NILPOTENT CLASSICAL MECHANICS

Abstract

The formalism of nilpotent mechanics is introduced in the Lagrangian and Hamiltonian form. Systems are described using nilpotent, commuting coordinates η. Necessary geometrical notions and elements of generalized differential η-calculus are introduced. The so-called s-geometry, in a special case when it is orthogonally related to a traceless symmetric form, shows some resemblances to the symplectic geometry. As an example of an η-system the nilpotent oscillator is introduced and its supersymmetrization considered. It is shown that the R-symmetry known for the graded superfield oscillator also present here for the supersymmetric η-system. The generalized Poisson bracket for (η, p)-variables satisfies modified Leibniz rule and has nontrivial Jacobiator.