) under NSF CA- REER Award CNS-1054246, NSF EXPEDITION CNS-0926181, NSF CNS-0931985, by DARPA under agreement number FA8750-12-2-029, as well as the Engineering and Physical Sciences Research Council (UK) under grant EP/I010335/1. The views and conclusions contained in this document are those of the authors and should not be inter- preted as representing the official policies, either expressed or implied, of any sponsoring institution or government. Keywords: algebraic invariant, deductive power, hierarchy, proof rules, differential equation, abstraction, continuous dynamics, formal verification, hybrid system Abstract This paper presents a theoretical and experimental comparison of sound proof rules for proving invariance of algebraic sets, that is, sets satisfying polynomial equalities, under the flow of poly- nomial ordinary differential equations. Problems of this nature arise in formal verification of con- tinuous and hybrid dynamical systems, where there is an increasing need for methods to expedite formal proofs. We study the trade-off between proof rule generality and practical performance and evaluate our theoretical observations on a set of heterogeneous benchmarks. The relationship between increased deductive power and running time performance of the proof rules is far from obvious; we discuss and illustrate certain classes of problems where this relationship is interesting.
    [Show full text]
  • Symmetry Transformation Groups and Differential Invariants — Eivind Schneider
    FACULTY OF SCIENCE AND TECHNOLOGY Department of Mathematics and Statistics Symmetry transformation groups and differential invariants — Eivind Schneider MAT-3900 Master’s Thesis in Mathematics November 2014 Abstract There exists a local classification of finite-dimensional Lie algebras of vector fields on C2. We lift the Lie algebras from this classification to the bundle C2 × C and compute differential invariants of these lifts. Acknowledgements I am very grateful to Boris Kruglikov for coming up with an interesting assignment and for guiding me through the work on it. I also extend my sincere gratitude to Valentin Lychagin for his help through many insightful lectures and conversations. A heartfelt thanks to you both, for sharing your knowledge and for the inspiration you give me. I would like to thank Henrik Winther for many helpful discussions. Finally, I thank my family, and especially my wife, Susann, for their support and encouragement. ii Contents 1 Introduction 1 2 Preliminaries 3 2.1 Jet spaces and prolongation of vector fields . 3 2.2 Classification of Lie algebras of vector fields in one and two dimensions . 5 2.3 Lifts . 7 2.3.1 Cohomology . 8 2.3.2 Coordinate change . 9 2.4 Differential invariants . 9 2.4.1 Determining the number of differential invariants of order k ........................... 10 2.4.2 Invariant derivations . 11 2.4.3 Tresse derivatives . 12 2.4.4 The Lie-Tresse theorem . 13 3 Warm-up: Differential invariants of lifts of Lie algebras in D(C) 15 3.1 g1 = h@xi .............................. 15 3.2 g2 = h@x; x@xi ..........................
    [Show full text]
  • The Invariant Variational Bicomplex
    Contemporary Mathematics The Invariant Variational Bicomplex Irina A. Kogan and Peter J. Olver Abstract. We establish a group-invariant version of the variational bicom- plex that is based on a general moving frame construction. The main appli- cation is an explicit group-invariant formula for the Euler-Lagrange equations of an invariant variational problem. 1. Introduction. Most modern physical theories begin by postulating a symmetry group and then formulating ¯eld equations based on a group-invariant variational principle. As ¯rst recognized by Sophus Lie, [17], every invariant variational problem can be written in terms of the di®erential invariants of the symmetry group. The associated Euler-Lagrange equations inherit the symmetry group of the variational problem, and so can also be written in terms of the di®erential invariants. Surprisingly, to date no-one has found a general group-invariant formula that enables one to directly construct the Euler-Lagrange equations from the invariant form of the variational problem. Only a few speci¯c examples, including plane curves and space curves and surfaces in Euclidean geometry, are worked out in Gri±ths, [12], and in Anderson's notes, [3], on the variational bicomplex. In this note, we summarize our recent solution to this problem; full details can be found in [16]. Our principal result is that, in all cases, the Euler-Lagrange equations have the invariant form (1) A¤E(L) ¡ B¤H(L) = 0; where E(L) is an invariantized version of the usual Euler-Lagrange expression or e e \Eulerian", while H(L) is an invariantized Hamiltonian, which in the multivariate context is,ein fact, a tensor, [20], and A¤; B¤ certain invariant di®erential operators, which we name the Euleriane and Hamiltonian operators.
