Jet groupoids, natural bundles and the Vessiot equivalence method [Elektronische Ressource] / vorgelegt von Arne Lorenz

Arnelorenz

Sujets

• Mathematics

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2009
• Nombre de lectures: 32
• Poids de l'ouvrage: 1 Mo

Extrait

Jet Groupoids, Natural Bundles
and the
Vessiot Equivalence Method
Von der Fakult¨at fu¨r Mathematik, Informatik und Naturwissenschaften der
RWTH Aachen University zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften genehmigte Dissertation
vorgelegt von
Diplom-Physiker
Arne Lorenz
aus Elmshorn
Berichter: Universit¨atsprofessor Dr. Wilhelm Plesken gen. Wigger
Universit¨atsprofessor Dr. Vladimir P. Gerdt
Tag der mu¨ndlichen Pru¨fung: 18. M¨arz 2009
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online
verfu¨gbar.2Contents
Introduction 7
1 Geometric Formulation of PDEs 15
1.1 Fibre Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.1.1 Exact Sequences of Fibre Bundles . . . . . . . . . . . . . . 18
1.2 Jet Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2.1 Prolongation and the Jet Bundle Functor . . . . . . . . . . 25
1.3 Systems of Partial Differential Equations . . . . . . . . . . . . . . . 28
1.3.1 Prolongation and Projection . . . . . . . . . . . . . . . . . 30
1.3.2 Formal Integrability . . . . . . . . . . . . . . . . . . . . . . 33
2 Lie Groupoids 35
2.1 Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Lie Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3 Jet Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Natural Bundles 47
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.1.1 Structure of Natural Bundles . . . . . . . . . . . . . . . . . 51
3.2 Natural Bundle Functors . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3 Natural Bundles and Jet Groupoids . . . . . . . . . . . . . . . . . 55
3.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 Prolongation and Projection . . . . . . . . . . . . . . . . . . . . . . 64
3.4.1 Prolongation . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4.2 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5 Integrability Conditions . . . . . . . . . . . . . . . . . . . . . . . . 73
3.5.1 Bundle Structure ofF . . . . . . . . . . . . . . . . . . . . 78(1)
3.5.2 Symbols of Jet Groupoids . . . . . . . . . . . . . . . . . . . 80
3.5.3 Janet Sequence . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.5.4 The Curvature Map . . . . . . . . . . . . . . . . . . . . . . 89
34 CONTENTS
4 Applications of Natural Bundles 91
4.1 Towards Formal Integrability . . . . . . . . . . . . . . . . . . . . . 92
4.2 Invariants on Natural Bundles . . . . . . . . . . . . . . . . . . . . . 98
4.3 Extending and Simplifying Vessiot’s Method. . . . . . . . . . . . . 103
4.3.1 Optimisation I: Minimal Bundles . . . . . . . . . . . . . . . 103
4.3.2 Proof of the Embedding Theorem . . . . . . . . . . . . . . 112
4.3.3 Optimisation II: Subgroupoids . . . . . . . . . . . . . . . . 122
5 Maple Examples 125
5.1 Riemannian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.1.1 The Natural Bundle . . . . . . . . . . . . . . . . . . . . . . 128
5.1.2 Spencer Cohomology . . . . . . . . . . . . . . . . . . . . . . 129
5.1.3 Prolongation and Projection . . . . . . . . . . . . . . . . . 130
5.1.4 Integrability Conditions . . . . . . . . . . . . . . . . . . . . 131
5.1.5 Vector Bundle Structure ofF . . . . . . . . . . . . . . . . 132(1)
5.1.6 Optimisation: Minimal Bundles . . . . . . . . . . . . . . . . 133
5.1.7 Invariants on Natural Bundles . . . . . . . . . . . . . . . . 134
5.2 Invariants for Lie Pseudogroup Actions . . . . . . . . . . . . . . . . 136
5.3 Second Order ODEs under Point Transformations . . . . . . . . . 141
6 The Vessiot Equivalence Method 147
6.1 The Equivalence Problem . . . . . . . . . . . . . . . . . . . . . . . 148
6.1.1 The Full Equivalence Problem . . . . . . . . . . . . . . . . 148
6.1.2 The Relative Equivalence Problem . . . . . . . . . . . . . . 152
6.1.3 The Vessiot Equivalence Method in Practice. . . . . . . . . 154
6.2 Cartan’s Equivalence Method . . . . . . . . . . . . . . . . . . . . . 158
6.2.1 G-Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.2.2 Cartan’s Equivalence Method at Work . . . . . . . . . . . . 163
6.2.3 Comparison Between Cartan’s and Vessiot’s Method . . . . 173
6.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.3.1 Third Order ODEs . . . . . . . . . . . . . . . . . . . . . . . 178
6.3.2 Fourth Order ODEs . . . . . . . . . . . . . . . . . . . . . . 183
7 Linear Partial Differential Operators 185
