ON FOUNDATION OF THE GENERALIZED NAMBU MECHANICS

We outline basic principles of a canonical formalism for the Nambu mec hanics - a generalization of Hamiltonian mechanics proposed by Yoichir o Nambu in 1973. It is based on the notion of a Nambu bracket, which g eneralizes the Poisson bracket a ''binary'' operation on classical obs ervables on the phase space - to the ''multiple'' operation of higher order n greater-than-or-equal-to 3. Nambu dynamics is described by the phase flow given by Nambu-Hamilton equations of motion - a system of ODE's which involves n - 1 ''Hamiltonians.'' We introduce the fundamen tal identity for the Nambu bracket - a generalization of the Jacobi id entity - as a consistency condition for the dynamics. We show that Nam bu bracket structure defines a hierarchy of infinite families of ''sub ordinated'' structures of lower order, including Poisson bracket struc ture, which satisfy certain matching conditions. The notion of Nambu b racket enables us to define Nambu Poisson manifolds - phase spaces for the Nambu mechanics, which turn out to be more ''rigid'' than Poisson manifolds - phase spaces for the Hamiltonian mechanics. We introduce the analog of the action form and the action principle for the Nambu m echanics. In its formulation, dynamics of loops (n - 2-dimensional cha ins for the general n-ary case) naturally appears. We discuss several approaches to the quantization of Nambu mechanics, based on the deform ation theory, path integral formulation and on Nambu-Heisenberg ''comm utation'' relations. In the latter formalism we present an explicit re presentation of the Nambu-Heisenberg relation in the n = 3 case. We em phasize the role ternary and higher order algebraic operations and mat hematical structures related to them play in passing from Hamilton's t o Nambu's dynamical picture.