  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.
