Gozzi CARTAN CALCULUS AND ITS GENERALIZATIONS VIA A PATH - INTEGRAL APPROACH TO CLASSICAL MECHANICS

In this paper we review the recently proposed path-integral counterpart of the Koopman-von Neumann operatorial approach to classical Hamiltonian mechanics. We identify in particular the geometrical variables entering this tbrmulation and show that they are essentially a basis of the cotangent bundle to the tangent bundle to phase-space. In this space we introduce an extended Poisson brackets structure which allows us to re-do ali the usuai Cartan calculus on symplectic manifolds via these brackets. We also briefly sketch how the Schouten-Nijenhuis, the Fròlicher-Nijenhuis and the Nijenhuis-Richardson brackets look in our formalism.