    [Show full text]
  • Differential Invariant Algebras
    Differential Invariant Algebras Peter J. Olver† School of Mathematics University of Minnesota Minneapolis, MN 55455 [email protected] http://www.math.umn.edu/∼olver Abstract. The equivariant method of moving frames provides a systematic, algorith- mic procedure for determining and analyzing the structure of algebras of differential in- variants for both finite-dimensional Lie groups and infinite-dimensional Lie pseudo-groups. This paper surveys recent developments, including a few surprises and several open ques- tions. 1. Introduction. Differential invariants are the fundamental building blocks for constructing invariant differential equations and invariant variational problems, as well as determining their ex- plicit solutions and conservation laws. The equivalence, symmetry and rigidity properties of submanifolds are all governed by their differential invariants. Additional applications abound in differential geometry and relativity, classical invariant theory, computer vision, integrable systems, geometric flows, calculus of variations, geometric numerical integration, and a host of other fields of both pure and applied mathematics. † Supported in part by NSF Grant DMS 11–08894. November 16, 2018 1 Therefore, the underlying structure of the algebras‡ of differential invariants for a broad range of transformation groups becomes a topic of great importance. Until recently, though, beyond the relatively easy case of curves, i.e., functions of a single independent variable, and a few explicit calculations for individual higher dimensional examples,
    [Show full text]
  • The Structure of Differential Invariant Algebras of Lie Groups and Lie
    The Structure of Di®erential Invariant Algebras of Lie Groups and Lie Pseudo{Groups Peter J. Olver University of Minnesota http://www.math.umn.edu/ » olver =) Juha Pohjanpelto Jeongoo Cheh à 0 Sur la th¶eorie, si importante sans doute, mais pour nous si obscure, des ¿groupes de Lie in¯nis, nous ne savons rien que ce qui trouve dans les m¶emoires de Cartan, premiere exploration a travers une jungle presque imp¶en¶etrable; mais celle-ci menace de se refermer sur les sentiers d¶eja trac¶es, si l'on ne procede bient^ot a un indispensable travail de d¶efrichement. | Andr¶e Weil, 1947 à 1 Pseudo-groups in Action ² Lie | Medolaghi | Vessiot ² Cartan : : : Guillemin, Sternberg ² Kuranishi, Spencer, Goldschmidt, Kumpera, : : : ² Relativity ² Noether's Second Theorem ² Gauge theory and ¯eld theories Maxwell, Yang{Mills, conformal, string, : : : ² Fluid Mechanics, Metereology Euler, Navier{Stokes, boundary layer, quasi-geostropic , : : : ² Linear and linearizable PDEs ² Solitons (in 2 + 1 dimensions) K{P, Davey-Stewartson, : : : ² Image processing ² Numerical methods | geometric integration ² Kac{Moody symmetry algebras ² Lie groups! à 2 What's New? Direct constructive algorithms for: ² Invariant Maurer{Cartan forms ² Structure equations ² Moving frames ² Di®erential invariants ² Invariant di®erential operators ² Constructive Basis Theorem ² Syzygies and recurrence formulae ² GrÄobner basis constructions ² Further applications =) Symmetry groups of di®erential equations =) Vessiot group splitting =) Gauge theories =) Calculus of variations à 3 Di®erential Invariants G | transformation group acting on p-dimensional submani- folds N = fu = f(x)g ½ M G(n) | prolonged action on the submanifold jet space Jn = Jn(M; p) Di®erential invariant I : Jn ! R I(g(n) ¢ (x; u(n))) = I(x; u(n)) =) curvature, torsion, : : : Invariant di®erential operators: D1; : : : ; Dp =) arc length derivative ? ? I(G) | the algebra of di®erential invariants ? ? à 4 The Basis Theorem Theorem.