7.1 LPDOs and Vessiot’s Equivalence Method . . . . . . . . . . . . . . 186
7.1.1 Linear Partial Differential Operators . . . . . . . . . . . . . 186
7.1.2 Modifications of Vessiot’s Approach . . . . . . . . . . . . . 188
7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.2.1 The Laplace Example withMaple . . . . . . . . . . . . . . 193
7.2.2 Third Order LPDOs on the Plane . . . . . . . . . . . . . . 195
7.2.3 A Third Order LPDO in Dimension Three. . . . . . . . . . 200
7.2.4 Fourth Order LPDOs on the Plane . . . . . . . . . . . . . . 202
Appendix 207CONTENTS 5
A Symbols and Spencer Cohomology 209
B Jet Groups 225
B.1 Her Majesty GL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225q
B.2 The Lie Algebra gl of GL . . . . . . . . . . . . . . . . . . . . . . 227qq
B.3 Subgroups G of GL . . . . . . . . . . . . . . . . . . . . . . . . . . 229q q
B.4 Lie Subalgebras g of gl . . . . . . . . . . . . . . . . . . . . . . . . 231q q
B.5 G -actions on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 233q
B.5.1 Prolongation . . . . . . . . . . . . . . . . . . . . . . . . . . 233
B.5.2 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
B.5.3 Equivariant Maps . . . . . . . . . . . . . . . . . . . . . . . 234
C Lie Algebroids 237
C.1 Lie Algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
C.1.1 The Lie Algebroid of a Lie Groupoid . . . . . . . . . . . . . 238
C.2 Jet Algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
C.2.1 The Algebroid of Π . . . . . . . . . . . . . . . . . . . . . . 2430
C.2.2 The Algebroid of Π . . . . . . . . . . . . . . . . . . . . . . 244q
C.2.3 Prolongation Formulae for ] and [ . . . . . . . . . . . . . . 244
D Implementation 251
D.1 TheMaple Package JetGroupoids . . . . . . . . . . . . . . . . . 252
D.1.1 Natural Bundle Commands . . . . . . . . . . . . . . . . . . 252
D.1.2 Invariants and Related Commands . . . . . . . . . . . . . . 253
D.1.3 Exterior Differential Forms . . . . . . . . . . . . . . . . . . 253
D.1.4 Tool Procedures . . . . . . . . . . . . . . . . . . . . . . . . 254
D.2 TheMaple Package jets . . . . . . . . . . . . . . . . . . . . . . . 255
D.2.1 Groupoid and Algebroid Commands . . . . . . . . . . . . . 255
D.2.2 Natural Bundle Commands . . . . . . . . . . . . . . . . . . 255
D.3 TheMaple Package Spencer . . . . . . . . . . . . . . . . . . . . . 256
D.3.1 JanetOre Extension for the Exterior Algebra . . . . . . . . 257
D.3.2 ZeroSets Extension for Involutive . . . . . . . . . . . . . . 258
D.4 Sample Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . 2586 CONTENTSIntroduction
The present thesis takes a work of Vessiot [Ves03] as the starting point to de-
velop a new equivalence method which has theoretical advantages over Cartan’s
method [Car08]. The development is focused on both theoretical and compu-
tational aspects. Equivalence means that two geometric objects on a manifold
can locally be mapped to each other by a smooth transformation. The main dif-
ference between Vessiot’s and Cartan’s approach is that Vessiot’s method works
on arbitrary geometric objects, whereas Cartan has to reduce all problems to a
coframe, which is a very special geometric object. A coframe is a basis of the
cotangent space.
The Vessiot equivalence method, developed in this thesis, is successfully ap-
plied to the example of linear partial differential operators (LPDOs) under gauge
transformations. For third and fourth order LPDOs in dimension two, generat-
ing sets of invariants have been calculated. They allow to decide equivalence of
LPDOs under gauge transformations. Furthermore they are of interest for fac-
torisation and the exact integration of the operators. At order three, this leads
to the improvement of several results from Mansfield and Shemyakova [MS08],
who use Cartan’s moving frame method. The fourth order results are completely
new.
In order to treat LPDOs with Cartan’s method, the problem must be for-
mulated in terms of coframes and this requires human interaction. Choosing a
coframe generally involves unnatural choices that have to be ruled out in the
end (see e.g. [Olv95, Ex. 9.2] on Riemannian metrics). In contrast to Cartan’s
method, as mentioned above, Vessiot’s approach works directly with the geo-
metric objects. Their transformation is encoded in natural bundles, which are
either given by the problem or constructed automatically. In case of LPDOs the
coordinates of the natural bundle are simply the coefficients of the operators.
For Cartan’s approach, all problems have to be transformed to first order