Nilpotent Classical Mechanics

TL;DR

This paper introduces nilpotent mechanics using commuting nilpotent coordinates, develops associated geometric structures, and explores supersymmetric extensions, revealing similarities to symplectic geometry and novel algebraic properties.

Contribution

It presents a new formalism of nilpotent mechanics with geometric and algebraic structures, including supersymmetrization and generalized Poisson brackets.

Findings

Introduction of nilpotent mechanics formalism

Development of s-geometry related to symplectic geometry

Supersymmetric nilpotent oscillator with R-symmetry

Abstract

The formalism of nilpotent mechanics is introduced in the Lagrangian and Hamiltonian form. Systems are described using nilpotent, commuting coordinates $\eta$. Necessary geometrical notions and elements of generalized differential $\eta$-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 $\eta$-system the nilpotent oscillator is introduced and its supersymmetrization considered. It is shown that the $R$-symmetry known for the Graded Superfield Oscillator (GSO) is present also here for the supersymmetric $\eta$-system. The generalized Poisson bracket for $(\eta,p)$-variables satisfies modified Leibniz rule and has nontrivial Jacobiator.