    [Show full text]
  • Bulletin De La S
    BULLETIN DE LA S. M. F. GLORIA MARÍ BEFFA The theory of differential invariants and KDV hamiltonian evolutions Bulletin de la S. M. F., tome 127, no 3 (1999), p. 363-391 <http://www.numdam.org/item?id=BSMF_1999__127_3_363_0> © Bulletin de la S. M. F., 1999, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http: //smf.emath.fr/Publications/Bulletin/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/ conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Bull. Soc. math. France, 127, 1999, p. 363-391. THE THEORY OF DIFFERENTIAL INVARIANTS AND KDV HAMILTONIAN EVOLUTIONS BY GLORIA MARI BEFFA (*) ABSTRACT. — In this paper I prove that the second KdV Hamiltonian evolution associated to SL(n, R) can be view as the most general evolution of projective curves, invariant under the SL(n,R)-projective action on RP"'"1, provided that certain integrability conditions are satisfied. This way, I establish a very close relationship between the theory of geometrical invariance, and KdV Hamiltonian evolutions. This relationship was conjectured in [4]. RESUME. — LA THEORIE DES INVARIANTS DIFFERENTIELS ET LES EVOLUTIONS HAMILTONIENNES DE KDV. — Dans cet article, je prouve que la seconde evolution hamiltonienne de KdV, associee au groupe SL(n,R), peut etre consideree comme 1'evolution la plus generale des courbes projectives qui sont invariantes par Faction projective de SL(n, R) sur RP71"1, si une certaine condition d'integrabilite est satisfaite.
    [Show full text]
  • Point Classification of 2Nd Order Odes: Tresse Classification
    Point classification of 2nd order ODEs: Tresse classification revisited and beyond Boris Kruglikov Abstract In 1896 Tresse gave a complete description of relative differential in- variants for the pseudogroup action of point transformations on the 2nd order ODEs. The purpose of this paper is to review, in light of modern geometric approach to PDEs, this classification and also discuss the role of absolute invariants and the equivalence problem.1 Introduction Second order scalar ordinary differential equations have been the classical tar- get of investigations and source of inspiration for complicated physical models. Under contact transformations all these equations are locally equivalent, but to find such a transformation for a pair of ODEs is the same hard problem as to find a general solution, which as we know from Ricatti equations is not always possible. Most integration methods for second order ODEs are related to another pseudogroup action – point transformations, which do not act transitively on the space of all such equations. All linear 2nd order ODEs are point equivalent. S. Lie noticed that ODEs linearizable via point transformations have neces- sarily cubic nonlinearity in the first derivatives and described a general test to construct this linearization map [Lie2]. Later R. Liouville found precise condi- tions for linearization [Lio]. But it was A. Tresse who first wrote the complete set of differential invariants for general 2nd order ODEs. arXiv:0809.4653v1 [math.CA] 26 Sep 2008 The paper [Tr2] is a milestone in the geometric theory of differential equa- tions, but mostly one result (linearization of S.Lie-R.Liouville-A.Tresse) from the manuscript is used nowadays.
