TITLE:
The Law of Conservation of Momentum and the Contribution of No Potential Forces to the Equations for Continuum Mechanics and Kinetics
AUTHORS:
Evelina Prozorova
KEYWORDS:
Angular Momentum, Non-Symmetrical Stress Tensor, Hamiltonian Formalism
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.10 No.6,
June
30,
2022
ABSTRACT: The most common systems are open non-stationary systems. From the previously formulated equations and some experiments, the connection between the gradients of physical quantities and the moment of momentum (force) is traced. The article investigates this trace. The use of the Hamiltonian formalism and the dependence of the force only on the distance between particles limit the study. In the collision integral, for example, for a rarefied gas, the Lennard-Jones potential is often used, which is not of the type considered. The foregoing forces us to turn to the study of the influence of forces of a more general form on the equations of mechanics. Hamilton’s formalism traces the behavior of closed systems. The general form of boundary conditions and forces changes the theory proposed in the works by N.N. Bogolyubov. The results of the reformulation are discussed. Even in classical theory, after taking moments, we arrive at Boltzmann’s theory at no symmetric stress tensor. The symmetric tensor is obtained after the assumption of a small effect of no symmetry and from the condition of the balance of forces. The requirement of simultaneous fulfillment of the laws of conservation of forces and moments of forces leads to the existence of two solutions. To take into account the moment, in addition to the conditions for the equilibrium of forces, the law of equilibrium of the moments of forces is required in the calculations. From it, the degree of no symmetry of the stress tensor is determined. The work illustrates the contribution of the distributed moment of force to the problems of continuum mechanics and the kinetic theory. Examples of the solution to the problem of fluid mechanics, the theory of elasticity and kinetic theory are given. A correspondence is established between the terms of the Liouville equation with more general and traditional forces. Previously, the influence of moments in boundary layer problems, jet problems, and the simplest problems of elasticity theory was considered. The work proves the important role of distributed moments in the formation of new structures.