Derivations with Leibniz defect

TL;DR

This paper introduces a generalized differentiation operator with a non-zero Leibniz defect, enabling new formulations of variational principles and dynamical equations, and revealing unique vibrational behaviors in conservative systems.

Contribution

It presents a novel formalism based on the kappa-operator with Leibniz defect, extending classical calculus and mechanics to include non-Leibniz differentiation.

Findings

Positive Leibniz defect causes red shift in vibrations.

Negative Leibniz defect causes blue shift in vibrations.

Hamiltonian remains constant despite frequency changes.

Abstract

The non-Leibniz formalism is introduced in this article. The formalism is based on the generalized differentiation operator (kappa-operator) with a non-zero Leibniz defect. The Leibniz defect of the introduced operator linearly depends on one scaling parameter. In a special case, if the Leibniz defect vanishes, the generalized differentiation operator reduces to the common differentiation operator. The kappa-operator allows the formulation of the variational principles and corresponding Lagrange and Hamiltonian equations. The solutions of some generalized dynamical equations are provided closed form.With a positive Leibniz defect the amplitude of free vibration remains constant with time with the fading frequency (<<red shift>>). The negative Leibniz defect leads the opposite behavior, demonstrating the growing frequency (<<blue shift>>). However, the Hamiltonian remains constant in time in both cases. Thus the introduction of non-zero Leibniz defect leads to an alternative mathematical description of the conservative systems.