    [Show full text]
  • Differential Invariants of Measurements, and Their Relation to Central Moments
    entropy Article Differential Invariants of Measurements, and Their Relation to Central Moments Eivind Schneider Faculty of Science, University of Hradec Králové, Rokitanského 62, 50003 Hradec Králové, Czech Republic; [email protected] Received: 31 August 2020; Accepted: 30 September 2020; Published: 3 October 2020 Abstract: Due to the principle of minimal information gain, the measurement of points in an affine ∗ space V determines a Legendrian submanifold of V × V × R. Such Legendrian submanifolds are equipped with additional geometric structures that come from the central moments of the underlying probability distributions and are invariant under the action of the group of affine transformations on V. We investigate the action of this group of affine transformations on Legendrian submanifolds ∗ of V × V × R by giving a detailed overview of the structure of the algebra of scalar differential invariants, and we show how the scalar differential invariants can be constructed from the central moments. In the end, we view the results in the context of equilibrium thermodynamics of gases, and notice that the heat capacity is one of the differential invariants. Keywords: measurement; information gain; thermodynamics; symplectic geometry; contact geometry; central moments; differential invariants; heat capacity 1. Introduction It has been known since the work of Gibbs [1,2] that thermodynamics can be formulated in the language of contact geometry. Because the fundamental thermodynamic relation takes the form of a contact structure on an odd-dimensional manifold, closed systems in thermal equilibrium correspond to Legendrian submanifolds with respect to this contact structure. More recently, some effort has been devoted towards studying an additional metric structure in the context of thermodynamics (see for example [3,4] and references therein).
    [Show full text]
  • Introduction to Differential Invariants
    Introduction to Differential Invariants Valeriy Yumaguzhin Opava, 2005 Contents 1. Introduction 4 Chapter 1. Jet bundles 5 1. Jets 5 2. Bundles 7 3. Jet bundles 9 4. Cartan distributions 10 5. Exercises 11 Chapter 2. Lie transformations 13 1. Point transformations 13 2. Contact transformations 16 3. Lie transformations 17 4. Exercises 18 Chapter 3. Lie vector fields 19 1. Liftings of vector fields 19 2. The coordinate description of lifted vector fields 20 3. The Lie algebra of lifted vector fields 21 4. Exercises 22 Chapter 4. Lie pseudogroups 23 1. Pseudogroups 23 2. Lie algebras of pseudogroups 24 3. Exercises 25 Chapter 5. Differential invariants 27 1. Differential invariants on the jet bundles. The equivalence problem 27 2. Scalar differential invariants 28 3. Symmetries and invariants 30 4. Example 30 5. Exercises 32 Chapter 6. Classification of linear ODEs up to equivalence 33 1. Reduction to the lesser pseudogroup 33 3 4 CONTENTS 2. The natural bundle of linear ODEs 34 3. Differential invariants of linear ODEs 35 4. The equivalence problem 40 5. The classification of linear ODEs 41 6. Exercises 43 Chapter 7. Differential invariants in natural bundles 45 1. Differential groups and geometric structures 45 2. Natural bundles 47 3. Formal vector fields. Spencer cohomologies 50 4. The construction of differential invariants of natural bundles 52 5. Exercises 64 Chapter 8. G-structures 65 1. Frame bundles 65 2. Geometric structures and their prolongations 67 3. G-structures and geometric structures. The equivalence problem 68 4. The integrability problem 69 5. Structure functions of G-structures and their prolongations 70 6.
    [Show full text]
  • Modern Developments in the Theory and Applications of Moving Frames
    London Mathematical Society Impact150 Stories 1 (2015) 14{50 C 2015 Author(s) doi:10.1112/l150lms/l.0001 e Modern Developments in the Theory and Applications of Moving Frames Peter J. Olver Abstract This article surveys recent advances in the equivariant approach to the method of moving frames, concentrating on finite-dimensional Lie group actions. A sampling from the many current applications | to geometry, invariant theory, and image processing | will be presented. 1. Introduction. According to Akivis, [2], the method of rep`eresmobiles, which was translated into English as moving framesy, can be traced back to the moving trihedrons introduced by the Estonian mathematician Martin Bartels (1769{1836), a teacher of both Gauß and Lobachevsky. The apotheosis of the classical development can be found in the seminal advances of Elie´ Cartan, [25, 26], who forged earlier contributions by Frenet, Serret, Darboux, Cotton, and others into a powerful tool for analyzing the geometric properties of submanifolds and their invariants under the action of transformation groups. An excellent English language treatment of the Cartan approach can be found in the book by Guggenheimer, [49]. The 1970's saw the first attempts, cf. [29, 45, 46, 64], to place Cartan's constructions on a firm theoretical foundation. However, the method remained constrained within classical geometries and homogeneous spaces, e.g. Euclidean, equi-affine, or projective, [35]. In the late 1990's, I began to investigate how moving frames and all their remarkable consequences might be adapted to more general, non-geometrically-based group actions that arise in a broad range of applications.
