In this paper we develop a path-integral formulation of classical Hamiltonian dynamics, that means we give a functional-integral representation of classical transition probabilities. This is done by giving weight ``one'' to the classical paths and weight ``zero'' to all the others. With the help of anticommuting ghosts this measure can be rewritten as the exponential of a certain action S~. Associated with this path integral there is an operatorial formalism that turns out to be an extension of the well-known operatorial approach of Liouville, Koopman, and von Neumann. The new formalism describes the evolution of scalar probability densities and of p-form densities on phase space in a unified framework. In this work we provide an interpretation for the ghost fields as being the well-known Jacobi fields of classical mechanics. With this interpretation the Hamiltonian H~, derived from the action S~, turns out to be the Lie derivative associated with the Hamiltonian flow. We also find that the action S~ presents a set of Becchi-Rouet-Stora- (BRS-)type invariances mixing the original phase-space variables with the ghosts. Together with a Sp(2) symmetry of the pure ghosts sector, they form a universal invariance group ISp(2) which is present in any Hamiltonian system. The physical and geometrical meaning of the ISp(2) generators is discussed in detail: in particular the conservation of one of the generators is shown to be equivalent to the Liouville theorem. The ISp(2) algebra is then used to give a modern operatorial reformulation of the old Cartan calculus on symplectic manifolds.