Pfaffian Systems, k-Symplectic Systems

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The geometrical view of mechanics is based on the study of certain exterior systems, the most classical of which are Pfaffian systems. This text presents the classification theorems (Frobenius, Darboux) and the local classification of Pfaffian systems of five variables, following Cartan. It also presents a new class of exterior systems, called k-symplectic systems, generalizing the notion of symplectic form. These systems permit the authors to write in the language of exterior forms the equations proposed by Nambu for a model of statistical mechanics. This work is aimed at graduate students and at research workers in the field of mathematics, differential geometry, statistical mechanics, mathematics of physics and lie algebras.