    [Show full text]
  • Selected Topic S in the Geometrica L Study of Differentia L Equations
    Selected Topic s in the Geometrica l Study o f Differentia l Equations In fron t o f the Founder s Librar y From lef t t o right: crouching : Gloria Mari-Beffa , Pete r Olver , Thierry Robart , Nik y Kamran; stand - ing (fron t row) : Pete r Hydon , Barbar a Shipman , Irin a Kogan , Evelyn e Hubert , Mik e Lacorte , Mohammad Mahmood , Da n Grossman , Pau l Kainen , Elizabet h Mansfield , Sylvi e Desjardins , Maya Chhetri , Joshu a Leslie , Debr a Lewis , Gabrie l Ayine , Zhoud e Shao , Jin g Pin g Wang , Mingxiang Chen , Ami r Maleki ; standin g (bac k row): Stephen Shipman , Mar k Fels , Rafael Heredero , Enrique Reyes , Stanle y Einstein-Matthews , Andr e Wehner , Vladimi r Itskov , Pete r Clarkson , Jeffrey Morton , Austi n Roche , Pete r va n de r Kamp , Al i Ayari, Rober t Milson , Matthe w Biesecker , Clement Lutterodt . http://dx.doi.org/10.1090/cbms/096 Conference Boar d o f the Mathematical Science s CBMS Regional Conference Series in Mathematics Number 9 6 Selected Topic s in the Geometrica l Study o f Differentia l Equations Niky Kamran Published fo r th e Conference Boar d of the Mathematica l Science s by the American Mathematica l Societ y * fkL'' ^ Providence, Rhod e Islan d with support fro m th e National Science Foundatio n CBMS Conferenc e o n the Geometrica l Stud y o f Differentia l Equation s hel d a t Howar d University, Washington , D.C . June 20-25 , 200 0 Partially supporte d b y th e National Scienc e Foundatio n 2000 Mathematics Subject Classification.
    [Show full text]
  • Lie Symmetry Group Methods for Differential Equations
    Lie symmetry group methods for differential equations F. G¨ung¨or∗1 1Department of Mathematics, Faculty of Science and Letters, Istanbul Technical University, 34469 Istanbul, Turkey Abstract Fundamentals on Lie group methods and applications to differ- ential equations are surveyed. Numerous examples are included to elucidate their extensive applicability for analytically solving both or- dinary and partial differential equations. arXiv:1901.01543v8 [math.CA] 1 Jun 2021 ∗[email protected] 1 Contents 1 Vector fields and integral curves 4 2 Differential equations and their symmetry group 7 3 Differential invariants 18 3.1 Invariant differentiation . 19 4 Reduction of order for ordinary differential equations 24 4.1 Group classification problem . 50 5 Group-invariant solutions 57 5.1 Linearization by symmetry structure . 81 5.2 Lie’s Linearization Theorem . 88 5.3 Concludingremarks........................ 90 2 Introduction This paper reviews a treatment of differential equations using methods from Lie group theory. Symmetry group methods are amongst the most pow- erful universal tools for the study of differential equations. There has been rapid progress on these methods over the last few decades. Methods and algo- rithms for classifying subalgebras of Lie algebras, new results on the structure and classification of abstract finite and infinite dimensional Lie algebras [48] and methods for solving group classification problems for differential equa- tions greatly facilitated to systematically obtain exact analytic solutions by quadratures to ordinary differential
    [Show full text]