Pfaffian Systems, k-Symplectic Systems [electronic resource] / by Azzouz Awane, Michel Goze

Author:: Awane, Azzouz
Published:: Dordrecht : Springer Netherlands : Imprint: Springer, 2000.
Physical Description:: XIII, 240 pages : online resource
Additional Creators:: Goze, Michel and SpringerLink (Online service)

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Contents:

1 Exterior Forms -- 2 Exterior Systems -- 3 k-Symplectic Exterior Systems -- 4 Pfaffian Systems -- 5 Classification of Pfaffian Systems -- 6 k-Symplectic Manifolds -- 7 k-Symplectic Affine Manifolds -- 8 Homogeneous k-Symplectic G-Spaces -- 9 Geometric Pre-Quantization.

Summary:

The theory of foliations and contact forms have experienced such great development recently that it is natural they have implications in the field of mechanics. They form part of the framework of what Jean Dieudonne calls "Elie Cartan's great theory of the Pfaffian systems", and which even nowadays is still far from being exhausted. The major reference work is without any doubt that of Elie Cartan on Pfaffian systems with five variables. In it one discovers there the bases of an algebraic classification of these systems, their methods of reduction, and the highlighting of the first fundamental invariants. This work opens to us, even today, a colossal field of investigation and the mystery of a ternary form containing the differential invariants of the systems with five variables always deligthts anyone who wishes to find out about them. One of the goals of this memorandum is to present this work of Cartan - which was treated even more analytically by Goursat in its lectures on Pfaffian systems - in order to expound the classifications currently known. The theory of foliations and contact forms appear in the study of completely integrable Pfaffian systems of rank one. In each of these situations there is a local model described either by Frobenius' theorem, or by Darboux' theorem. It is this type of theorem which it would be desirable to have for a non-integrable Pfaffian system which may also be of rank greater than one.

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ISBN:

9789401595261

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