TY - JOUR

T1 - Some Consequences of Mathematical Inaccuracies in Mechanics

AU - Prozorova, Evelina

PY - 2022/9/16

Y1 - 2022/9/16

N2 - The article is devoted to the study of some violations of the known laws of mathematics and classical mechanics in continuum mechanics and kinetics. The most common is open non-stationary systems. From the equations formulated earlier and some experiments, a connection traced between the gradients of physical quantities and the angular momentum (force). The use of Hamilton's formalism and the dependence of force only on the distance between particles limits the study. In the collision integral, for example, for a rarefied gas, the Lennard-Jones potential, which not related to the type under consideration, is often used. Hamilton's formalism traces the behavior of closed systems. The general form of boundary conditions and forces changes the theory proposed in the works of N.N. Bogolyubov. The results of the reformulation discussed. Even in the classical theory, after taking into account the moments, we come to the absence of symmetric stress tensor in Boltzmann theory. The symmetric tensor obtains after assumption of small influence from absence of symmetry at the condition of the forces balance. No symmetric tensor leads to the existence of two solutions. New examples of solving problems on hydromechanics, elasticity theory and kinetic theory are given. A correspondence between the terms of the Liouville equation with more general and traditional forces established for continues mechanics. Previously considered boundary layer problems, jet problems and the simplest problems of elasticity theory. The paper proposes a method for finding the second solution for no symmetric problems, if the solution of the symmetric problem we know. The mathematical inaccuracies of the theory of continuum mechanics and kinetics discussed.

AB - The article is devoted to the study of some violations of the known laws of mathematics and classical mechanics in continuum mechanics and kinetics. The most common is open non-stationary systems. From the equations formulated earlier and some experiments, a connection traced between the gradients of physical quantities and the angular momentum (force). The use of Hamilton's formalism and the dependence of force only on the distance between particles limits the study. In the collision integral, for example, for a rarefied gas, the Lennard-Jones potential, which not related to the type under consideration, is often used. Hamilton's formalism traces the behavior of closed systems. The general form of boundary conditions and forces changes the theory proposed in the works of N.N. Bogolyubov. The results of the reformulation discussed. Even in the classical theory, after taking into account the moments, we come to the absence of symmetric stress tensor in Boltzmann theory. The symmetric tensor obtains after assumption of small influence from absence of symmetry at the condition of the forces balance. No symmetric tensor leads to the existence of two solutions. New examples of solving problems on hydromechanics, elasticity theory and kinetic theory are given. A correspondence between the terms of the Liouville equation with more general and traditional forces established for continues mechanics. Previously considered boundary layer problems, jet problems and the simplest problems of elasticity theory. The paper proposes a method for finding the second solution for no symmetric problems, if the solution of the symmetric problem we know. The mathematical inaccuracies of the theory of continuum mechanics and kinetics discussed.

KW - Angular Momentum

KW - Hamiltonian formalism

KW - Non-Symmetrical Stress Tensor

UR - https://www.mendeley.com/catalogue/aef90666-6db4-3006-889a-b8f131a09b5f/

U2 - 10.37394/232011.2022.17.17

DO - 10.37394/232011.2022.17.17

M3 - Article

VL - 17

SP - 124

EP - 131

JO - WSEAS Transactions on Applied and Theoretical Mechanics

JF - WSEAS Transactions on Applied and Theoretical Mechanics

SN - 1991-8747

ER -