Derivations with Leibniz defect

V. Kobelev

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arXiv · 1710.00160

Derivations with Leibniz defect

V. Kobelev

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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 (< >). The negative Leibniz defect leads the opposite behavior, demonstrating the growing frequency (< >). 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.

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V. Kobelev. 2017-09-30. Derivations with Leibniz defect. https://arxiv.org/abs/1710.00160

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