Contact Geometry and Nonlinear Differential EquationsCambridge University Press, 2007 - 496 pages Methods from contact and symplectic geometry can be used to solve highly non-trivial nonlinear partial and ordinary differential equations without resorting to approximate numerical methods or algebraic computing software. This book explains how it's done. It combines the clarity and accessibility of an advanced textbook with the completeness of an encyclopedia. The basic ideas that Lie and Cartan developed at the end of the nineteenth century to transform solving a differential equation into a problem in geometry or algebra are here reworked in a novel and modern way. Differential equations are considered as a part of contact and symplectic geometry, so that all the machinery of Hodge-deRham calculus can be applied. In this way a wide class of equations can be tackled, including quasi-linear equations and Monge-Ampere equations (which play an important role in modern theoretical physics and meteorology). |
Table des matières
Section 1 | 3 |
Section 2 | 18 |
Section 3 | 32 |
Section 4 | 35 |
Section 5 | 36 |
Section 6 | 62 |
Section 7 | 63 |
Section 8 | 64 |
Section 12 | 147 |
Section 13 | 183 |
Section 14 | 201 |
Section 15 | 214 |
Section 16 | 215 |
Section 17 | 224 |
Section 18 | 263 |
Section 19 | 269 |
Section 9 | 65 |
Section 10 | 103 |
Section 11 | 119 |
Section 20 | 273 |
Section 21 | 371 |
Expressions et termes fréquents
A²(V Ann(P bundle called canonical basis Cartan distribution characteristic numbers commutator complex structure conservation law contact manifold contact transformation contact vector field corresponding decomposition defines denote diffeomorphism differential 1-forms differential equations differential operator direct sum dp₁ e-structure effective differential 2-form effective form elliptic equivalent Example function f Hamiltonian hyperbolic integral manifold invariant isomorphism J¹M Jacobi equation Lagrangian submanifold Lagrangian subspace Lemma Lie algebra linear symmetry Monge-Ampère equation Monge-Ampère operators multivalued solution non-holonomic non-zero Note obtain the following one-dimensional P₁ parabolic Pf(w Pfaffian Proof representation respect shuffling symmetries skew-orthogonal smooth function smooth manifold Sp(V Sym(P symplectic manifold symplectic space symplectic structure symplectic transformation symplectic vector space tangent space tensor Theorem two-dimensional w₁ Λω ди др дрг дрі ду дх მ მ მ
Références à ce livre
Symmetry and Perturbation Theory: Proceedings of the International ... Giuseppe Gaeta,Raffaele Vitolo,Sebastian Walcher Aucun aperçu disponible - 2007 |