Cohomological Analysis of Partial Differential Equations and Secondary CalculusAmerican Mathematical Soc., 16. okt. 2001 This book is dedicated to fundamentals of a new theory, which is an analog of affine algebraic geometry for (nonlinear) partial differential equations. This theory grew up from the classical geometry of PDE's originated by S. Lie and his followers by incorporating some nonclassical ideas from the theory of integrable systems, the formal theory of PDE's in its modern cohomological form given by D. Spencer and H. Goldschmidt and differential calculus over commutative algebras (Primary Calculus). The main result of this synthesis is Secondary Calculus on diffieties, new geometrical objects which are analogs of algebraic varieties in the context of (nonlinear) PDE's. Secondary Calculus surprisingly reveals a deep cohomological nature of the general theory of PDE's and indicates new directions of its further progress. Recent developments in quantum field theory showed Secondary Calculus to be its natural language, promising a nonperturbative formulation of the theory. In addition to PDE's themselves, the author describes existing and potential applications of Secondary Calculus ranging from algebraic geometry to field theory, classical and quantum, including areas such as characteristic classes, differential invariants, theory of geometric structures, variational calculus, control theory, etc. This book, focused mainly on theoretical aspects, forms a natural dipole with Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Volume 182 in this same series, Translations of Mathematical Monographs, and shows the theory "in action". |
Indhold
0 | 18 |
Elements of Differential Calculus | 25 |
Geometry of FiniteOrder Contact Structures | 57 |
Geometry of Infinitely Prolonged Differential | 95 |
Cspectral Sequence and Some Applications | 127 |
Introduction to Secondary Calculus | 189 |
| 237 | |
| 240 | |
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ê acyclic bundle C-field C-spectral sequence C-theory C-transformations CA¹ called canonical Cartan distribution CD(O cochain mapping cohomology class coincides coker complex compute conservation laws Consider coordinates COROLLARY corresponding defined definition denoted derivation Diff Diffeo diffeomorphism differential calculus differential forms differential invariants differential operators diffiety elements Euler operator Euler-Lagrange Euler-Lagrange equation fact fibers filtration formula functions functor geometric structures Green formula Hence homomorphism homotopy infinitesimal isomorphism Jk E,n Lagrangian lemma Lie algebra Lie derivative Lie field Lie transformations linear manifold morphism n-dimensional natural S-module Noether nonlinear notation Note obtained obviously partial differential equations problem projective PROOF PROPOSITION proved representing object resp restriction result Rham scalar Secondary Calculus secondary module secondary vector smbl smooth solution spectral submanifold submodule Subsection symmetries tangent term Theorem theory trivial V₁ vector fields
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