Zitterbewegung as purely classical phenomenon

Nonrelativistic formalism is developed, which allows describing systems with internal degrees of freedom in the scalar potential field $U$, which is a function both on relative coordinates and time, and on relative speed and accelerations. The equation for energy, which is an integral of motion when $U$ satisfies to certain differential condition, is derived for the general case. For a free mass point all solutions of the equations of motion in the center-of-inertia reference frame, moving with constant velocity, are found. As a result, the center of mass follows along helical and more complicated trajectories round a direction of motion of the center of inertia. This motion can be interpreted as trembling movement (Zitterbewegung). On this basis a conclusion is done that Zitterbewegung has purely classical origin, arising even in a nonrelativistic case if internal degrees of freedom are taken into account. The general equation of motion for a spin which can be interpreted from positions of a classical mechanics is written down. Application of the obtained results to the electron leads to new conception of electric charge, sign of which corresponds to a sign of spin polarization.Comment: 24 pages, 4 figure

Mathematical models of optimal allocation of logistical capacities in the regional system of commodity flows Математические модели оптимального размещения логистических мощностей в региональной системе товарных потоков

The article considers tasks of allocation and innovation development of logistical capacities. The essence of the first task lies in optimal (from the position of the criterion of minimum expenditures) formation of the regional system of logistical capacities of various types and purposes in possible places of their location. These capacities should treat the set commodity flows, which possess a complex structure and are produced by different centres. The original conditions of the second task are a certain, already existing, system of logistical capacities. It is necessary to conduct its modification by a rational method using methods with various degree of innovation: simple accumulation of capacities of original productivity, modernisation of existing technological units and introduction of new more efficient capacities. In order to solve the specified tasks, the article built optimisation models, which correspond with setting mathematical tasks of partially integer-valued linear programming.В статье рассматриваются задачи размещения и инновационного развития логистических мощностей. Суть первой задачи состоит в оптимальном (с позиций критерия минимума затрат) формировании региональной системы логистических мощностей различного вида и назначения в возможных пунктах их дислокации. Данные мощности должны обрабатывать заданные товарные потоки, которые обладают сложной структурой и продуцируются различными центрами. Исходными условиями второй задачи является некоторая уже действующая система логистических мощностей. Необходимо рациональным образом выполнить ее модификацию, используя методы различной степени инновационности: простое наращивание мощностей исходной продуктивности, модернизация имеющихся технологических блоков и ввод в действие новых, более эффективных мощностей. Для решения указанных задач построены оптимизационные модели, которые соответствуют математическим постановкам задач частично целочисленного линейного программирования