Contents
Preface i x
Acknowledgments xii i
Chapter 1. Differentia l equation s an d thei r geometr y 1
1. Th e Cauch y proble m fo r first-orde r partia l differentia l equation s 1
2. Hyperboli c equation s integrabl e b y th e metho d o f Darboux 3
3. External , interna l an d generalize d symmetrie s 5
4. Th e invers e proble m o f the calculu s o f variations 6
5. Som e importan t topic s no t covere d i n thes e lecture s 7
Chapter 2 . Externa l an d generalize d symmetrie s 9
1. Je t bundle s 9
2. System s o f differential equation s 12
3. Externa l symmetrie s 13
4. Classica l symmetr y reductio n 15
5. Contac t transformation s an d Backlund' s Theore m 17
6. Generalize d symmetrie s o f differential equation s 19
7. Generalize d symmetrie s an d conservatio n law s 2 2
Chapter 3 . Internal , externa l an d generalize d symmetrie s 2 7
1. Interna l symmetrie s 2 7
2. Norma l system s o f ordinary differentia l equation s 2 8
3. Under-determine d system s o f ordinary differentia l equation s 2 9
4. Contac t condition s fo r ordinar y differentia l equation s 3 2
5. Contac t condition s fo r partia l differentia l equation s 3 5
Chapter 4 . Transformation s o f surfaces 3 9
1. Th e metho d o f Laplace 3 9
2. Th e Laplac e transformatio n fo r surface s 4 2
3. A n applicatio n t o th e sf e Tod a fiel d theor y 4 4
Chapter 5 . Transformation s o f submanifold s 4 7
1. A multi-dimensional geometri c Laplac e transformatio n 4 7
2. Transformation s o f systems o f partial differentia l equation s 5 1
Chapter 6 . Hamiltonia n system s o f conservation law s 5 7
1. System s o f conservatio n law s an d thei r loca l geometr y 5 7
2. Strongl y hyperboli c system s ric h i n conservatio n law s 6 0
3. Th e Laplac e transformatio n fo r strongl y hyperboli c system s 6 3
Chapter 7 . Th e variationa l bi-comple x 6 7
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