2017/09/30 by Kobelev, V.
#37Jxx #37K05 #37Lxx #70H08 #70S05 #FOS: Physical sciences #General Physics (physics.gen-ph)
paper · doi:10.48550/arxiv.1710.00160
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.